Set-based Differential Evolution with Exponential and Binomial Crossovers for Discrete Optimization Problem

This paper presents set-based differential evolution with exponential and binomial crossovers for discrete optimization problems. While differential evolution has been traditionally applied to continuous optimization problems, it can be adopted to discrete optimization problem by extending techniques in continuous space into discrete space by employing set-based representation. In our approach, a candidate solution is defined as a crisp set, and all arithmetic operations in mutation are redefined through novel operations. The mutation operator in our algorithm adds two different solutions selected randomly to the current solution and the new solution is constructed probabilistically. For two crossovers, the number of inherited mutant parameters follows an exponential distribution in exponential crossover and that of inherited mutant parameters follows a binomial distribution in binomial crossover. This study investigates exponential and binomial crossover mechanisms within the framework of set-based differential evolution. Specifically, we examine the influence of crossover rate on solution accuracy in traveling salesman problem, conducted in two phases: broad search and narrow search. To demonstrate the effectiveness of our algorithms, we examine numerical experiments and compare results with existing algorithms.