A Modification of the PBIL Algorithm Inspired by the CMA-ES Algorithm in Discrete Knapsack Problem

Alicja Poturała,Stanisław Kozdrowski,Jarosław Arabas,Maria Konieczka

doi:10.3390/app11199136

Abstract

The subject of this paper is the comparison of two algorithms belonging to the class of evolutionary algorithms. The first one is the well-known Population-Based Incremental Learning (PBIL) algorithm, while the second one, proposed by us, is a modification of it and based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm. In the proposed Covariance Matrix Adaptation Population-Based Incremental Learning (CMA-PBIL) algorithm, the probability distribution of population is described by two parameters: the covariance matrix and the probability vector. The comparison of algorithms was performed in the discrete domain of the solution space, where we used the well-known knapsack problem in a variety of data correlations. The results obtained show that the proposed CMA-PBIL algorithm can perform better than standard PBIL in some cases. Therefore, the proposed algorithm can be a reasonable alternative to the PBIL algorithm in the discrete space domain.

Highlights

Evolutionary algorithms (EAs) are a valuable tool for solving many multidimensional and NP-hard [1] practical problems, mainly because they outperform traditional methods, whose high space complexity often disqualifies them from being used to solve complex problems
This paper investigated the impact of modifying the Population-Based Incremental Learning (PBIL) algorithm to include dependencies between variables on convergence and performance
The idea behind the CMA-PBIL algorithm was to introduce a covariance matrix to describe the probability distribution that represents the populations, which was inspired by the Covariance Matrix Adaptation Evolution Strategy (CMA-evolutionary strategies (ES)) algorithm

Summary

Introduction

Evolutionary algorithms (EAs) are a valuable tool for solving many multidimensional and NP-hard [1] practical problems, mainly because they outperform traditional methods, whose high space complexity often disqualifies them from being used to solve complex problems. One of the more straightforward and well-known EDAs initially used to solve discrete problems is the Population-Based Incremental Learning (PBIL) algorithm, first proposed by Baluja in [3]. This algorithm owes its simplicity to the fact that the probability distribution of the subsequent bits in the chromosome is independent, so both the point generation and the learning process can be performed separately for each variable. PBIL is popular and various modifications of the algorithm have been developed, introducing, among others, the probability vector multiplication [4,5], elite strategy [6] and non-parametric approach [7]

Methods

Results

Conclusion