Using first-order information in Direct Multisearch for multiobjective optimization

TL;DR

This paper investigates the use of first-order derivative information in the derivative-free Direct MultiSearch algorithm for multiobjective optimization, showing that incorporating derivatives can enhance performance in approximating Pareto fronts.

Contribution

The study introduces a variant of DMS that uses first-order information to improve pruning and search directions, demonstrating competitive results against derivative-based methods.

Findings

Adding first-order info improves DMS efficiency

The derivative-enhanced DMS is competitive with state-of-the-art algorithms

Small evaluation budgets favor the derivative-augmented approach

Abstract

Derivatives are an important tool for single-objective optimization. In fact, it is commonly accepted that derivative-based methods present a better performance than derivative-free optimization approaches. In this work, we will show that the same does not apply to multiobjective derivative-based optimization, when the goal is to compute an approximation to the complete Pareto front of a given problem. The competitiveness of Direct MultiSearch (DMS), a robust and efficient derivative-free optimization algorithm, will be stated for derivative-based multiobjective optimization problems. We will then assess the potential enrichment of adding first-order information to the DMS framework. Derivatives will be used to prune the positive spanning sets considered at the poll step of the algorithm, highlighting the role that ascent directions, that conform to the geometry of the nearby feasible region, can have. Both variants of DMS show to be competitive against a state-of-art derivative-based algorithm. Moreover, for reasonable small budgets of function evaluations, the new variant is not only competitive with the derivative-based solver but also with the original implementation of DMS.