Methods for multi-objective optimization: An analysis
Methods for multi-objective optimization: An analysis
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DOI：
10.1016/j.ins.2014.08.071
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发表时间：
2015-02-01
影响因子：
8.1
通讯作者：
Fleming, P. J.
中科院分区：
文献类型：
--
作者：
Giagkiozis, I.;Fleming, P. J.
基于分解的方法通常被认为是目标数量增加的多目标非凸优化问题的解决方案。这些方法采用标量化函数将多目标问题简化为一组单目标问题，这些问题在求解时会产生一组最佳解的良好近似。该集合通常称为帕累托前沿。在这项工作中，我们从概率的角度探讨了使用基于分解的方法相对于基于帕累托的方法对算法收敛的影响。也就是说，我们研究使用基于分解的方法（例如使用切比雪夫标量函数）相对于基于帕累托的方法是否具有优势。我们发现，在目标函数的温和条件下，当我们考虑为遵循平衡轨迹的算法找到最优解的概率时，切比雪夫标量函数与帕累托支配关系具有几乎相同的效果。我们提出这样的假设：与当前可用的经验证据相比，这一看似矛盾的结果表明基于帕累托的方法和基于分解的方法之间的性能差异是由于前一类算法无法遵循平衡的轨迹。我们还将广义分解与这项工作的结果联系起来，并展示了如何根据帕累托前沿几何的先验假设获得给定问题的最佳标量化函数。 (C) 2014 Elsevier Inc. 保留所有权利。
Decomposition-based methods are often cited as the solution to multi-objective nonconvex optimization problems with an increased number of objectives. These methods employ a scalarizing function to reduce the multi-objective problem into a set of single objective problems, which upon solution yield a good approximation of the set of optimal solutions. This set is commonly referred to as Pareto front. In this work we explore the implications of using decomposition-based methods over Pareto-based methods on algorithm convergence from a probabilistic point of view. Namely, we investigate whether there is an advantage of using a decomposition-based method, for example using the Chebyshev scalarizing function, over Pareto-based methods. We find that, under mild conditions on the objective function, the Chebyshev scalarizing function has an almost identical effect to Pareto-dominance relations when we consider the probability of finding superior solutions for algorithms that follow a balanced trajectory. We propose the hypothesis that this seemingly contradicting result compared with currently available empirical evidence, signals that the disparity in performance between Pareto-based and decomposition-based methods is due to the inability of the former class of algorithms to follow a balanced trajectory. We also link generalized decomposition to the results in this work and show how to obtain optimal scalarizing functions for a given problem, subject to prior assumptions on the Pareto front geometry. (C) 2014 Elsevier Inc. All rights reserved.