The importance of being constrained: dealing with infeasible solutions in Differential Evolution and beyond

TL;DR

This paper emphasizes the critical importance of explicitly defining how heuristic optimization algorithms handle infeasible solutions outside constraints, demonstrating its impact on performance, diversity, and reproducibility, especially in Differential Evolution.

Contribution

It introduces the concept of a strategy for managing infeasible solutions as a new algorithmic component, advocating for its formalization, study, and inclusion in heuristic optimization methods.

Findings

Handling infeasible solutions significantly affects algorithm performance and diversity.

The importance of this handling increases with problem dimensionality.

Different Evolution algorithms are similarly affected by this issue.

Abstract

We argue that results produced by a heuristic optimisation algorithm cannot be considered reproducible unless the algorithm fully specifies what should be done with solutions generated outside the domain, even in the case of simple box constraints. Currently, in the field of heuristic optimisation, such specification is rarely mentioned or investigated due to the assumed triviality or insignificance of this question. Here, we demonstrate that, at least in algorithms based on Differential Evolution, this choice induces notably different behaviours - in terms of performance, disruptiveness and population diversity. This is shown theoretically (where possible) for standard Differential Evolution in the absence of selection pressure and experimentally for the standard and state-of-the-art Differential Evolution variants on special test function $f_0$ and BBOB benchmarking suite, respectively. Moreover, we demonstrate that the importance of this choice quickly grows with problem's dimensionality. Different Evolution is not at all special in this regard - there is no reason to presume that other heuristic optimisers are not equally affected by the aforementioned algorithmic choice. Thus, we urge the field of heuristic optimisation to formalise and adopt the idea of a new algorithmic component in heuristic optimisers, which we call here a strategy of dealing with infeasible solutions. This component needs to be consistently (a) specified in algorithmic descriptions to guarantee reproducibility of results, (b) studied to better understand its impact on algorithm's performance in a wider sense and (c) included in the (automatic) algorithmic design. All of these should be done even for problems with box constraints.