Multimodal Optimization by Covariance Matrix Self-Adaptation Evolution Strategy with Repelling Subpopulations

Abstract

During the recent decades, many niching methods have been proposed and empirically verified on some available test problems. They often rely on some particular assumptions associated with the distribution, shape, and size of the basins, which can seldom be made in practical optimization problems. This study utilizes several existing concepts and techniques, such as taboo points, normalized Mahalanobis distance, and the Ursem's hill-valley function in order to develop a new tool for multimodal optimization, which does not make any of these assumptions. In the proposed method, several subpopulations explore the search space in parallel. Offspring of a subpopulation are forced to maintain a sufficient distance to the center of fitter subpopulations and the previously identified basins, which are marked as taboo points. The taboo points repel the subpopulation to prevent convergence to the same basin. A strategy to update the repelling power of the taboo points is proposed to address the challenge of basins of dissimilar size. The local shape of a basin is also approximated by the distribution of the subpopulation members converging to that basin. The proposed niching strategy is incorporated into the covariance matrix self-adaptation evolution strategy (CMSA-ES), a potent global optimization method. The resultant method, called the covariance matrix self-adaptation with repelling subpopulations (RS-CMSA), is assessed and compared to several state-of-the-art niching methods on a standard test suite for multimodal optimization. An organized procedure for parameter setting is followed which assumes a rough estimation of the desired/expected number of minima available. Performance sensitivity to the accuracy of this estimation is also studied by introducing the concept of robust mean peak ratio. Based on the numerical results using the available and the introduced performance measures, RS-CMSA emerges as the most successful method when robustness and efficiency are considered at the same time.

Real-valued evolutionary multi-modal optimization driven by hill-valley clustering

“…In RS-CMSA, the equidistant test points are replaced by a golden section search, with a maximum of N t = 10 test points. More test points could be used due to its limited application only as post-processing step to filter out indistinct global optima [2].…”

“…CMSA is terminated using the recommended criteria [7], and parameters set as in RS-CMSA [2], that is, if the improvement in fitness value over the last 10 + ⌊30d/N c ⌋ generations is less than TOL = 10 −5 . Tolerance TOL corresponds to the desired accuracy of optima in the CEC2013 niching benchmark suite, and should be adapted if a different accuracy is required for the problem at hand.…”

Niching an estimation-of-distribution algorithm by hierarchical Gaussian mixture learning

“…As a secondary goal, it is interesting to see how well HGML can aid in true multi-modal optimization, without further netuning. We therefore show its performance in nding all global optima on the benchmark problems from the CEC'2013 special session on multi-modal optimization [12] in Table 1, and repeat the experiment as presented in [1]. As performance measure, the peak ratio is used, which is the number of global optima, referred to as peaks, correctly determined within the computational budget (MaxEvals).…”

“…For reference, the results of RS-CMSA [1], the winner of the GECCO'16 benchmark, and NEA2 [16], which is based on the nearest be er tree, are shown.…”

“…ese approaches are generally a wrapper around a core search algorithm that orchestrates the use of multiple instances of said algorithm in distinct locally interesting regions, or niches, of the search space. A recent overview and comparison of niching approaches is given in [1], where the Covariance Matrix Self-Adaptation Evolution Strategy with Repelling Sub-populations (RS-CMSA) is introduced, which is the winner of the GECCO'16 Competition on Niching Methods for Multimodal Optimization based on the CEC'2013 niching benchmark [12]. In RS-CMSA, but also in LIPS [17], NSDE [18], PNPCDE [3], NEA2 [16], and IPOP-CMA-ES [2], the user has to estimate the desired number of optima or a minimum distance between niches, and a considerable amount of e ort is spent on calibration and sensitivity analysis of this parameter for di erent benchmark problems.…”