Differential Evolution for linear equality constraint satisfaction via unconstrained search in the null space

Abstract

Evolutionary algorithms (EAs) are widely used for a variety of optimization problems, most of them with the presence of constraints. As move operators are usually blind to the constraints, (i.e. when operating upon feasible individuals they do not necessarily generate feasible offspring) standard EAs must be equipped with a constraint handling technique. This paper focuses on exactly satisfying the linear equality constraints present in continuous optimization problems that may also include additional non-linear equality and inequality constraints. The proposed method, named DELEqC-III, is an extension of two other previously developed methods. In this work, the original constrained problem (in ${\mathbb {R}}^n$) is transformed into a lower-dimensional (${\mathbb {R}}^{n-m}$) unconstrained optimization problem, where n is the number of variables and m is the number of linear equality constraints. DELEqC-III performs the search in the null space associated with the linear equality constraints allowing the method to exactly satisfy such constraints. In order to show the efficiency of the method, scalable test-problems are used to analyze the performance of the new proposal.