[论文解读] Maximizing Non-monotone/Non-submodular Functions by Multi-objective Evolutionary Algorithms.

Chao Qian, Yang Yu+3 more|arXiv (Cornell University)|Nov 20, 2017

Metaheuristic Optimization Algorithms Research参考文献 22被引用 4

一句话总结

本文提出了对多目标进化算法（EAs）在最大化非单调且非子模目标函数方面的理论分析，证明了GSEMO算法在多项式期望运行时间内，对两类一般问题类别的非单调子模最大化问题和带大小约束的单调非子模最大化问题，均能实现良好的近似保证。

ABSTRACT

Evolutionary algorithms (EAs) are a kind of nature-inspired general-purpose optimization algorithm, and have shown empirically good performance in solving various real-word optimization problems. However, due to the highly randomized and complex behavior, the theoretical analysis of EAs is difficult and is an ongoing challenge, which has attracted a lot of research attentions. During the last two decades, promising results on the running time analysis (one essential theoretical aspect) of EAs have been obtained, while most of them focused on isolated combinatorial optimization problems, which do not reflect the general-purpose nature of EAs. To provide a general theoretical explanation of the behavior of EAs, it is desirable to study the performance of EAs on a general class of combinatorial optimization problems. To the best of our knowledge, this direction has been rarely touched and the only known result is the provably good approximation guarantees of EAs for the problem class of maximizing monotone submodular set functions with matroid constraints, which includes many NP-hard combinatorial optimization problems. The aim of this work is to contribute to this line of research. As many combinatorial optimization problems also involve non-monotone or non-submodular objective functions, we consider these two general problem classes, maximizing non-monotone submodular functions without constraints and maximizing monotone non-submodular functions with a size constraint. We prove that a simple multi-objective EA called GSEMO can generally achieve good approximation guarantees in polynomial expected running time.

研究动机与目标

• 为进化算法行为提供超越孤立问题的一般性理论解释。
• 填补组合优化中非单调且非子模目标函数理论分析的空白。
• 将现有关于带拟阵约束的单调子模函数结果扩展至更广泛的功能类别。
• 在一般问题类别上建立多目标EAs的可证明良好近似性能。

提出的方法

• 该研究分析了GSEMO算法——一种简单的多目标进化算法——在组合优化中的应用。
• 重点研究两类问题：无约束的非单调子模函数最大化问题，以及带大小约束的单调非子模函数最大化问题。
• 通过理论分析评估GSEMO的期望运行时间与近似质量。
• 利用子模性和非子模性的性质，对算法的收敛行为进行边界控制。
• 该方法建立了实现良好近似比的多项式期望运行时间。
• 该方法依赖于随机过程分析和搜索空间中期望进展的估计。

实验结果

研究问题

• RQ1 多目标EAs能否在无约束条件下，对非单调子模最大化问题实现良好的近似保证？
• RQ2 EAs能否在大小约束下，为单调非子模函数提供可证明的性能边界？
• RQ3 GSEMO在这些一般函数类中的期望运行时间如何扩展？
• RQ4 在非子模问题上EAs经验成功背后的理论依据是什么？
• RQ5 针对单调子模函数的理论框架能否扩展到非单调和非子模情形？

主要发现

• GSEMO算法在无约束非单调子模函数最大化问题中，以多项式期望运行时间实现了良好的近似保证。
• 对于带大小约束的单调非子模函数，GSEMO同样在多项式期望时间内实现了良好的近似比。
• 理论分析证实，EAs在单调子模问题之外也具有可证明的有效性。
• 研究结果将EAs理论理解的范围扩展至包含非单调和非子模目标函数。
• 本研究提供了针对这两类一般问题类别的EA理论结果的首个已知发现。
• 研究结果通过证明EAs在更广泛的组合优化问题上的理论有效性，支持了其通用性特征。