CMA-PAES: Pareto archived evolution strategy using covariance matrix adaptation for Multi-Objective Optimisation

Abstract

Abstract-The quality of Evolutionary Multi-Objective Optimisation (EMO) approximation sets can be measured by their proximity, diversity and pertinence. In this paper we introduce a modular and extensible Multi-Objective Evolutionary Algorithm (MOEA) capable of converging to the Pareto-optimal front in a minimal number of function evaluations and producing a diverse approximation set. This algorithm, called the Covariance Matrix Adaptation Pareto Archived Evolution Strategy (CMA-PAES), is a form of (µ + λ) Evolution Strategy which uses an online archive of previously found Pareto-optimal solutions (maintained by a bounded Pareto-archiving scheme) as well as a population of solutions which are subjected to variation using Covariance Matrix Adaptation. The performance of CMA-PAES is compared to NSGA-II (currently considered the benchmark MOEA in the literature) on the ZDT test suite of bi-objective optimisation problems and the significance of the results are analysed using randomisation testing.

Covariance matrix adaptation pareto archived evolution strategy with hypervolume-sorted adaptive grid algorithm

“…The resulting algorithm, namely the Covariance Matrix Adaptation Pareto Archived Evolution Strategy with Hypervolume-sorted Adaptive Grid Algorithm (CMA-PAES-HAGA) makes use of the search logic of the Covariance Matrix Adaptation Evolution Strategy (CMAES) [30] and the archive of the PAES structure. A preliminary version of the proposed algorithm without the newly proposed selection mechanism has been proposed in [72], however, its primitive selection mechanism restricts its application to two-objective problems only.…”

“…Covariance Matrix Adaptation (CMA) is a powerful approach to parameter variation which has demonstrated promising results in the optimisation of singleobjective problems with CMA-ES [36], and multiobjective problems with MO-CMA-ES [85] and CMA-PAES [72]. These optimisation algorithms rely on CMA entirely for the variation of solution parameters, and therefore they do not suffer from the curse of dimensionality which affects many optimisation algorithms which rely on reproduction operators.…”

“…The approach to selection employed by MO-CMA-ES relies on the Contributing Hypervolume Indicator (CHV) as a second-level sorting criterion, the calculation of which increases significantly with the number of objectives considered. In contrast, the approach to selection employed by CMA-PAES is driven by an AGA which incurs little algorithm overhead in its calculation [72]. This AGA becomes ineffective when applied to many-objective problems as the number of grid divisions which controls the division of the objective space and the size of the sub-populations becomes increasingly sensitive with each additional objective [70].…”

“…Whilst CMA-PAES has shown promising results in comparison to PAES, NSGA-II, and MO-CMA-ES [70,72], it suffers from the curse of dimensionality when moving to many-objective problems. This is because the process of the AGA for diversity preservation was designed with the two-objective case in mind.…”

A novel preference articulation operator for the Evolutionary Multi-Objective Optimisation of classifiers in concealed weapons detection

“…As a demonstration, in Figure 10 the WZ preference articulation operator has been combined with an initial population for the ZDT4 synthetic test problem from the ZDT test suite, generated by CMA-PAES [30], an algorithm which uses covariance matrix adaptation for search and adaptive grid archiving for selection and maintenance of a population. After 2000 function evaluations CMA-PAES combined with the WZ preference articulation operator has produced the results illustrated in Figure 11.…”