On the Extension of the DIRECT Algorithm to Multiple Objectives

Abstract

Deterministic global optimization algorithms like Piyavskii-Shubert, direct, ego and many more, have a recognized standing, for problems with many local optima. Although many single objective optimization algorithms have been extended to multiple objectives, completely deterministic algorithms for nonlinear problems with guarantees of convergence to global Pareto optimality are still missing. For instance, deterministic algorithms usually make use of some form of scalarization, which may lead to incomplete representations of the Pareto optimal set. Thus, all global Pareto optima may not be obtained, especially in nonconvex cases. On the other hand, algorithms attempting to produce representations of the globally Pareto optimal set are usually based on heuristics. We analyze the concept of global convergence for multiobjective optimization algorithms and propose a convergence criterion based on the Hausdorff distance in the decision space. Under this light, we consider the well-known global optimization algorithm direct, analyze the available algorithms in the literature that extend direct to multiple objectives and discuss possible alternatives. In particular, we propose a novel definition for the notion of potential Pareto optimality extending the notion of potential optimality defined in direct. We also discuss its advantages and disadvantages when compared with algorithms existing in the literature.

Algorithm 1028: VTMOP: Solver for Blackbox Multiobjective Optimization Problems

“…DIRECT is globally convergent for nonconvex functions if the objective function is Lipschitz continuous with a Lipschitz constant that is bounded in the feasible design space X. Multiobjective DIRECT (MODIR) [19] extends DIRECT to the multiobjective case by defining and subdividing potentially Pareto optimal hyperintervals. Multiobjective DIRECT algorithms of this form offer similar convergence guarantees as Jones' DIRECT when every component function of the objective is Lipschitz continuous [50]. However, in practice, many function evaluations are needed to achieve high-quality solution sets, so [19] recommend combining MODIR with a faster locally convergent MOA, such as the derivativefree multiobjective line search algorithm of [49].…”