Multiobjective Optimization Through a Series of Single-Objective Formulations

Abstract

This work deals with bound constrained multiobjective optimization (MOP) of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited, or are absent. Typical situations arise when the functions are computed as the result of a computer simulation. We first present definitions and optimality conditions as well as two families of single-objective formulations of MOP. Next, we propose a new algorithm called for the biobjective optimization (BOP) problem (i.e., MOP with two objective functions). The property that Pareto points may be ordered in BOP and not in MOP is exploited by our algorithm. generates an approximation of the Pareto front by solving a series of single-objective formulations of BOP. These single-objective problems are solved using the recent (mesh adaptive direct search) algorithm for nonsmooth optimization. The Pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the Clarke calculus. Finally, is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization, such as a nonconvex or disjoint Pareto front, local Pareto fronts, or a nonuniform Pareto front.