{"title":"Local convergence analysis of an inexact trust-region method for nonsmooth optimization","authors":"Robert J. Baraldi, Drew P. Kouri","doi":"10.1007/s11590-023-02092-8","DOIUrl":null,"url":null,"abstract":"<p>In Baraldi (Math Program 20:1–40, 2022), we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated excellent performance and scalability with problem size. In this paper, we enrich the convergence analysis for this algorithm, proving strong convergence of the iterates with guaranteed rates. In particular, we demonstrate that the trust-region algorithm recovers superlinear, even quadratic, convergence rates when using a second-order Taylor approximation of the smooth objective function term.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"8 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Optimization Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11590-023-02092-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In Baraldi (Math Program 20:1–40, 2022), we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated excellent performance and scalability with problem size. In this paper, we enrich the convergence analysis for this algorithm, proving strong convergence of the iterates with guaranteed rates. In particular, we demonstrate that the trust-region algorithm recovers superlinear, even quadratic, convergence rates when using a second-order Taylor approximation of the smooth objective function term.
在《Baraldi》（Math Program 20:1-40, 2022）中，我们介绍了一种用于最小化希尔伯特空间中平滑非凸函数与非平滑凸函数之和的非精确信任区域算法--这类问题在数据科学、学习、最优控制和逆问题中无处不在。该算法表现出卓越的性能，并可随着问题规模的增大而扩展。在本文中，我们丰富了该算法的收敛性分析，证明了迭代的强收敛性和保证率。特别是，我们证明了当使用二阶泰勒近似平滑目标函数项时，信任区域算法能恢复超线性甚至二次收敛率。
期刊介绍：
Optimization Letters is an international journal covering all aspects of optimization, including theory, algorithms, computational studies, and applications, and providing an outlet for rapid publication of short communications in the field. Originality, significance, quality and clarity are the essential criteria for choosing the material to be published.
Optimization Letters has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time one of the most striking trends in optimization is the constantly increasing interdisciplinary nature of the field.
Optimization Letters aims to communicate in a timely fashion all recent developments in optimization with concise short articles (limited to a total of ten journal pages). Such concise articles will be easily accessible by readers working in any aspects of optimization and wish to be informed of recent developments.
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