Constrained multiobjective optimization of expensive black-box functions using a heuristic branch-and-bound approach

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-01-02 DOI:10.1007/s10898-023-01336-2

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引用次数: 0

Abstract

While constrained, multiobjective optimization is generally very difficult, there is a special case in which such problems can be solved with a simple, elegant branch-and-bound algorithm. This special case is when the objective and constraint functions are Lipschitz continuous with known Lipschitz constants. Given these Lipschitz constants, one can compute lower bounds on the functions over subregions of the search space. This allows one to iteratively partition the search space into rectangles, deleting those rectangles which—based on the lower bounds—contain points that are all provably infeasible or provably dominated by previously sampled point(s). As the algorithm proceeds, the rectangles that have not been deleted provide a tight covering of the Pareto set in the input space. Unfortunately, for black-box optimization this elegant algorithm cannot be applied, as we would not know the Lipschitz constants. In this paper, we show how one can heuristically extend this branch-and-bound algorithm to the case when the problem functions are black-box using an approach similar to that used in the well-known DIRECT global optimization algorithm. We call the resulting method “simDIRECT.” Initial experience with simDIRECT on test problems suggests that it performs similar to, or better than, multiobjective evolutionary algorithms when solving problems with small numbers of variables (up to 12) and a limited number of runs (up to 600).

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使用启发式分支和边界方法对昂贵的黑盒子函数进行有约束的多目标优化

摘要虽然有约束的多目标优化一般都非常困难，但有一种特殊情况，即这类问题可以用一种简单、优雅的分支和边界算法来解决。这种特殊情况是目标函数和约束函数都是具有已知 Lipschitz 常量的 Lipschitz 连续函数。给定这些 Lipschitz 常量，就可以计算出搜索空间子区域的函数下限。这样，我们就可以将搜索空间迭代分割成矩形区域，删除那些根据下限计算出的矩形区域中包含的点，这些点都是证明不可行的，或者是证明被先前采样点支配的。随着算法的进行，未被删除的矩形将紧密覆盖输入空间中的帕累托集合。遗憾的是，这种优雅的算法无法应用于黑箱优化，因为我们不知道 Lipschitz 常量。在本文中，我们展示了如何利用一种类似于著名的 DIRECT 全局优化算法的方法，启发式地将这种分支与边界算法扩展到问题函数为黑箱的情况。我们将由此产生的方法称为 "simDIRECT"。simDIRECT 在测试问题上的初步经验表明，在解决变量数量较少（最多 12 个）、运行次数有限（最多 600 次）的问题时，它的表现与多目标进化算法相似，甚至更好。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

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.