Benchmarking Parameter Control Methods in Differential Evolution for Mixed-Integer Black-Box Optimization

TL;DR

This work analyzes parameter control methods (PCMs) for differential evolution (DE) applied to mixed-integer black-box optimization using the bbob-mixint suite. By evaluating nine PCMs across eight mutation strategies and two repair methods, the study reveals that PCM performance hinges on the mutation-repair combination, with simple PCMs like P-CoBi and P-Co often outperform the SHADE PCM, and DE with a suitable PCM can surpass CMA-ES variants at larger budgets. It also shows that P-SHA, despite its success in numerical BB optimization, fails on bbob-mixint due to maladaptive memory updates that cause early stagnation. The findings highlight the potential for targeted PCM selection or algorithm portfolios to enhance performance in mixed-integer BB optimization, particularly for high budgets and high dimensions.

Abstract

Differential evolution (DE) generally requires parameter control methods (PCMs) for the scale factor and crossover rate. Although a better understanding of PCMs provides a useful clue to designing an efficient DE, their effectiveness is poorly understood in mixed-integer black-box optimization. In this context, this paper benchmarks PCMs in DE on the mixed-integer black-box optimization benchmarking function (bbob-mixint) suite in a component-wise manner. First, we demonstrate that the best PCM significantly depends on the combination of the mutation strategy and repair method. Although the PCM of SHADE is state-of-the-art for numerical black-box optimization, our results show its poor performance for mixed-integer black-box optimization. In contrast, our results show that some simple PCMs (e.g., the PCM of CoDE) perform the best in most cases. Then, we demonstrate that a DE with a suitable PCM performs significantly better than CMA-ES with integer handling for larger budgets of function evaluations. Finally, we show how the adaptation in the PCM of SHADE fails.

Figures (13)

• Figure 1: Comparison of the nine PCMs and NOPCM using the rand/1 strategy and Baldwinian repair method on the 24 bbob-mixint functions with $n \in \{10, 80, 160\}$.
• Figure 4: Comparison of P-j and P-CoBi with the three CMA-ES variants on the 24 bbob-mixint functions with $n \in \{10, 80, 160\}$.
• Figure 5: Analysis results of a typical single run of P-SHA on $f_3$ with $n=80$.
• Figure S.1: Comparison of the nine PCMs and NOPCM with the rand/1 mutation strategy and the Baldwinian repair method on the 24 bbob-mixint functions for $n \in \{5, 10, 20, 40, 80, 160\}$.
• Figure S.3: Comparison of the nine PCMs and NOPCM with the rand/2 mutation strategy and the Baldwinian repair method on the 24 bbob-mixint functions for $n \in \{5, 10, 20, 40, 80, 160\}$.