@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},
}

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.

@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}, 
}

@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{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. }
}

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{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.}
}

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.

@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},
}

@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},
}

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

@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},
}

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.

@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{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},
}

@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},
}