Worst-case Complexity Bounds of Directional Direct-search Methods for Multiobjective Optimization

TL;DR

This paper analyzes the worst-case complexity of Direct Multisearch, a derivative-free algorithm for multiobjective optimization, providing bounds that depend on the problem's dimensions and the acceptance criteria.

Contribution

It offers the first worst-case complexity bounds for Direct Multisearch in unconstrained multiobjective optimization, including a refined bound for a specific acceptance criterion.

Findings

Complexity bound proportional to the square of the inverse threshold for general Direct Multisearch.

Refined complexity bound proportional to the square of the inverse threshold for a stricter acceptance criterion.

Results apply to nonconvex smooth functions in unconstrained settings.

Abstract

Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.