Worst-case complexity analysis of derivative-free methods for multi-objective optimization

TL;DR

This paper analyzes the worst-case complexity of two derivative-free algorithms, DFMOnew and DFMOlight, for unconstrained multi-objective optimization with black-box functions, providing iteration and evaluation bounds.

Contribution

It introduces two new algorithms based on linesearch expansion and derives their worst-case complexity, extending understanding of derivative-free methods in multi-objective optimization.

Findings

DFMOlight's complexity matches recent literature results.

Provided bounds on iterations with stationarity above a threshold.

Analyzed algorithms for black-box multi-objective problems.

Abstract

In this work, we are concerned with the worst case complexity analysis of "a posteriori" methods for unconstrained multi-objective optimization problems where objective function values can only be obtained by querying a black box. We present two main algorithms, namely DFMOnew and DFMOlight which are based on a linesearch expansion technique. In particular, \DFMOnew, requires a complete exploration of the points in the current set of non-dominated solutions, whereas DFMOlight only requires the exploration around a single point in the set of non-dominated solutions. For these algorithms, we derive worst case iteration and evaluation complexity results. In particular, the complexity results for DFMOlight aligns with those recently proved in the literature for a directional multisearch method. Furthermore, exploiting an expansion technique of the step, we are also able to give further complexity results concerning the number of iterations with a measure of stationarity above a prefixed tolerance.