var bibbase_data = {"data":"<img src=\"//bibbase.org/img/ajax-loader.gif\" id=\"spinner\"\n  style=\"display: none;\" alt=\"Loading..\" />\n\n<div id=\"bibbase\">\n\n  <script type=\"text/javascript\">\n    var bibbase = {\n      params: {\"bib\":\"https://pxie98.github.io/xpcpublication.bib\",\"jsonp\":\"1\",\"theme\":\"side\",\"host\":\"bibbase.org\",\"ssl\":false},\n      data: [{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"ReMU: Regional Minimal Updating for Model-Based Derivative-Free Optimization\",\"author\":[{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Stefan\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Wild\"],\"suffixes\":[]}],\"year\":\"2025\",\"eprint\":\"2504.03606\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.OC\",\"url\":\"https://arxiv.org/abs/2504.03606\",\"bibtex\":\"@misc{xie2025remuregionalminimalupdating,\\n      title={ReMU: Regional Minimal Updating for Model-Based Derivative-Free Optimization}, \\n      author={Pengcheng Xie and Stefan M. Wild},\\n      year={2025},\\n      eprint={2504.03606},\\n      archivePrefix={arXiv},\\n      primaryClass={math.OC},\\n      url={https://arxiv.org/abs/2504.03606}, \\n}\\n\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Wild, S. M.\"],\"key\":\"xie2025remuregionalminimalupdating\",\"id\":\"xie2025remuregionalminimalupdating\",\"bibbaseid\":\"xie-wild-remuregionalminimalupdatingformodelbasedderivativefreeoptimization-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://arxiv.org/abs/2504.03606\"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"A novel numerical method tailored for unconstrained optimization problems\",\"author\":[{\"firstnames\":[\"Lin\"],\"propositions\":[],\"lastnames\":[\"Li\"],\"suffixes\":[]},{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Li\"],\"propositions\":[],\"lastnames\":[\"Zhang\"],\"suffixes\":[]}],\"year\":\"2025\",\"eprint\":\"2504.02832\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.OC\",\"url\":\"https://arxiv.org/abs/2504.02832\",\"bibtex\":\"@misc{li2025novelnumericalmethodtailored,\\n      title={A novel numerical method tailored for unconstrained optimization problems}, \\n      author={Lin Li and Pengcheng Xie and Li Zhang},\\n      year={2025},\\n      eprint={2504.02832},\\n      archivePrefix={arXiv},\\n      primaryClass={math.OC},\\n      url={https://arxiv.org/abs/2504.02832}, \\n}\\n\\n\\n\\n\\n\\n\",\"author_short\":[\"Li, L.\",\"Xie, P.\",\"Zhang, L.\"],\"key\":\"li2025novelnumericalmethodtailored\",\"id\":\"li2025novelnumericalmethodtailored\",\"bibbaseid\":\"li-xie-zhang-anovelnumericalmethodtailoredforunconstrainedoptimizationproblems-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://arxiv.org/abs/2504.02832\"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Ya-xiang\"],\"propositions\":[],\"lastnames\":[\"Yuan\"],\"suffixes\":[]}],\"title\":\"Least H2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms\",\"journal\":\"IMA Journal of Numerical Analysis\",\"pages\":\"drae106\",\"year\":\"2025\",\"month\":\"03\",\"abstract\":\"One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the $H^\\\\{2\\\\}$ norm of the difference between neighboring quadratic models. The motivation for applying the $H^\\\\{2\\\\}$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of $H^\\\\{2\\\\}$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least $H^\\\\{2\\\\}$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.\",\"issn\":\"0272-4979\",\"doi\":\"10.1093/imanum/drae106\",\"url\":\"https://doi.org/10.1093/imanum/drae106\",\"eprint\":\"https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drae106/62210513/drae106.pdf\",\"bibtex\":\"@article{10.1093/imanum/drae106,\\n    author = {Pengcheng Xie and Ya-xiang Yuan},\\n    title = {Least H2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms},\\n    journal = {IMA Journal of Numerical Analysis},\\n    pages = {drae106},\\n    year = {2025},\\n    month = {03},\\n    abstract = {One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the \\\\$H^\\\\{2\\\\}\\\\$ norm of the difference between neighboring quadratic models. The motivation for applying the \\\\$H^\\\\{2\\\\}\\\\$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of \\\\$H^\\\\{2\\\\}\\\\$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least \\\\$H^\\\\{2\\\\}\\\\$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.},\\n    issn = {0272-4979},\\n    doi = {10.1093/imanum/drae106},\\n    url = {https://doi.org/10.1093/imanum/drae106},\\n    eprint = {https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drae106/62210513/drae106.pdf},\\n}\\n\\n\\n\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Yuan, Y.\"],\"key\":\"10.1093/imanum/drae106\",\"id\":\"10.1093/imanum/drae106\",\"bibbaseid\":\"xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://doi.org/10.1093/imanum/drae106\"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems\",\"author\":[{\"firstnames\":[\"Lin\"],\"propositions\":[],\"lastnames\":[\"Li\"],\"suffixes\":[]},{\"firstnames\":[\"Yuheng\"],\"propositions\":[],\"lastnames\":[\"Zhou\"],\"suffixes\":[]},{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Huiyuan\"],\"propositions\":[],\"lastnames\":[\"Li\"],\"suffixes\":[]}],\"year\":\"2025\",\"eprint\":\"2503.01912\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.OC\",\"url\":\"https://arxiv.org/abs/2503.01912\",\"bibtex\":\"@misc{li2025spectrallevenbergmarquardtdeflationmethodmultiple,\\n      title={A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems}, \\n      author={Lin Li and Yuheng Zhou and Pengcheng Xie and Huiyuan Li},\\n      year={2025},\\n      eprint={2503.01912},\\n      archivePrefix={arXiv},\\n      primaryClass={math.OC},\\n      url={https://arxiv.org/abs/2503.01912}, \\n}\\n\\n\\n\\n\",\"author_short\":[\"Li, L.\",\"Zhou, Y.\",\"Xie, P.\",\"Li, H.\"],\"key\":\"li2025spectrallevenbergmarquardtdeflationmethodmultiple\",\"id\":\"li2025spectrallevenbergmarquardtdeflationmethodmultiple\",\"bibbaseid\":\"li-zhou-xie-li-aspectrallevenbergmarquardtdeflationmethodformultiplesolutionsofsemilinearellipticsystems-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://arxiv.org/abs/2503.01912\"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains\",\"author\":[{\"firstnames\":[\"Yangyi\"],\"propositions\":[],\"lastnames\":[\"Ye\"],\"suffixes\":[]},{\"firstnames\":[\"Lin\"],\"propositions\":[],\"lastnames\":[\"Li\"],\"suffixes\":[]},{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Haijun\"],\"propositions\":[],\"lastnames\":[\"Yu\"],\"suffixes\":[]}],\"year\":\"2025\",\"eprint\":\"2503.07624\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.NA\",\"url\":\"https://arxiv.org/abs/2503.07624\",\"bibtex\":\"@misc{ye2025improvedadaptiveorthogonalbasis,\\n      title={An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains}, \\n      author={Yangyi Ye and Lin Li and Pengcheng Xie and Haijun Yu},\\n      year={2025},\\n      eprint={2503.07624},\\n      archivePrefix={arXiv},\\n      primaryClass={math.NA},\\n      url={https://arxiv.org/abs/2503.07624}, \\n}\\n\\n\\n\\n\",\"author_short\":[\"Ye, Y.\",\"Li, L.\",\"Xie, P.\",\"Yu, H.\"],\"key\":\"ye2025improvedadaptiveorthogonalbasis\",\"id\":\"ye2025improvedadaptiveorthogonalbasis\",\"bibbaseid\":\"ye-li-xie-yu-animprovedadaptiveorthogonalbasisdeflationmethodformultiplesolutionswithapplicationstononlinearellipticequationsinvaryingdomains-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://arxiv.org/abs/2503.07624\"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Stefan\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Wild\"],\"suffixes\":[]}],\"title\":\"Barycenter of Weight Coefficient Region of Least Weighted $H^2$ Norm Updating Quadratic Models with Vanishing Trust-region Radius\",\"year\":\"2024\",\"journal\":\"SIAM NCC 2024, Early Career Travel Award\",\"keywords\":\"optimization\",\"bibtex\":\"@article{xie2024bary,\\nauthor={Pengcheng Xie and Stefan M. Wild},\\n  title = {Barycenter of Weight Coefficient Region of Least Weighted $H^2$ Norm Updating Quadratic Models with Vanishing Trust-region Radius},\\n  year = {2024},\\n  journal = {SIAM NCC 2024, Early Career Travel Award},\\nkeywords = {optimization},\\n}\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Wild, S. M.\"],\"key\":\"xie2024bary\",\"id\":\"xie2024bary\",\"bibbaseid\":\"xie-wild-barycenterofweightcoefficientregionofleastweightedh2normupdatingquadraticmodelswithvanishingtrustregionradius-2024\",\"role\":\"author\",\"urls\":{},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Pengcheng\",\"Xie\"],\"firstnames\":[\"etc.\"],\"suffixes\":[]}],\"title\":\"A Novel Local Analysis of Objectives Approximated by Neural Network: L-Change\",\"year\":\"2024\",\"journal\":\"International Conference on Mathematical Theory of Deep Learning (MTDL)\",\"keywords\":\"machine learning, neural network\",\"bibtex\":\"@misc{xie2024lchange,\\nauthor={Pengcheng Xie, etc.},\\n  title = {A Novel Local Analysis of Objectives Approximated by Neural Network: L-Change},\\n  year = {2024},\\n  journal = {International Conference on Mathematical Theory of Deep Learning (MTDL)},\\nkeywords = {machine learning, neural network},\\n}\\n\\n\\n\",\"author_short\":[\"Pengcheng Xie, e.\"],\"key\":\"xie2024lchange\",\"id\":\"xie2024lchange\",\"bibbaseid\":\"pengchengxie-anovellocalanalysisofobjectivesapproximatedbyneuralnetworklchange-2024\",\"role\":\"author\",\"urls\":{},\"keyword\":[\"machine learning\",\"neural network\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"On the Relationship between $Λ$-poisedness in Derivative-Free Optimization and Outliers in Local Outlier Factor\",\"author\":[{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]}],\"year\":\"2024\",\"eprint\":\"2407.17529\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.OC\",\"url\":\"https://arxiv.org/abs/2407.17529\",\"keywords\":\"optimization, approximation\",\"bibtex\":\"@misc{zhang2024relationshiplambdapoisednessderivativefreeoptimization,\\n      title={On the Relationship between $\\\\Lambda$-poisedness in Derivative-Free Optimization and Outliers in Local Outlier Factor}, \\n      author={Pengcheng Xie},\\n      year={2024},\\n      eprint={2407.17529},\\n      archivePrefix={arXiv},\\n      primaryClass={math.OC},\\n      url={https://arxiv.org/abs/2407.17529}, \\nkeywords = {optimization, approximation},\\n}\\n\\n\\n\",\"author_short\":[\"Xie, P.\"],\"key\":\"zhang2024relationshiplambdapoisednessderivativefreeoptimization\",\"id\":\"zhang2024relationshiplambdapoisednessderivativefreeoptimization\",\"bibbaseid\":\"xie-ontherelationshipbetweenpoisednessinderivativefreeoptimizationandoutliersinlocaloutlierfactor-2024\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://arxiv.org/abs/2407.17529\"},\"keyword\":[\"optimization\",\"approximation\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"K.\",\"J.\"],\"propositions\":[],\"lastnames\":[\"Dzahini\"],\"suffixes\":[]},{\"firstnames\":[\"S.\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Wild\"],\"suffixes\":[]},{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]}],\"title\":\"Optimization approaches for solving inverse problems must account for uncertainty in both data and downstream decisions\",\"year\":\"2025\",\"note\":\"Position paper, Inverse Methods for Complex Systems under Uncertainty Workshop, Sponsored by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research\",\"url\":\"https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf\",\"bibtex\":\"@misc{postionpaper2025optimization,\\n  author       = {K. J. Dzahini and S. M. Wild and P. Xie},\\n  title        = {Optimization approaches for solving inverse problems must account for uncertainty in both data and downstream decisions},\\n  year         = {2025},\\n  note         = {Position paper, Inverse Methods for Complex Systems under Uncertainty Workshop, Sponsored by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research},\\n  url          = {https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf}\\n}\\n\\n\\n\",\"author_short\":[\"Dzahini, K. J.\",\"Wild, S. M.\",\"Xie, P.\"],\"key\":\"postionpaper2025optimization\",\"id\":\"postionpaper2025optimization\",\"bibbaseid\":\"dzahini-wild-xie-optimizationapproachesforsolvinginverseproblemsmustaccountforuncertaintyinbothdataanddownstreamdecisions-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf\"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"W.\"],\"propositions\":[],\"lastnames\":[\"Hu\"],\"suffixes\":[]},{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"L.\"],\"propositions\":[],\"lastnames\":[\"Zhang\"],\"suffixes\":[]},{\"firstnames\":[\"Y.-x.\"],\"propositions\":[],\"lastnames\":[\"Yuan\"],\"suffixes\":[]}],\"title\":\"A Low Computation Cost Cubic Regularized Quasi-Newton Method for Distributed Optimization: LC3RQN\",\"year\":\"2024\",\"url\":\"https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf\",\"keywords\":\"optimization\",\"bibtex\":\"@article{hu2024lc3rqn,\\n  author = {W. Hu and P. Xie and L. Zhang and Y.-x. Yuan},\\n  title = {A Low Computation Cost Cubic Regularized Quasi-Newton Method for Distributed Optimization: LC3RQN},\\n  year = {2024},\\n  url = {https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf},\\nkeywords = {optimization},\\n}\\n\\n\",\"author_short\":[\"Hu, W.\",\"Xie, P.\",\"Zhang, L.\",\"Yuan, Y.\"],\"key\":\"hu2024lc3rqn\",\"id\":\"hu2024lc3rqn\",\"bibbaseid\":\"hu-xie-zhang-yuan-alowcomputationcostcubicregularizedquasinewtonmethodfordistributedoptimizationlc3rqn-2024\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":23,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Y.-x.\"],\"propositions\":[],\"lastnames\":[\"Yuan\"],\"suffixes\":[]}],\"title\":\"A new two-dimensional model-based subspace method for large-scale unconstrained derivative-free optimization: 2D-MoSub\",\"year\":\"2023\",\"note\":\"Under review\",\"doi\":\"10.48550/arXiv.2309.14855\",\"url\":\"https://arxiv.org/pdf/2309.14855.pdf\",\"keywords\":\"optimization\",\"bibtex\":\"@article{xie2023twodimensional,\\n  author = {P. Xie and Y.-x. Yuan},\\n  title = {A new two-dimensional model-based subspace method for large-scale unconstrained derivative-free optimization: 2D-MoSub},\\n  year = {2023},\\n  note = {Under review},\\n  doi = {10.48550/arXiv.2309.14855},\\n  url = {https://arxiv.org/pdf/2309.14855.pdf},\\nkeywords = {optimization},\\n}\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Yuan, Y.\"],\"key\":\"xie2023twodimensional\",\"id\":\"xie2023twodimensional\",\"bibbaseid\":\"xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://arxiv.org/pdf/2309.14855.pdf\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":7,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Xie\"],\"firstnames\":[\"Pengcheng\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Yuan\"],\"firstnames\":[\"Ya-xiang\"],\"suffixes\":[]}],\"title\":\"A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model\",\"journal\":\"SIAM Journal on Optimization\",\"volume\":\"35\",\"number\":\"2\",\"pages\":\"1110-1133\",\"year\":\"2025\",\"doi\":\"10.1137/23M1582023\",\"url\":\"https://doi.org/10.1137/23M1582023 \",\"eprint\":\"https://doi.org/10.1137/23M1582023 \",\"abstract\":\"Abstract. We analyze the least norm type underdetermined quadratic interpolation model proposed by Conn and Toint [An algorithm using quadratic interpolation for unconstrained derivative free optimization, 1996] from the perspective of the property of trust-region iteration. We find the Karush–Kuhn–Tucker multiplier’s nondeterminacy when constructing a quadratic model considering the trust-region iteration in the case where the current iteration point is on the boundary of the trust region. The lack of the quadratic model’s uniqueness caused by the Karush–Kuhn–Tucker multiplier’s nondeterminacy leads us to propose a new model to consequently improve the model by selectively treating the previously obtained underdetermined quadratic model as a quadratic model or a linear one. A new derivative-free method is given by introducing the improved underdetermined quadratic interpolation model considering the optimality of the model based on the trust-region iteration. The theoretical motivation, property, computational details, and the quadratic model’s formula derived from the Karush–Kuhn–Tucker conditions are discussed. The formula is implementation-friendly for the existing model-based derivative-free methods. The numerical results with released codes support the advantages of our quadratic model in the derivative-free optimization methods. To the best of our knowledge, this is the first work considering the property of trust-region iteration and the model’s optimality when constructing the underdetermined quadratic model for derivative-free trust-region methods. \",\"bibtex\":\"@article{xieyuannew,\\nauthor = {Xie, Pengcheng and Yuan, Ya-xiang},\\ntitle = {A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model},\\njournal = {SIAM Journal on Optimization},\\nvolume = {35},\\nnumber = {2},\\npages = {1110-1133},\\nyear = {2025},\\ndoi = {10.1137/23M1582023},\\n\\nURL = { \\n    \\n        https://doi.org/10.1137/23M1582023\\n    \\n    \\n\\n},\\neprint = { \\n    \\n        https://doi.org/10.1137/23M1582023\\n    \\n    \\n\\n}\\n,\\n    abstract = { Abstract. We analyze the least norm type underdetermined quadratic interpolation model proposed by Conn and Toint [An algorithm using quadratic interpolation for unconstrained derivative free optimization, 1996] from the perspective of the property of trust-region iteration. We find the Karush–Kuhn–Tucker multiplier’s nondeterminacy when constructing a quadratic model considering the trust-region iteration in the case where the current iteration point is on the boundary of the trust region. The lack of the quadratic model’s uniqueness caused by the Karush–Kuhn–Tucker multiplier’s nondeterminacy leads us to propose a new model to consequently improve the model by selectively treating the previously obtained underdetermined quadratic model as a quadratic model or a linear one. A new derivative-free method is given by introducing the improved underdetermined quadratic interpolation model considering the optimality of the model based on the trust-region iteration. The theoretical motivation, property, computational details, and the quadratic model’s formula derived from the Karush–Kuhn–Tucker conditions are discussed. The formula is implementation-friendly for the existing model-based derivative-free methods. The numerical results with released codes support the advantages of our quadratic model in the derivative-free optimization methods. To the best of our knowledge, this is the first work considering the property of trust-region iteration and the model’s optimality when constructing the underdetermined quadratic model for derivative-free trust-region methods. }\\n}\\n\\n\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Yuan, Y.\"],\"key\":\"xieyuannew\",\"id\":\"xieyuannew\",\"bibbaseid\":\"xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://doi.org/10.1137/23M1582023 \"},\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Y.\"],\"propositions\":[],\"lastnames\":[\"Yuan\"],\"suffixes\":[]}],\"title\":\"Derivative-free optimization with transformed objective functions (DFOTO) and the algorithm based on the least Frobenius norm updating quadratic model\",\"year\":\"2023\",\"journal\":\"Journal of the Operations Research Society of China\",\"url\":\"https://link.springer.com/article/10.1007/s40305-023-00532-x\",\"keywords\":\"optimization\",\"bibtex\":\"@article{xie2023dfoto,\\n  author = {P. Xie and Y. Yuan},\\n  title = {Derivative-free optimization with transformed objective functions {(DFOTO)} and the algorithm based on the least {Frobenius} norm updating quadratic model},\\n  year = {2023},\\n  journal = {Journal of the Operations Research Society of China},\\n  url = {https://link.springer.com/article/10.1007/s40305-023-00532-x},\\nkeywords = {optimization},\\n}\\n\\n\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Yuan, Y.\"],\"key\":\"xie2023dfoto\",\"id\":\"xie2023dfoto\",\"bibbaseid\":\"xie-yuan-derivativefreeoptimizationwithtransformedobjectivefunctionsdfotoandthealgorithmbasedontheleastfrobeniusnormupdatingquadraticmodel-2023\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://link.springer.com/article/10.1007/s40305-023-00532-x\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"Y.-x.\"],\"propositions\":[],\"lastnames\":[\"Yuan\"],\"suffixes\":[]}],\"title\":\"A derivative-free optimization algorithm combining line-search and trust-region techniques\",\"journal\":\"Chinese Annals of Mathematics, Series B\",\"volume\":\"44\",\"number\":\"5\",\"pages\":\"719-734\",\"year\":\"2023\",\"url1\":\"https://link.springer.com/article/10.1007/s11401-023-0040-y\",\"url2\":\"https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&flag=1\",\"url3\":\"https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&flag=1\",\"keywords\":\"optimization\",\"bibtex\":\"@article{xie2023linesearch,\\n  author = {P. Xie and Y.-x. Yuan},\\n  title = {A derivative-free optimization algorithm combining line-search and trust-region techniques},\\n  journal = {Chinese Annals of Mathematics, Series B},\\n  volume = {44},\\n  number = {5},\\n  pages = {719-734},\\n  year = {2023},\\n  url1 = {https://link.springer.com/article/10.1007/s11401-023-0040-y},\\n  url2 = {https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&flag=1},\\n  url3 = {https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&flag=1},\\nkeywords = {optimization},\\n}\\n\\n\",\"author_short\":[\"Xie, P.\",\"Yuan, Y.\"],\"key\":\"xie2023linesearch\",\"id\":\"xie2023linesearch\",\"bibbaseid\":\"xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023\",\"role\":\"author\",\"urls\":{\"1\":\"https://link.springer.com/article/10.1007/s11401-023-0040-y\",\"2\":\"https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&flag=1\",\"3\":\"https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&flag=1\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]}],\"title\":\"A derivative-free trust-region method for optimization on the ellipsoid\",\"journal\":\"Journal of Physics: Conference Series\",\"volume\":\"2620\",\"pages\":\"012007\",\"year\":\"2023\",\"url\":\"https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf\",\"note\":\"Presented in 2023 International Conference on Advances in Computer Science and Engineering Technology (ACSE2023)\",\"keywords\":\"optimization\",\"bibtex\":\"@article{xie2023ellipsoid,\\n  author = {P. Xie},\\n  title = {A derivative-free trust-region method for optimization on the ellipsoid},\\n  journal = {Journal of Physics: Conference Series},\\n  volume = {2620},\\n  pages = {012007},\\n  year = {2023},\\n  url = {https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf},\\n  note = {Presented in 2023 International Conference on Advances in Computer Science and Engineering Technology (ACSE2023)},\\nkeywords = {optimization},\\n}\\n\\n\",\"author_short\":[\"Xie, P.\"],\"key\":\"xie2023ellipsoid\",\"id\":\"xie2023ellipsoid\",\"bibbaseid\":\"xie-aderivativefreetrustregionmethodforoptimizationontheellipsoid-2023\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":3,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Sufficient conditions for error distance reduction in the ℓ2-norm trust region between minimizers of local nonconvex multivariate quadratic approximates\",\"journal\":\"Journal of Computational and Applied Mathematics\",\"volume\":\"453\",\"pages\":\"116146\",\"year\":\"2025\",\"issn\":\"0377-0427\",\"doi\":\"https://doi.org/10.1016/j.cam.2024.116146\",\"url\":\"https://www.sciencedirect.com/science/article/pii/S0377042724003959\",\"author\":[{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]}],\"keywords\":\"optimization\",\"abstract\":\"This paper analyzes the sufficient conditions for distance reduction between minimizers of local nonconvex quadratic approximate functions with diagonal Hessian in the ℓ2-norm trust regions after two iterations. Some examples illustrate the theoretical results of this study.\",\"bibtex\":\"@article{XIE2025116146,\\ntitle = {Sufficient conditions for error distance reduction in the ℓ2-norm trust region between minimizers of local nonconvex multivariate quadratic approximates},\\njournal = {Journal of Computational and Applied Mathematics},\\nvolume = {453},\\npages = {116146},\\nyear = {2025},\\nissn = {0377-0427},\\ndoi = {https://doi.org/10.1016/j.cam.2024.116146},\\nurl = {https://www.sciencedirect.com/science/article/pii/S0377042724003959},\\nauthor = {Pengcheng Xie},\\nkeywords = {optimization},\\nabstract = {This paper analyzes the sufficient conditions for distance reduction between minimizers of local nonconvex quadratic approximate functions with diagonal Hessian in the ℓ2-norm trust regions after two iterations. Some examples illustrate the theoretical results of this study.}\\n}\\n\\n\",\"author_short\":[\"Xie, P.\"],\"key\":\"XIE2025116146\",\"id\":\"XIE2025116146\",\"bibbaseid\":\"xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://www.sciencedirect.com/science/article/pii/S0377042724003959\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":19,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"An efficient derivative-free method for finding multiple solutions\",\"year\":\"2024\",\"note\":\"To be posted on arXiv\",\"keywords\":\"numerical PDE\",\"bibtex\":\"@misc{2023multiple,\\n  title = {An efficient derivative-free method for finding multiple solutions},\\n  year = {2024},\\n  note = {To be posted on arXiv},\\nkeywords = {numerical PDE},\\n}\\n\\n\",\"key\":\"2023multiple\",\"id\":\"2023multiple\",\"bibbaseid\":\"anonymous-anefficientderivativefreemethodforfindingmultiplesolutions-2024\",\"role\":\"\",\"urls\":{},\"keyword\":[\"numerical PDE\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"inproceedings\",\"type\":\"inproceedings\",\"author\":[{\"firstnames\":[\"S.\"],\"propositions\":[],\"lastnames\":[\"Li\"],\"suffixes\":[]},{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Z.\",\"Zhou\"],\"firstnames\":[\"et\",\"al.\"],\"suffixes\":[]}],\"title\":\"Simulation of interaction of folded waveguide space traveling wave tubes with derivative-free mixed-integer based NEWUOA algorithm\",\"booktitle\":\"2021 7th International Conference on Computer and Communications (ICCC)\",\"year\":\"2021\",\"pages\":\"1215–1219\",\"doi\":\"10.1109/ICCC54389.2021.9674410\",\"url\":\"https://ieeexplore.ieee.org/document/9674410\",\"keywords\":\"optimization\",\"bibtex\":\"@inproceedings{li2021newuoa,\\n  author = {S. Li and P. Xie and Z. Zhou, et al.},\\n  title = {Simulation of interaction of folded waveguide space traveling wave tubes with derivative-free mixed-integer based NEWUOA algorithm},\\n  booktitle = {2021 7th International Conference on Computer and Communications (ICCC)},\\n  year = {2021},\\n  pages = {1215--1219},\\n  doi = {10.1109/ICCC54389.2021.9674410},\\n  url = {https://ieeexplore.ieee.org/document/9674410},\\nkeywords = {optimization},\\n}\\n\\n\",\"author_short\":[\"Li, S.\",\"Xie, P.\",\"Z. Zhou, e. a.\"],\"key\":\"li2021newuoa\",\"id\":\"li2021newuoa\",\"bibbaseid\":\"li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://ieeexplore.ieee.org/document/9674410\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"inproceedings\",\"type\":\"inproceedings\",\"author\":[{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]},{\"firstnames\":[\"M.\"],\"propositions\":[],\"lastnames\":[\"Tao\"],\"suffixes\":[]}],\"title\":\"Parametric resonant control of macroscopic behaviors of multiple oscillators\",\"booktitle\":\"2019 American Control Conference (ACC)\",\"year\":\"2019\",\"pages\":\"1898–1905\",\"doi\":\"10.23919/ACC.2019.8814709\",\"url\":\"https://ieeexplore.ieee.org/document/8814709\",\"keywords\":\"control\",\"bibtex\":\"@inproceedings{xie2019acc,\\n  author = {P. Xie and M. Tao},\\n  title = {Parametric resonant control of macroscopic behaviors of multiple oscillators},\\n  booktitle = {2019 American Control Conference (ACC)},\\n  year = {2019},\\n  pages = {1898--1905},\\n  doi = {10.23919/ACC.2019.8814709},\\n  url = {https://ieeexplore.ieee.org/document/8814709},\\nkeywords = {control},\\n}\\n\\n\\n\",\"author_short\":[\"Xie, P.\",\"Tao, M.\"],\"key\":\"xie2019acc\",\"id\":\"xie2019acc\",\"bibbaseid\":\"xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://ieeexplore.ieee.org/document/8814709\"},\"keyword\":[\"control\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"P.\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]}],\"title\":\"A note on the invariant distribution of a stochastic dynamical system\",\"year\":\"2024\",\"doi\":\"10.12074/202304.01048\",\"url\":\"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398\",\"keywords\":\"control\",\"bibtex\":\"@misc{xie2023invariant,\\n  author = {P. Xie},\\n  title = {A note on the invariant distribution of a stochastic dynamical system},\\n  year = {2024},\\n  doi = {10.12074/202304.01048},\\n  url = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398},\\nkeywords = {control},\\n}\\n\\n\\n\",\"author_short\":[\"Xie, P.\"],\"key\":\"xie2023invariant\",\"id\":\"xie2023invariant\",\"bibbaseid\":\"xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398\"},\"keyword\":[\"control\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Pengcheng\"],\"propositions\":[],\"lastnames\":[\"Xie\"],\"suffixes\":[]}],\"title\":\"The Modeling and Optimization of a Multi-dam System\",\"journal\":\"Applied and Computational Mathematics\",\"volume\":\"13\",\"number\":\"5\",\"pages\":\"140-152\",\"doi\":\"10.11648/j.acm.20241305.13\",\"url\":\"https://doi.org/10.11648/j.acm.20241305.13\",\"eprint\":\"https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241305.13\",\"abstract\":\"In this paper, we present a maintenance plan for the Kariba Dam in Africa. The Kariba Dam, a double curvature concrete arch dam in Zambia's capital, is crucial for regional energy and water management. Ensuring its safety and functionality is essential. This study provides a comprehensive analysis and maintenance plan to safeguard the dam's long-term viability. Our mathematical analysis begins with a threshold evaluation of three proposed options, considering costs such as relocation, dam removal, construction of new dams, repairs, and ecological damage, as well as benefits like energy generation, flood prevention, employment, tourism, and ecological protection. The data-driven analysis indicates that our option is the most economically viable. We assess water management capabilities, using them as a safety coefficient for the dams. We selected 30 seed points along the riverbank to establish dams. Our recommendation is to increase the number of dams and ensure their strategic distribution. An assessment model based on the analytic hierarchy process was then developed, focusing on three factors: safety, economy, and population. We determined the weights of each factor. The optimal scheme was identified through this model, and the sensitivity of the results was also evaluated. The greatest impact under extreme conditions was found. This paper provides the details of the theoretical analysis and the numerical experiments, which include the use of modeling, optimization, mathematical programming, and so on.\",\"year\":\"2024\",\"keywords\":\"optimization\",\"bibtex\":\"@article{10.11648/j.acm.20241305.13,\\n  author = {Pengcheng Xie},\\n  title = {The Modeling and Optimization of a Multi-dam System},\\n  journal = {Applied and Computational Mathematics},\\n  volume = {13},\\n  number = {5},\\n  pages = {140-152},\\n  doi = {10.11648/j.acm.20241305.13},\\n  url = {https://doi.org/10.11648/j.acm.20241305.13},\\n  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241305.13},\\n  abstract = {In this paper, we present a maintenance plan for the Kariba Dam in Africa. The Kariba Dam, a double curvature concrete arch dam in Zambia's capital, is crucial for regional energy and water management. Ensuring its safety and functionality is essential. This study provides a comprehensive analysis and maintenance plan to safeguard the dam's long-term viability. Our mathematical analysis begins with a threshold evaluation of three proposed options, considering costs such as relocation, dam removal, construction of new dams, repairs, and ecological damage, as well as benefits like energy generation, flood prevention, employment, tourism, and ecological protection. The data-driven analysis indicates that our option is the most economically viable. We assess water management capabilities, using them as a safety coefficient for the dams. We selected 30 seed points along the riverbank to establish dams. Our recommendation is to increase the number of dams and ensure their strategic distribution. An assessment model based on the analytic hierarchy process was then developed, focusing on three factors: safety, economy, and population. We determined the weights of each factor. The optimal scheme was identified through this model, and the sensitivity of the results was also evaluated. The greatest impact under extreme conditions was found. This paper provides the details of the theoretical analysis and the numerical experiments, which include the use of modeling, optimization, mathematical programming, and so on.},\\n year = {2024},\\nkeywords = {optimization},\\n}\\n\\n\",\"author_short\":[\"Xie, P.\"],\"key\":\"10.11648/j.acm.20241305.13\",\"id\":\"10.11648/j.acm.20241305.13\",\"bibbaseid\":\"xie-themodelingandoptimizationofamultidamsystem-2024\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://doi.org/10.11648/j.acm.20241305.13\"},\"keyword\":[\"optimization\"],\"metadata\":{\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\"html\":\"\"}],\n      metadata: {\"owner\":\"xie, p\",\"authorlinks\":{\"xie, p\":\"https://pxie98.github.io/publication.html\"}},\n      ownerID: \"se4YnehBangxagBX4\"\n    };\n  </script>\n\n  <script type=\"text/javascript\">\n  var site$;\n  if (typeof $ != 'undefined') {\n    console.log('existing jquery: storing and then restoring');\n    site$ = $;\n    $ = null;\n  }\n  </script>\n\n  <script src=\"https://ajax.googleapis.com/ajax/libs/jquery/2.2.4/jquery.min.js\">\n  </script>\n  <script type=\"text/javascript\">\n  if (site$) {\n    console.log('existing jquery: avoiding conflict and restoring');\n    bb$ = $.noConflict(true);\n    $ = site$;\n  } else {\n    bb$ = $;\n  }\n  </script>\n\n  <script src=\"//bibbase.org/js/bibbase.min.js\"\n          type=\"text/javascript\">\n  </script>\n\n  <script type=\"text/javascript\"\n          src=\"https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">\n  </script>\n  <script type=\"text/x-mathjax-config\">\n    MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\\\(','\\\\)']]},\n                        skipTags: [\"body\"],\n                        processClass: \"bibbase_paper\"});\n  </script>\n\n  <style>\n  @media print {\n    .dontprint {\n        display: none !important;\n    }\n\n  .bibbase_group {\n    background-color: #E0E0E0;\n    font-style: italic;\n    font-size: larger;\n    text-shadow: 0px 0px 3px #FFFFFF;\n    border-radius: 4px 4px 4px 4px;\n    -moz-border-radius: 4px 4px 4px 4px;\n    -webkit-border-radius: 4px;\n    padding: 5px 40px 5px;\n    margin-top: 10px;\n    margin-bottom: 5px;\n    cursor: pointer;\n  }\n\n  .bibbase_paper {\n    margin-bottom: 5px;\n  }\n\n  }\n  </style>\n\n  \n  <div style=\"float: right; margin-right: 10px; margin-top: 10px; text-align: right; font-size: 12px\">\n    generated by\n    <a href=\"//bibbase.org\"\n       onclick=\"trackOutboundLink('bibbase', 'logo clicked', window.location.href, '//bibbase.org'); return false;\">\n      <img src=\"//bibbase.org/img/logo.svg\"\n           style=\"vertical-align: middle; margin-left: 3px; height: 16px;\"/\n           alt=\"bibbase.org\"\n           title=\"BibBase – The easiest way to maintain your publications page.\"\n        ></a>\n    \n  </div>\n  \n\n  <div id=\"bibbase_header\" class=\"dontprint\" style=\"\">\n\n    <ul class=\"nav nav-pills\" style=\"margin: 0px;\">\n\n      <li class=\"dropdown\" id=\"groupby_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\">\n          Group by\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li class=\"groupby year\"><a onclick=\"groupby('year')\">\n            Year</a>\n        </li>\n        <li class=\"groupby author\"><a onclick=\"groupby('author_short')\">\n            Author</a>\n        </li>\n        <li class=\"groupby type\"><a onclick=\"groupby('type')\">\n            Type</a>\n        </li>\n        <li class=\"groupby keyword\"><a onclick=\"groupby('keyword')\">\n            Keyword</a>\n        </li>\n        <li class=\"groupby downloads\"><a onclick=\"groupby('downloads')\">\n            Downloads</a>\n        </li>\n      </ul>\n      </li>\n\n\n      <li class=\"dropdown\" id=\"menu_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\" alt=\"controls\">\n          <i class=\"fa fa-reorder\" alt=\"controls\"></i>\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li>\n          <a onclick=\"toggleAll()\">\n            <i class=\"fa fa-chevron-down fa-fw\"></i> Expand/Collapse All\n          </a>\n        </li>\n        <li>\n          <a download=\"xpcpublication.bib\" href=\"data:text/bibtex;base64,@misc{xie2025remuregionalminimalupdating,
      title={ReMU: Regional Minimal Updating for Model-Based Derivative-Free Optimization}, 
      author={Pengcheng Xie and Stefan M. Wild},
      year={2025},
      eprint={2504.03606},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2504.03606}, 
}





@misc{li2025novelnumericalmethodtailored,
      title={A novel numerical method tailored for unconstrained optimization problems}, 
      author={Lin Li and Pengcheng Xie and Li Zhang},
      year={2025},
      eprint={2504.02832},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2504.02832}, 
}







@article{10.1093/imanum/drae106,
    author = {Pengcheng Xie and Ya-xiang Yuan},
    title = {Least H2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms},
    journal = {IMA Journal of Numerical Analysis},
    pages = {drae106},
    year = {2025},
    month = {03},
    abstract = {One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the \$H^\{2\}\$ norm of the difference between neighboring quadratic models. The motivation for applying the \$H^\{2\}\$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of \$H^\{2\}\$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least \$H^\{2\}\$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.},
    issn = {0272-4979},
    doi = {10.1093/imanum/drae106},
    url = {https://doi.org/10.1093/imanum/drae106},
    eprint = {https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drae106/62210513/drae106.pdf},
}







@misc{li2025spectrallevenbergmarquardtdeflationmethodmultiple,
      title={A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems}, 
      author={Lin Li and Yuheng Zhou and Pengcheng Xie and Huiyuan Li},
      year={2025},
      eprint={2503.01912},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2503.01912}, 
}





@misc{ye2025improvedadaptiveorthogonalbasis,
      title={An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains}, 
      author={Yangyi Ye and Lin Li and Pengcheng Xie and Haijun Yu},
      year={2025},
      eprint={2503.07624},
      archivePrefix={arXiv},
      primaryClass={math.NA},
      url={https://arxiv.org/abs/2503.07624}, 
}





@article{xie2024bary,
author={Pengcheng Xie and Stefan M. Wild},
  title = {Barycenter of Weight Coefficient Region of Least Weighted $H^2$ Norm Updating Quadratic Models with Vanishing Trust-region Radius},
  year = {2024},
  journal = {SIAM NCC 2024, Early Career Travel Award},
keywords = {optimization},
}




@misc{xie2024lchange,
author={Pengcheng Xie, etc.},
  title = {A Novel Local Analysis of Objectives Approximated by Neural Network: L-Change},
  year = {2024},
  journal = {International Conference on Mathematical Theory of Deep Learning (MTDL)},
keywords = {machine learning, neural network},
}




@misc{zhang2024relationshiplambdapoisednessderivativefreeoptimization,
      title={On the Relationship between $\Lambda$-poisedness in Derivative-Free Optimization and Outliers in Local Outlier Factor}, 
      author={Pengcheng Xie},
      year={2024},
      eprint={2407.17529},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2407.17529}, 
keywords = {optimization, approximation},
}




@misc{postionpaper2025optimization,
  author       = {K. J. Dzahini and S. M. Wild and P. Xie},
  title        = {Optimization approaches for solving inverse problems must account for uncertainty in both data and downstream decisions},
  year         = {2025},
  note         = {Position paper, Inverse Methods for Complex Systems under Uncertainty Workshop, Sponsored by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research},
  url          = {https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf}
}




@article{hu2024lc3rqn,
  author = {W. Hu and P. Xie and L. Zhang and Y.-x. Yuan},
  title = {A Low Computation Cost Cubic Regularized Quasi-Newton Method for Distributed Optimization: LC3RQN},
  year = {2024},
  url = {https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf},
keywords = {optimization},
}



@article{xie2023twodimensional,
  author = {P. Xie and Y.-x. Yuan},
  title = {A new two-dimensional model-based subspace method for large-scale unconstrained derivative-free optimization: 2D-MoSub},
  year = {2023},
  note = {Under review},
  doi = {10.48550/arXiv.2309.14855},
  url = {https://arxiv.org/pdf/2309.14855.pdf},
keywords = {optimization},
}




@article{xieyuannew,
author = {Xie, Pengcheng and Yuan, Ya-xiang},
title = {A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model},
journal = {SIAM Journal on Optimization},
volume = {35},
number = {2},
pages = {1110-1133},
year = {2025},
doi = {10.1137/23M1582023},

URL = { 
    
        https://doi.org/10.1137/23M1582023
    
    

},
eprint = { 
    
        https://doi.org/10.1137/23M1582023
    
    

}
,
    abstract = { Abstract. We analyze the least norm type underdetermined quadratic interpolation model proposed by Conn and Toint [An algorithm using quadratic interpolation for unconstrained derivative free optimization, 1996] from the perspective of the property of trust-region iteration. We find the Karush–Kuhn–Tucker multiplier’s nondeterminacy when constructing a quadratic model considering the trust-region iteration in the case where the current iteration point is on the boundary of the trust region. The lack of the quadratic model’s uniqueness caused by the Karush–Kuhn–Tucker multiplier’s nondeterminacy leads us to propose a new model to consequently improve the model by selectively treating the previously obtained underdetermined quadratic model as a quadratic model or a linear one. A new derivative-free method is given by introducing the improved underdetermined quadratic interpolation model considering the optimality of the model based on the trust-region iteration. The theoretical motivation, property, computational details, and the quadratic model’s formula derived from the Karush–Kuhn–Tucker conditions are discussed. The formula is implementation-friendly for the existing model-based derivative-free methods. The numerical results with released codes support the advantages of our quadratic model in the derivative-free optimization methods. To the best of our knowledge, this is the first work considering the property of trust-region iteration and the model’s optimality when constructing the underdetermined quadratic model for derivative-free trust-region methods. }
}






@article{xie2023dfoto,
  author = {P. Xie and Y. Yuan},
  title = {Derivative-free optimization with transformed objective functions {(DFOTO)} and the algorithm based on the least {Frobenius} norm updating quadratic model},
  year = {2023},
  journal = {Journal of the Operations Research Society of China},
  url = {https://link.springer.com/article/10.1007/s40305-023-00532-x},
keywords = {optimization},
}






@article{xie2023linesearch,
  author = {P. Xie and Y.-x. Yuan},
  title = {A derivative-free optimization algorithm combining line-search and trust-region techniques},
  journal = {Chinese Annals of Mathematics, Series B},
  volume = {44},
  number = {5},
  pages = {719-734},
  year = {2023},
  url1 = {https://link.springer.com/article/10.1007/s11401-023-0040-y},
  url2 = {https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&flag=1},
  url3 = {https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&flag=1},
keywords = {optimization},
}



@article{xie2023ellipsoid,
  author = {P. Xie},
  title = {A derivative-free trust-region method for optimization on the ellipsoid},
  journal = {Journal of Physics: Conference Series},
  volume = {2620},
  pages = {012007},
  year = {2023},
  url = {https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf},
  note = {Presented in 2023 International Conference on Advances in Computer Science and Engineering Technology (ACSE2023)},
keywords = {optimization},
}



@article{XIE2025116146,
title = {Sufficient conditions for error distance reduction in the ℓ2-norm trust region between minimizers of local nonconvex multivariate quadratic approximates},
journal = {Journal of Computational and Applied Mathematics},
volume = {453},
pages = {116146},
year = {2025},
issn = {0377-0427},
doi = {https://doi.org/10.1016/j.cam.2024.116146},
url = {https://www.sciencedirect.com/science/article/pii/S0377042724003959},
author = {Pengcheng Xie},
keywords = {optimization},
abstract = {This paper analyzes the sufficient conditions for distance reduction between minimizers of local nonconvex quadratic approximate functions with diagonal Hessian in the ℓ2-norm trust regions after two iterations. Some examples illustrate the theoretical results of this study.}
}



@misc{2023multiple,
  title = {An efficient derivative-free method for finding multiple solutions},
  year = {2024},
  note = {To be posted on arXiv},
keywords = {numerical PDE},
}



@inproceedings{li2021newuoa,
  author = {S. Li and P. Xie and Z. Zhou, et al.},
  title = {Simulation of interaction of folded waveguide space traveling wave tubes with derivative-free mixed-integer based NEWUOA algorithm},
  booktitle = {2021 7th International Conference on Computer and Communications (ICCC)},
  year = {2021},
  pages = {1215--1219},
  doi = {10.1109/ICCC54389.2021.9674410},
  url = {https://ieeexplore.ieee.org/document/9674410},
keywords = {optimization},
}



@inproceedings{xie2019acc,
  author = {P. Xie and M. Tao},
  title = {Parametric resonant control of macroscopic behaviors of multiple oscillators},
  booktitle = {2019 American Control Conference (ACC)},
  year = {2019},
  pages = {1898--1905},
  doi = {10.23919/ACC.2019.8814709},
  url = {https://ieeexplore.ieee.org/document/8814709},
keywords = {control},
}




@misc{xie2023invariant,
  author = {P. Xie},
  title = {A note on the invariant distribution of a stochastic dynamical system},
  year = {2024},
  doi = {10.12074/202304.01048},
  url = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398},
keywords = {control},
}




@article{10.11648/j.acm.20241305.13,
  author = {Pengcheng Xie},
  title = {The Modeling and Optimization of a Multi-dam System},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {5},
  pages = {140-152},
  doi = {10.11648/j.acm.20241305.13},
  url = {https://doi.org/10.11648/j.acm.20241305.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241305.13},
  abstract = {In this paper, we present a maintenance plan for the Kariba Dam in Africa. The Kariba Dam, a double curvature concrete arch dam in Zambia's capital, is crucial for regional energy and water management. Ensuring its safety and functionality is essential. This study provides a comprehensive analysis and maintenance plan to safeguard the dam's long-term viability. Our mathematical analysis begins with a threshold evaluation of three proposed options, considering costs such as relocation, dam removal, construction of new dams, repairs, and ecological damage, as well as benefits like energy generation, flood prevention, employment, tourism, and ecological protection. The data-driven analysis indicates that our option is the most economically viable. We assess water management capabilities, using them as a safety coefficient for the dams. We selected 30 seed points along the riverbank to establish dams. Our recommendation is to increase the number of dams and ensure their strategic distribution. An assessment model based on the analytic hierarchy process was then developed, focusing on three factors: safety, economy, and population. We determined the weights of each factor. The optimal scheme was identified through this model, and the sensitivity of the results was also evaluated. The greatest impact under extreme conditions was found. This paper provides the details of the theoretical analysis and the numerical experiments, which include the use of modeling, optimization, mathematical programming, and so on.},
 year = {2024},
keywords = {optimization},
}

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M.\n</span>\n\n  \n\n</span>\n\n   2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2504.03606\"\n     onclick=\"log_download('xie-wild-remuregionalminimalupdatingformodelbasedderivativefreeoptimization-2025', 'https://arxiv.org/abs/2504.03606', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"ReMU: Regional Minimal Updating for Model-Based Derivative-Free Optimization [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-wild-remuregionalminimalupdatingformodelbasedderivativefreeoptimization-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-wild-remuregionalminimalupdatingformodelbasedderivativefreeoptimization-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{xie2025remuregionalminimalupdating,\n      title={ReMU: Regional Minimal Updating for Model-Based Derivative-Free Optimization}, \n      author={Pengcheng Xie and Stefan M. Wild},\n      year={2025},\n      eprint={2504.03606},\n      archivePrefix={arXiv},\n      primaryClass={math.OC},\n      url={https://arxiv.org/abs/2504.03606}, \n}\n\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"li-xie-zhang-anovelnumericalmethodtailoredforunconstrainedoptimizationproblems-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2504.02832\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'li-xie-zhang-anovelnumericalmethodtailoredforunconstrainedoptimizationproblems-2025', 'https://arxiv.org/abs/2504.02832', event);\">\n    A novel numerical method tailored for unconstrained optimization problems.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Li, L.; <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Zhang, L.\n</span>\n\n  \n\n</span>\n\n   2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2504.02832\"\n     onclick=\"log_download('li-xie-zhang-anovelnumericalmethodtailoredforunconstrainedoptimizationproblems-2025', 'https://arxiv.org/abs/2504.02832', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A novel numerical method tailored for unconstrained optimization problems [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/li-xie-zhang-anovelnumericalmethodtailoredforunconstrainedoptimizationproblems-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('li-xie-zhang-anovelnumericalmethodtailoredforunconstrainedoptimizationproblems-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{li2025novelnumericalmethodtailored,\n      title={A novel numerical method tailored for unconstrained optimization problems}, \n      author={Lin Li and Pengcheng Xie and Li Zhang},\n      year={2025},\n      eprint={2504.02832},\n      archivePrefix={arXiv},\n      primaryClass={math.OC},\n      url={https://arxiv.org/abs/2504.02832}, \n}\n\n\n\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://doi.org/10.1093/imanum/drae106\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025', 'https://doi.org/10.1093/imanum/drae106', event);\">\n    Least H2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Yuan, Y.\n</span>\n\n  \n\n</span>\n\n  <i>IMA Journal of Numerical Analysis</i>,drae106. 03 2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://doi.org/10.1093/imanum/drae106\"\n     onclick=\"log_download('xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025', 'https://doi.org/10.1093/imanum/drae106', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Least H2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1093/imanum/drae106\"\n     onclick=\"log_download('xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025', 'http://doi.org/10.1093/imanum/drae106', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-yuan-leasth2normupdatingofquadraticinterpolationmodelsforderivativefreetrustregionalgorithms-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{10.1093/imanum/drae106,\n    author = {Pengcheng Xie and Ya-xiang Yuan},\n    title = {Least H2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms},\n    journal = {IMA Journal of Numerical Analysis},\n    pages = {drae106},\n    year = {2025},\n    month = {03},\n    abstract = {One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the \\$H^\\{2\\}\\$ norm of the difference between neighboring quadratic models. The motivation for applying the \\$H^\\{2\\}\\$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of \\$H^\\{2\\}\\$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least \\$H^\\{2\\}\\$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.},\n    issn = {0272-4979},\n    doi = {10.1093/imanum/drae106},\n    url = {https://doi.org/10.1093/imanum/drae106},\n    eprint = {https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drae106/62210513/drae106.pdf},\n}\n\n\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the $H^\\{2\\}$ norm of the difference between neighboring quadratic models. The motivation for applying the $H^\\{2\\}$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of $H^\\{2\\}$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least $H^\\{2\\}$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"li-zhou-xie-li-aspectrallevenbergmarquardtdeflationmethodformultiplesolutionsofsemilinearellipticsystems-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2503.01912\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'li-zhou-xie-li-aspectrallevenbergmarquardtdeflationmethodformultiplesolutionsofsemilinearellipticsystems-2025', 'https://arxiv.org/abs/2503.01912', event);\">\n    A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Li, L.; Zhou, Y.; <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Li, H.\n</span>\n\n  \n\n</span>\n\n   2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2503.01912\"\n     onclick=\"log_download('li-zhou-xie-li-aspectrallevenbergmarquardtdeflationmethodformultiplesolutionsofsemilinearellipticsystems-2025', 'https://arxiv.org/abs/2503.01912', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/li-zhou-xie-li-aspectrallevenbergmarquardtdeflationmethodformultiplesolutionsofsemilinearellipticsystems-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('li-zhou-xie-li-aspectrallevenbergmarquardtdeflationmethodformultiplesolutionsofsemilinearellipticsystems-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{li2025spectrallevenbergmarquardtdeflationmethodmultiple,\n      title={A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems}, \n      author={Lin Li and Yuheng Zhou and Pengcheng Xie and Huiyuan Li},\n      year={2025},\n      eprint={2503.01912},\n      archivePrefix={arXiv},\n      primaryClass={math.OC},\n      url={https://arxiv.org/abs/2503.01912}, \n}\n\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"ye-li-xie-yu-animprovedadaptiveorthogonalbasisdeflationmethodformultiplesolutionswithapplicationstononlinearellipticequationsinvaryingdomains-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2503.07624\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'ye-li-xie-yu-animprovedadaptiveorthogonalbasisdeflationmethodformultiplesolutionswithapplicationstononlinearellipticequationsinvaryingdomains-2025', 'https://arxiv.org/abs/2503.07624', event);\">\n    An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Ye, Y.; Li, L.; <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Yu, H.\n</span>\n\n  \n\n</span>\n\n   2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2503.07624\"\n     onclick=\"log_download('ye-li-xie-yu-animprovedadaptiveorthogonalbasisdeflationmethodformultiplesolutionswithapplicationstononlinearellipticequationsinvaryingdomains-2025', 'https://arxiv.org/abs/2503.07624', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/ye-li-xie-yu-animprovedadaptiveorthogonalbasisdeflationmethodformultiplesolutionswithapplicationstononlinearellipticequationsinvaryingdomains-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('ye-li-xie-yu-animprovedadaptiveorthogonalbasisdeflationmethodformultiplesolutionswithapplicationstononlinearellipticequationsinvaryingdomains-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{ye2025improvedadaptiveorthogonalbasis,\n      title={An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains}, \n      author={Yangyi Ye and Lin Li and Pengcheng Xie and Haijun Yu},\n      year={2025},\n      eprint={2503.07624},\n      archivePrefix={arXiv},\n      primaryClass={math.NA},\n      url={https://arxiv.org/abs/2503.07624}, \n}\n\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"dzahini-wild-xie-optimizationapproachesforsolvinginverseproblemsmustaccountforuncertaintyinbothdataanddownstreamdecisions-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'dzahini-wild-xie-optimizationapproachesforsolvinginverseproblemsmustaccountforuncertaintyinbothdataanddownstreamdecisions-2025', 'https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf', event);\">\n    Optimization approaches for solving inverse problems must account for uncertainty in both data and downstream decisions.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Dzahini, K. J.; Wild, S. M.;  and <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>\n</span>\n\n  \n\n</span>\n\n   2025.\n  <span class=\"note\">Position paper, Inverse Methods for Complex Systems under Uncertainty Workshop, Sponsored by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf\"\n     onclick=\"log_download('dzahini-wild-xie-optimizationapproachesforsolvinginverseproblemsmustaccountforuncertaintyinbothdataanddownstreamdecisions-2025', 'https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Optimization approaches for solving inverse problems must account for uncertainty in both data and downstream decisions [pdf]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/dzahini-wild-xie-optimizationapproachesforsolvinginverseproblemsmustaccountforuncertaintyinbothdataanddownstreamdecisions-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('dzahini-wild-xie-optimizationapproachesforsolvinginverseproblemsmustaccountforuncertaintyinbothdataanddownstreamdecisions-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{postionpaper2025optimization,\n  author       = {K. J. Dzahini and S. M. Wild and P. Xie},\n  title        = {Optimization approaches for solving inverse problems must account for uncertainty in both data and downstream decisions},\n  year         = {2025},\n  note         = {Position paper, Inverse Methods for Complex Systems under Uncertainty Workshop, Sponsored by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research},\n  url          = {https://www.orau.gov/support_files/2025InverseMethods/WildS.pdf}\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://doi.org/10.1137/23M1582023 \"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025', 'https://doi.org/10.1137/23M1582023 ', event);\">\n    A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Yuan, Y.\n</span>\n\n  \n\n</span>\n\n  <i>SIAM Journal on Optimization</i>, 35(2): 1110-1133.  2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://doi.org/10.1137/23M1582023 \"\n     onclick=\"log_download('xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025', 'https://doi.org/10.1137/23M1582023 ', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1137/23M1582023\"\n     onclick=\"log_download('xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025', 'http://doi.org/10.1137/23M1582023', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-yuan-aderivativefreemethodusinganewunderdeterminedquadraticinterpolationmodel-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{xieyuannew,\nauthor = {Xie, Pengcheng and Yuan, Ya-xiang},\ntitle = {A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model},\njournal = {SIAM Journal on Optimization},\nvolume = {35},\nnumber = {2},\npages = {1110-1133},\nyear = {2025},\ndoi = {10.1137/23M1582023},\n\nURL = { \n    \n        https://doi.org/10.1137/23M1582023\n    \n    \n\n},\neprint = { \n    \n        https://doi.org/10.1137/23M1582023\n    \n    \n\n}\n,\n    abstract = { Abstract. We analyze the least norm type underdetermined quadratic interpolation model proposed by Conn and Toint [An algorithm using quadratic interpolation for unconstrained derivative free optimization, 1996] from the perspective of the property of trust-region iteration. We find the Karush–Kuhn–Tucker multiplier’s nondeterminacy when constructing a quadratic model considering the trust-region iteration in the case where the current iteration point is on the boundary of the trust region. The lack of the quadratic model’s uniqueness caused by the Karush–Kuhn–Tucker multiplier’s nondeterminacy leads us to propose a new model to consequently improve the model by selectively treating the previously obtained underdetermined quadratic model as a quadratic model or a linear one. A new derivative-free method is given by introducing the improved underdetermined quadratic interpolation model considering the optimality of the model based on the trust-region iteration. The theoretical motivation, property, computational details, and the quadratic model’s formula derived from the Karush–Kuhn–Tucker conditions are discussed. The formula is implementation-friendly for the existing model-based derivative-free methods. The numerical results with released codes support the advantages of our quadratic model in the derivative-free optimization methods. To the best of our knowledge, this is the first work considering the property of trust-region iteration and the model’s optimality when constructing the underdetermined quadratic model for derivative-free trust-region methods. }\n}\n\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Abstract. We analyze the least norm type underdetermined quadratic interpolation model proposed by Conn and Toint [An algorithm using quadratic interpolation for unconstrained derivative free optimization, 1996] from the perspective of the property of trust-region iteration. We find the Karush–Kuhn–Tucker multiplier’s nondeterminacy when constructing a quadratic model considering the trust-region iteration in the case where the current iteration point is on the boundary of the trust region. The lack of the quadratic model’s uniqueness caused by the Karush–Kuhn–Tucker multiplier’s nondeterminacy leads us to propose a new model to consequently improve the model by selectively treating the previously obtained underdetermined quadratic model as a quadratic model or a linear one. A new derivative-free method is given by introducing the improved underdetermined quadratic interpolation model considering the optimality of the model based on the trust-region iteration. The theoretical motivation, property, computational details, and the quadratic model’s formula derived from the Karush–Kuhn–Tucker conditions are discussed. The formula is implementation-friendly for the existing model-based derivative-free methods. The numerical results with released codes support the advantages of our quadratic model in the derivative-free optimization methods. To the best of our knowledge, this is the first work considering the property of trust-region iteration and the model’s optimality when constructing the underdetermined quadratic model for derivative-free trust-region methods. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.sciencedirect.com/science/article/pii/S0377042724003959\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025', 'https://www.sciencedirect.com/science/article/pii/S0377042724003959', event);\">\n    Sufficient conditions for error distance reduction in the ℓ2-norm trust region between minimizers of local nonconvex multivariate quadratic approximates.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Journal of Computational and Applied Mathematics</i>, 453: 116146.  2025.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.sciencedirect.com/science/article/pii/S0377042724003959\"\n     onclick=\"log_download('xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025', 'https://www.sciencedirect.com/science/article/pii/S0377042724003959', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Sufficient conditions for error distance reduction in the ℓ2-norm trust region between minimizers of local nonconvex multivariate quadratic approximates [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/https://doi.org/10.1016/j.cam.2024.116146\"\n     onclick=\"log_download('xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025', 'http://doi.org/https://doi.org/10.1016/j.cam.2024.116146', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>19 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-sufficientconditionsforerrordistancereductioninthe2normtrustregionbetweenminimizersoflocalnonconvexmultivariatequadraticapproximates-2025')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{XIE2025116146,\ntitle = {Sufficient conditions for error distance reduction in the ℓ2-norm trust region between minimizers of local nonconvex multivariate quadratic approximates},\njournal = {Journal of Computational and Applied Mathematics},\nvolume = {453},\npages = {116146},\nyear = {2025},\nissn = {0377-0427},\ndoi = {https://doi.org/10.1016/j.cam.2024.116146},\nurl = {https://www.sciencedirect.com/science/article/pii/S0377042724003959},\nauthor = {Pengcheng Xie},\nkeywords = {optimization},\nabstract = {This paper analyzes the sufficient conditions for distance reduction between minimizers of local nonconvex quadratic approximate functions with diagonal Hessian in the ℓ2-norm trust regions after two iterations. Some examples illustrate the theoretical results of this study.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This paper analyzes the sufficient conditions for distance reduction between minimizers of local nonconvex quadratic approximate functions with diagonal Hessian in the ℓ2-norm trust regions after two iterations. Some examples illustrate the theoretical results of this study.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2024\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2024')\"\n      onkeypress=\"toggleGroup('group_2024')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2024</span>\n      <span class=\"bibbase_group_count\">\n        \n        (7)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"xie-wild-barycenterofweightcoefficientregionofleastweightedh2normupdatingquadraticmodelswithvanishingtrustregionradius-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  Barycenter of Weight Coefficient Region of Least Weighted $H^2$ Norm Updating Quadratic Models with Vanishing Trust-region Radius.\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Wild, S. M.\n</span>\n\n  \n\n</span>\n\n  <i>SIAM NCC 2024, Early Career Travel Award</i>.  2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-wild-barycenterofweightcoefficientregionofleastweightedh2normupdatingquadraticmodelswithvanishingtrustregionradius-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-wild-barycenterofweightcoefficientregionofleastweightedh2normupdatingquadraticmodelswithvanishingtrustregionradius-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{xie2024bary,\nauthor={Pengcheng Xie and Stefan M. Wild},\n  title = {Barycenter of Weight Coefficient Region of Least Weighted $H^2$ Norm Updating Quadratic Models with Vanishing Trust-region Radius},\n  year = {2024},\n  journal = {SIAM NCC 2024, Early Career Travel Award},\nkeywords = {optimization},\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"pengchengxie-anovellocalanalysisofobjectivesapproximatedbyneuralnetworklchange-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  A Novel Local Analysis of Objectives Approximated by Neural Network: L-Change.\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Pengcheng Xie, e.\n</span>\n\n  \n\n</span>\n\n   2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/pengchengxie-anovellocalanalysisofobjectivesapproximatedbyneuralnetworklchange-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('pengchengxie-anovellocalanalysisofobjectivesapproximatedbyneuralnetworklchange-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{xie2024lchange,\nauthor={Pengcheng Xie, etc.},\n  title = {A Novel Local Analysis of Objectives Approximated by Neural Network: L-Change},\n  year = {2024},\n  journal = {International Conference on Mathematical Theory of Deep Learning (MTDL)},\nkeywords = {machine learning, neural network},\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-ontherelationshipbetweenpoisednessinderivativefreeoptimizationandoutliersinlocaloutlierfactor-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2407.17529\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-ontherelationshipbetweenpoisednessinderivativefreeoptimizationandoutliersinlocaloutlierfactor-2024', 'https://arxiv.org/abs/2407.17529', event);\">\n    On the Relationship between $Λ$-poisedness in Derivative-Free Optimization and Outliers in Local Outlier Factor.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>\n</span>\n\n  \n\n</span>\n\n   2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2407.17529\"\n     onclick=\"log_download('xie-ontherelationshipbetweenpoisednessinderivativefreeoptimizationandoutliersinlocaloutlierfactor-2024', 'https://arxiv.org/abs/2407.17529', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On the Relationship between $Λ$-poisedness in Derivative-Free Optimization and Outliers in Local Outlier Factor [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-ontherelationshipbetweenpoisednessinderivativefreeoptimizationandoutliersinlocaloutlierfactor-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-ontherelationshipbetweenpoisednessinderivativefreeoptimizationandoutliersinlocaloutlierfactor-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{zhang2024relationshiplambdapoisednessderivativefreeoptimization,\n      title={On the Relationship between $\\Lambda$-poisedness in Derivative-Free Optimization and Outliers in Local Outlier Factor}, \n      author={Pengcheng Xie},\n      year={2024},\n      eprint={2407.17529},\n      archivePrefix={arXiv},\n      primaryClass={math.OC},\n      url={https://arxiv.org/abs/2407.17529}, \nkeywords = {optimization, approximation},\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"hu-xie-zhang-yuan-alowcomputationcostcubicregularizedquasinewtonmethodfordistributedoptimizationlc3rqn-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'hu-xie-zhang-yuan-alowcomputationcostcubicregularizedquasinewtonmethodfordistributedoptimizationlc3rqn-2024', 'https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf', event);\">\n    A Low Computation Cost Cubic Regularized Quasi-Newton Method for Distributed Optimization: LC3RQN.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Hu, W.; <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>; Zhang, L.;  and Yuan, Y.\n</span>\n\n  \n\n</span>\n\n  .  2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf\"\n     onclick=\"log_download('hu-xie-zhang-yuan-alowcomputationcostcubicregularizedquasinewtonmethodfordistributedoptimizationlc3rqn-2024', 'https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"A Low Computation Cost Cubic Regularized Quasi-Newton Method for Distributed Optimization: LC3RQN [pdf]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/hu-xie-zhang-yuan-alowcomputationcostcubicregularizedquasinewtonmethodfordistributedoptimizationlc3rqn-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>23 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('hu-xie-zhang-yuan-alowcomputationcostcubicregularizedquasinewtonmethodfordistributedoptimizationlc3rqn-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{hu2024lc3rqn,\n  author = {W. Hu and P. Xie and L. Zhang and Y.-x. Yuan},\n  title = {A Low Computation Cost Cubic Regularized Quasi-Newton Method for Distributed Optimization: LC3RQN},\n  year = {2024},\n  url = {https://lsec.cc.ac.cn/~moa2024/Poster_List.pdf},\nkeywords = {optimization},\n}\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"anonymous-anefficientderivativefreemethodforfindingmultiplesolutions-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  An efficient derivative-free method for finding multiple solutions.\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  \n</span>\n\n  \n\n</span>\n\n   2024.\n  <span class=\"note\">To be posted on arXiv</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/anonymous-anefficientderivativefreemethodforfindingmultiplesolutions-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('anonymous-anefficientderivativefreemethodforfindingmultiplesolutions-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{2023multiple,\n  title = {An efficient derivative-free method for finding multiple solutions},\n  year = {2024},\n  note = {To be posted on arXiv},\nkeywords = {numerical PDE},\n}\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024', 'https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398', event);\">\n    A note on the invariant distribution of a stochastic dynamical system.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>\n</span>\n\n  \n\n</span>\n\n   2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398\"\n     onclick=\"log_download('xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024', 'https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A note on the invariant distribution of a stochastic dynamical system [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.12074/202304.01048\"\n     onclick=\"log_download('xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024', 'http://doi.org/10.12074/202304.01048', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-anoteontheinvariantdistributionofastochasticdynamicalsystem-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{xie2023invariant,\n  author = {P. Xie},\n  title = {A note on the invariant distribution of a stochastic dynamical system},\n  year = {2024},\n  doi = {10.12074/202304.01048},\n  url = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4910398},\nkeywords = {control},\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-themodelingandoptimizationofamultidamsystem-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://doi.org/10.11648/j.acm.20241305.13\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-themodelingandoptimizationofamultidamsystem-2024', 'https://doi.org/10.11648/j.acm.20241305.13', event);\">\n    The Modeling and Optimization of a Multi-dam System.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Applied and Computational Mathematics</i>, 13(5): 140-152.  2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://doi.org/10.11648/j.acm.20241305.13\"\n     onclick=\"log_download('xie-themodelingandoptimizationofamultidamsystem-2024', 'https://doi.org/10.11648/j.acm.20241305.13', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"The Modeling and Optimization of a Multi-dam System [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.11648/j.acm.20241305.13\"\n     onclick=\"log_download('xie-themodelingandoptimizationofamultidamsystem-2024', 'http://doi.org/10.11648/j.acm.20241305.13', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-themodelingandoptimizationofamultidamsystem-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-themodelingandoptimizationofamultidamsystem-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{10.11648/j.acm.20241305.13,\n  author = {Pengcheng Xie},\n  title = {The Modeling and Optimization of a Multi-dam System},\n  journal = {Applied and Computational Mathematics},\n  volume = {13},\n  number = {5},\n  pages = {140-152},\n  doi = {10.11648/j.acm.20241305.13},\n  url = {https://doi.org/10.11648/j.acm.20241305.13},\n  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241305.13},\n  abstract = {In this paper, we present a maintenance plan for the Kariba Dam in Africa. The Kariba Dam, a double curvature concrete arch dam in Zambia&#x27;s capital, is crucial for regional energy and water management. Ensuring its safety and functionality is essential. This study provides a comprehensive analysis and maintenance plan to safeguard the dam&#x27;s long-term viability. Our mathematical analysis begins with a threshold evaluation of three proposed options, considering costs such as relocation, dam removal, construction of new dams, repairs, and ecological damage, as well as benefits like energy generation, flood prevention, employment, tourism, and ecological protection. The data-driven analysis indicates that our option is the most economically viable. We assess water management capabilities, using them as a safety coefficient for the dams. We selected 30 seed points along the riverbank to establish dams. Our recommendation is to increase the number of dams and ensure their strategic distribution. An assessment model based on the analytic hierarchy process was then developed, focusing on three factors: safety, economy, and population. We determined the weights of each factor. The optimal scheme was identified through this model, and the sensitivity of the results was also evaluated. The greatest impact under extreme conditions was found. This paper provides the details of the theoretical analysis and the numerical experiments, which include the use of modeling, optimization, mathematical programming, and so on.},\n year = {2024},\nkeywords = {optimization},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper, we present a maintenance plan for the Kariba Dam in Africa. The Kariba Dam, a double curvature concrete arch dam in Zambia's capital, is crucial for regional energy and water management. Ensuring its safety and functionality is essential. This study provides a comprehensive analysis and maintenance plan to safeguard the dam's long-term viability. Our mathematical analysis begins with a threshold evaluation of three proposed options, considering costs such as relocation, dam removal, construction of new dams, repairs, and ecological damage, as well as benefits like energy generation, flood prevention, employment, tourism, and ecological protection. The data-driven analysis indicates that our option is the most economically viable. We assess water management capabilities, using them as a safety coefficient for the dams. We selected 30 seed points along the riverbank to establish dams. Our recommendation is to increase the number of dams and ensure their strategic distribution. An assessment model based on the analytic hierarchy process was then developed, focusing on three factors: safety, economy, and population. We determined the weights of each factor. The optimal scheme was identified through this model, and the sensitivity of the results was also evaluated. The greatest impact under extreme conditions was found. This paper provides the details of the theoretical analysis and the numerical experiments, which include the use of modeling, optimization, mathematical programming, and so on.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2023\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2023')\"\n      onkeypress=\"toggleGroup('group_2023')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2023</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/pdf/2309.14855.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023', 'https://arxiv.org/pdf/2309.14855.pdf', event);\">\n    A new two-dimensional model-based subspace method for large-scale unconstrained derivative-free optimization: 2D-MoSub.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Yuan, Y.\n</span>\n\n  \n\n</span>\n\n  .  2023.\n  <span class=\"note\">Under review</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/pdf/2309.14855.pdf\"\n     onclick=\"log_download('xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023', 'https://arxiv.org/pdf/2309.14855.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"A new two-dimensional model-based subspace method for large-scale unconstrained derivative-free optimization: 2D-MoSub [pdf]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.48550/arXiv.2309.14855\"\n     onclick=\"log_download('xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023', 'http://doi.org/10.48550/arXiv.2309.14855', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>7 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-yuan-anewtwodimensionalmodelbasedsubspacemethodforlargescaleunconstrainedderivativefreeoptimization2dmosub-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{xie2023twodimensional,\n  author = {P. Xie and Y.-x. Yuan},\n  title = {A new two-dimensional model-based subspace method for large-scale unconstrained derivative-free optimization: 2D-MoSub},\n  year = {2023},\n  note = {Under review},\n  doi = {10.48550/arXiv.2309.14855},\n  url = {https://arxiv.org/pdf/2309.14855.pdf},\nkeywords = {optimization},\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-yuan-derivativefreeoptimizationwithtransformedobjectivefunctionsdfotoandthealgorithmbasedontheleastfrobeniusnormupdatingquadraticmodel-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://link.springer.com/article/10.1007/s40305-023-00532-x\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-yuan-derivativefreeoptimizationwithtransformedobjectivefunctionsdfotoandthealgorithmbasedontheleastfrobeniusnormupdatingquadraticmodel-2023', 'https://link.springer.com/article/10.1007/s40305-023-00532-x', event);\">\n    Derivative-free optimization with transformed objective functions (DFOTO) and the algorithm based on the least Frobenius norm updating quadratic model.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Yuan, Y.\n</span>\n\n  \n\n</span>\n\n  <i>Journal of the Operations Research Society of China</i>.  2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://link.springer.com/article/10.1007/s40305-023-00532-x\"\n     onclick=\"log_download('xie-yuan-derivativefreeoptimizationwithtransformedobjectivefunctionsdfotoandthealgorithmbasedontheleastfrobeniusnormupdatingquadraticmodel-2023', 'https://link.springer.com/article/10.1007/s40305-023-00532-x', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Derivative-free optimization with transformed objective functions (DFOTO) and the algorithm based on the least Frobenius norm updating quadratic model [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-yuan-derivativefreeoptimizationwithtransformedobjectivefunctionsdfotoandthealgorithmbasedontheleastfrobeniusnormupdatingquadraticmodel-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-yuan-derivativefreeoptimizationwithtransformedobjectivefunctionsdfotoandthealgorithmbasedontheleastfrobeniusnormupdatingquadraticmodel-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{xie2023dfoto,\n  author = {P. Xie and Y. Yuan},\n  title = {Derivative-free optimization with transformed objective functions {(DFOTO)} and the algorithm based on the least {Frobenius} norm updating quadratic model},\n  year = {2023},\n  journal = {Journal of the Operations Research Society of China},\n  url = {https://link.springer.com/article/10.1007/s40305-023-00532-x},\nkeywords = {optimization},\n}\n\n\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://link.springer.com/article/10.1007/s11401-023-0040-y\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023', 'https://link.springer.com/article/10.1007/s11401-023-0040-y', event);\">\n    A derivative-free optimization algorithm combining line-search and trust-region techniques.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Yuan, Y.\n</span>\n\n  \n\n</span>\n\n  <i>Chinese Annals of Mathematics, Series B</i>, 44(5): 719-734.  2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://link.springer.com/article/10.1007/s11401-023-0040-y\"\n     onclick=\"log_download('xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023', 'https://link.springer.com/article/10.1007/s11401-023-0040-y', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A derivative-free optimization algorithm combining line-search and trust-region techniques [link]\"\n       title=\"1\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">1</span></a>\n  &nbsp;\n  \n  <a href=\"https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&amp;flag=1\"\n     onclick=\"log_download('xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023', 'https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&amp;flag=1', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A derivative-free optimization algorithm combining line-search and trust-region techniques [link]\"\n       title=\"2\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">2</span></a>\n  &nbsp;\n  \n  <a href=\"https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&amp;flag=1\"\n     onclick=\"log_download('xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023', 'https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&amp;flag=1', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A derivative-free optimization algorithm combining line-search and trust-region techniques [link]\"\n       title=\"3\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">3</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-yuan-aderivativefreeoptimizationalgorithmcombininglinesearchandtrustregiontechniques-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{xie2023linesearch,\n  author = {P. Xie and Y.-x. Yuan},\n  title = {A derivative-free optimization algorithm combining line-search and trust-region techniques},\n  journal = {Chinese Annals of Mathematics, Series B},\n  volume = {44},\n  number = {5},\n  pages = {719-734},\n  year = {2023},\n  url1 = {https://link.springer.com/article/10.1007/s11401-023-0040-y},\n  url2 = {https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=202305005&amp;flag=1},\n  url3 = {https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=202305005&amp;flag=1},\nkeywords = {optimization},\n}\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"xie-aderivativefreetrustregionmethodforoptimizationontheellipsoid-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-aderivativefreetrustregionmethodforoptimizationontheellipsoid-2023', 'https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf', event);\">\n    A derivative-free trust-region method for optimization on the ellipsoid.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Journal of Physics: Conference Series</i>, 2620: 012007.  2023.\n  <span class=\"note\">Presented in 2023 International Conference on Advances in Computer Science and Engineering Technology (ACSE2023)</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf\"\n     onclick=\"log_download('xie-aderivativefreetrustregionmethodforoptimizationontheellipsoid-2023', 'https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A derivative-free trust-region method for optimization on the ellipsoid [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-aderivativefreetrustregionmethodforoptimizationontheellipsoid-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>3 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-aderivativefreetrustregionmethodforoptimizationontheellipsoid-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{xie2023ellipsoid,\n  author = {P. Xie},\n  title = {A derivative-free trust-region method for optimization on the ellipsoid},\n  journal = {Journal of Physics: Conference Series},\n  volume = {2620},\n  pages = {012007},\n  year = {2023},\n  url = {https://iopscience.iop.org/article/10.1088/1742-6596/2620/1/012007/pdf},\n  note = {Presented in 2023 International Conference on Advances in Computer Science and Engineering Technology (ACSE2023)},\nkeywords = {optimization},\n}\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2021\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2021')\"\n      onkeypress=\"toggleGroup('group_2021')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2021</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://ieeexplore.ieee.org/document/9674410\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021', 'https://ieeexplore.ieee.org/document/9674410', event);\">\n    Simulation of interaction of folded waveguide space traveling wave tubes with derivative-free mixed-integer based NEWUOA algorithm.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Li, S.; <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Z. Zhou, e. a.\n</span>\n\n  \n\n</span>\n\n  In <i>2021 7th International Conference on Computer and Communications (ICCC)</i>, pages 1215–1219,  2021. \n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://ieeexplore.ieee.org/document/9674410\"\n     onclick=\"log_download('li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021', 'https://ieeexplore.ieee.org/document/9674410', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Simulation of interaction of folded waveguide space traveling wave tubes with derivative-free mixed-integer based NEWUOA algorithm [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1109/ICCC54389.2021.9674410\"\n     onclick=\"log_download('li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021', 'http://doi.org/10.1109/ICCC54389.2021.9674410', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('li-xie-zzhou-simulationofinteractionoffoldedwaveguidespacetravelingwavetubeswithderivativefreemixedintegerbasednewuoaalgorithm-2021')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@inproceedings{li2021newuoa,\n  author = {S. Li and P. Xie and Z. Zhou, et al.},\n  title = {Simulation of interaction of folded waveguide space traveling wave tubes with derivative-free mixed-integer based NEWUOA algorithm},\n  booktitle = {2021 7th International Conference on Computer and Communications (ICCC)},\n  year = {2021},\n  pages = {1215--1219},\n  doi = {10.1109/ICCC54389.2021.9674410},\n  url = {https://ieeexplore.ieee.org/document/9674410},\nkeywords = {optimization},\n}\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2019\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2019')\"\n      onkeypress=\"toggleGroup('group_2019')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2019</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://ieeexplore.ieee.org/document/8814709\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019', 'https://ieeexplore.ieee.org/document/8814709', event);\">\n    Parametric resonant control of macroscopic behaviors of multiple oscillators.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://pxie98.github.io/publication.html\">Xie, P.</a>;  and Tao, M.\n</span>\n\n  \n\n</span>\n\n  In <i>2019 American Control Conference (ACC)</i>, pages 1898–1905,  2019. \n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://ieeexplore.ieee.org/document/8814709\"\n     onclick=\"log_download('xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019', 'https://ieeexplore.ieee.org/document/8814709', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Parametric resonant control of macroscopic behaviors of multiple oscillators [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.23919/ACC.2019.8814709\"\n     onclick=\"log_download('xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019', 'http://doi.org/10.23919/ACC.2019.8814709', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('xie-tao-parametricresonantcontrolofmacroscopicbehaviorsofmultipleoscillators-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@inproceedings{xie2019acc,\n  author = {P. Xie and M. Tao},\n  title = {Parametric resonant control of macroscopic behaviors of multiple oscillators},\n  booktitle = {2019 American Control Conference (ACC)},\n  year = {2019},\n  pages = {1898--1905},\n  doi = {10.23919/ACC.2019.8814709},\n  url = {https://ieeexplore.ieee.org/document/8814709},\nkeywords = {control},\n}\n\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n\n\n\n  </div>\n\n\n  <!-- Modal -->\n\n  <!-- <iframe id=\"bibbase_network_frame\" -->\n  <!--         src=\"//bibbase.org/network/control\" -->\n  <!--         style=\"display: none;\"> -->\n  <!-- </iframe> -->\n\n</div>\n"}; document.write(bibbase_data.data);