A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model. Xie, P. & Yuan, Y. SIAM Journal on Optimization, 35(2):1110-1133, 2025. ![A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex 1 download 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{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. }
}
Downloads: 1
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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. 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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. 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