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

IF 1.8 3区数学 Q1 Mathematics Journal of Global Optimization Pub Date : 2024-01-02 DOI:10.1007/s10898-023-01336-2

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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。

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来源期刊

Journal of Global Optimization 数学-应用数学

CiteScore

0.10

自引率

5.60%

发文量

137

审稿时长

6 months

期刊介绍： The Journal of Global Optimization publishes carefully refereed papers that encompass theoretical, computational, and applied aspects of global optimization. While the focus is on original research contributions dealing with the search for global optima of non-convex, multi-extremal problems, the journal’s scope covers optimization in the widest sense, including nonlinear, mixed integer, combinatorial, stochastic, robust, multi-objective optimization, computational geometry, and equilibrium problems. Relevant works on data-driven methods and optimization-based data mining are of special interest. In addition to papers covering theory and algorithms of global optimization, the journal publishes significant papers on numerical experiments, new testbeds, and applications in engineering, management, and the sciences. Applications of particular interest include healthcare, computational biochemistry, energy systems, telecommunications, and finance. Apart from full-length articles, the journal features short communications on both open and solved global optimization problems. It also offers reviews of relevant books and publishes special issues.