Q-fully quadratic modeling and its application in a random subspace derivative-free method

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-06-20 DOI:10.1007/s10589-024-00590-8
Yiwen Chen, Warren Hare, Amy Wiebe
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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.

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全二次方建模及其在随机子空间无导数方法中的应用
基于模型的无导数优化(DFO)方法是一类重要的无导数优化方法,众所周知,这类方法在解决高维优化问题时比较吃力。最近的研究表明,将随机子空间纳入基于模型的无导数优化方法有可能提高其在高维问题上的性能。然而,由于二次近似模型的复杂性,目前大多数理论和实践成果都是基于线性近似模型的。本文提出了一种基于二次逼近的随机子空间信任区域算法。与大多数前辈算法不同,该算法不需要任何特殊形式的目标函数。我们研究了样本集的几何形状、近似的误差边界以及子空间的质量。特别是,我们提供了一种构建 Q 全二次模型的技术,这种技术易于分析和实现。我们提出了算法几乎可以确定的全局收敛结果,并给出了找到足够小梯度的预期迭代次数上限。我们还进行了数值实验,使用线性和二次逼近模型比较了我们算法的性能。数值结果表明了使用二次逼近的优缺点。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
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.
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