Least H 2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms

Journal Article · Sat Mar 01 00:00:00 EST 2025 · IMA Journal of Numerical Analysis

DOI:https://doi.org/10.1093/imanum/drae106· OSTI ID:2526285

Xie, Pengcheng; Yuan, Ya-xiang

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.

• Research Organization: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
• Sponsoring Organization: US Department of Energy; USDOE; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
• Grant/Contract Number: AC02-05CH11231
• OSTI ID: 2526285
• Alternate ID(s): OSTI ID: 2568497
• Journal Information: IMA Journal of Numerical Analysis, Journal Name: IMA Journal of Numerical Analysis; ISSN 0272-4979
• Publisher: Oxford University Press
• Country of Publication: United Kingdom
• Language: English