Pareto Front Approximation Using a Hybrid Approach

Abstract

A new method is proposed for approximating a Pareto front of a bound constrained biobjective optimization problem (BOP) where the evaluation of the objective functions is very expensive and/or the structure of the objective functions either cannot be exploited or not known. The method employs a hybrid optimization approach using two direct search techniques (dividing rectangles and mesh adaptive direct search). The algorithm iteratively formulates and solves several single objective optimization problems of the original BOP by using an adaptive weighting scheme, and moves closer to the true Pareto front at the end of each iteration. The method is tested on problems from the literature designed to illustrate some of the inherent difficulties in biobjective optimization such as a nonconvex or disjoint Pareto front, local Pareto front, or a nonuniform Pareto front. Finally, the algorithm is compared with a recent biobjective optimization algorithm, BiMADS, that generates an approximation of the Pareto front by solving a series of single objective formulations of the original BOP. Results show that the proposed algorithm efficiently generates a set of evenly distributed globally Pareto optimal solutions for a diverse types of problems. The accuracy and the distribution of the solutions obtained can be improved further with a relaxed budget in terms of true function evaluations.