{"title":"Q-fully quadratic modeling and its application in a random subspace derivative-free method","authors":"Yiwen Chen, Warren Hare, Amy Wiebe","doi":"10.1007/s10589-024-00590-8","DOIUrl":null,"url":null,"abstract":"<p>Model-based derivative-free optimization (DFO) methods are an important class of DFO methods that are known to struggle with solving high-dimensional optimization problems. Recent research has shown that incorporating random subspaces into model-based DFO methods has the potential to improve their performance on high-dimensional problems. However, most of the current theoretical and practical results are based on linear approximation models due to the complexity of quadratic approximation models. This paper proposes a random subspace trust-region algorithm based on quadratic approximations. Unlike most of its precursors, this algorithm does not require any special form of objective function. We study the geometry of sample sets, the error bounds for approximations, and the quality of subspaces. In particular, we provide a technique to construct <i>Q</i>-fully quadratic models, which is easy to analyze and implement. We present an almost-sure global convergence result of our algorithm and give an upper bound on the expected number of iterations to find a sufficiently small gradient. We also develop numerical experiments to compare the performance of our algorithm using both linear and quadratic approximation models. The numerical results demonstrate the strengths and weaknesses of using quadratic approximations.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10589-024-00590-8","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Model-based derivative-free optimization (DFO) methods are an important class of DFO methods that are known to struggle with solving high-dimensional optimization problems. Recent research has shown that incorporating random subspaces into model-based DFO methods has the potential to improve their performance on high-dimensional problems. However, most of the current theoretical and practical results are based on linear approximation models due to the complexity of quadratic approximation models. This paper proposes a random subspace trust-region algorithm based on quadratic approximations. Unlike most of its precursors, this algorithm does not require any special form of objective function. We study the geometry of sample sets, the error bounds for approximations, and the quality of subspaces. In particular, we provide a technique to construct Q-fully quadratic models, which is easy to analyze and implement. We present an almost-sure global convergence result of our algorithm and give an upper bound on the expected number of iterations to find a sufficiently small gradient. We also develop numerical experiments to compare the performance of our algorithm using both linear and quadratic approximation models. The numerical results demonstrate the strengths and weaknesses of using quadratic approximations.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.