An indicator-based evolutionary algorithm for the numerical treatment of equality constrained multi-objective optimisation problems

Abstract

In many applications, especially those of the real world, we find problems that require for several conflicting objectives to be optimised simultaneously; moreover, these problems may require the consideration of limitations that restrict the space of decisions. These problems arise in the scope of Constrained Optimisation that needs for optimal solutions to follow a set of equality and inequality constraints to be considered valid.

While Evolutionary approaches have proven themselves a useful tool for tackling Multi-objective Optimisation Problems (MOPs), they are incapable of accurately approximate the solution when considering Equality Constraints as part of the problem. At the same time, many state-of-the- art algorithms try to incorporate ways to handle Equality Constrained MOPs (ECMOPs) little to none, take into consideration the usage of performance indicators as means for solving this kind of problems.

Here, we designed and implemented an EMOA for tackling Equality Constrained MOPs (EC- MOPs). Using a performance indicator as a density estimator, based on an artificially con- structed Reference set that closely resembles the feasible area of a particular ECMOP, the algorithm was able to find Pareto-optimal solutions that both lie within the feasible region and improve the quality of the final approximation.

We make an empirical study of our proposed algorithm, testing its capabilities over a set of benchmarking functions composed of bi and three-objective optimisation problems, each with one equality constraint. To give validity to this project, we compare the obtained results against those obtained by two state-of-the-art algorithms. To quantify and compare the performance of each algorithm, we calculated the average Hausdorff distance (∆p) using the actual Pareto front of the benchmark problems, and calculated the ratio of feasible solutions within the final population.

The obtained results over the problem set demonstrate that it is possible to approximate the Pareto front of a given ECMOP using only an evolutionary algorithm. We obtain this candidate solution by approximating the shape of the front using an artificially constructed set, which takes into account the information of the constraints to modify the shape. This whole process required no gradient information, preserving the advantages of applying an evolutionary approach to the problems.