Covariance Matrix Adaptation for Multi-objective Optimization

Abstract

The covariancematrix adaptation evolution strategy (CMA-ES) is one of themost powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.

A Hypervolume-Based Optimizer for High-Dimensional Objective Spaces

“…The first MOEA tailored specifically to the hypervolume indicator was described in Emmerich et al (2005); it combines nondominated sorting with the hypervolume indicator and considers one offspring per generation (steady state). Similar fitness assignment strategies were later adopted in Zitzler et al (2007) and Igel et al (2007), and also other search algorithms were proposed where the hypervolume indicator is partially used for search guidance (Nicolini 2005;Mostaghim et al 2007). Moreover, specific aspects like hypervolume-based environmental selection (Bradstreet et al 2006), cf.…”

“…Starting with the lowest dominance depth level, the partitions are moved one by one to the new population as long as the first partition is reached that cannot be transferred completely. This corresponds to the scheme used in most hypervolume-based multiobjective optimizers (Emmerich et al 2005;Igel et al 2007;Brockhoff and Zitzler 2007). 2.…”

“…The scheme of first applying non-dominated sorting is similar to other algorithms (e.g., Igel et al 2007;Emmerich et al 2005). The differences are: (1) the fitness assignment scheme for mating, (2) the one for environmental selection, and (3) the method how the fitness values are determined.…”

“…Several hypervolume-based MOEAs have been proposed meanwhile (e.g., Emmerich et al 2005;Igel et al 2007;Brockhoff and Zitzler 2007), but their main drawback is their extreme computational overhead. Although there have been recent studies presenting improved algorithms for hypervolume calculation, currently high-dimensional problems with six or more objectives are infeasible for these MOEAs.…”

Genetic Algorithm Optimization of Point Charges in Force Field Development: Challenges and Insights

“…However, as we already discussed, the covariance matrix of the GA solutions is numerically equivalent to the Hessian, and, in fact, this useful property of the covariance matrices is utilized in some recently developed advanced evolutionary methods such as the covariance matrix adaptation evolution strategy (CMA-ES) approach. [88][89][90][91] Like other evolutionary strategy (ES) techniques, CMA-ES differs from less sophisticated classical GA methods in the implementation of the crossover and mutation operations; in some cases (CMA-ES included), new candidate solutions/offspring are sampled from the multivariate normal distribution, rather than produced by the traditional crossover operator. However, the most important CMA-ES feature in the context of this discussion is that a new set of solutions is generated using an approximate covariance matrix, which is updated at every step of the optimization.…”