Effective Front-Descent Algorithms with Convergence Guarantees

TL;DR

This paper introduces a generalized class of Front Descent algorithms for multi-objective optimization, providing convergence guarantees, complexity bounds, and demonstrating superior performance over existing methods.

Contribution

It offers a comprehensive analysis of Front Descent methods, including novel convergence results for the sequence of solution sets and validation through extensive experiments.

Findings

Convergence of iterate sets to stationarity is proven.

The approach outperforms state-of-the-art methods in benchmarks.

Finite precision analysis shows exploration steps only enrich solution sets.

Abstract

In this manuscript, we address continuous unconstrained multi-objective optimization problems and we discuss descent type methods for the reconstruction of the Pareto set. Specifically, we analyze the class of Front Descent methods, which generalizes the Front Steepest Descent algorithm allowing the employment of suitable, effective search directions (e.g., Newton, Quasi-Newton, Barzilai-Borwein). We provide a deep characterization of the behavior and the mechanisms of the algorithmic framework, and we prove that, under reasonable assumptions, standard convergence results and some complexity bounds hold for the generalized approach. Moreover, we prove that popular search directions can indeed be soundly used within the framework. Then, we provide a completely novel type of convergence results, concerning the sequence of sets produced by the procedure. In particular, iterate sets are shown to asymptotically approach stationarity for all of their points; the convergence result is accompanied by a worst-case iteration complexity bound; additionally, in finite precision settings, the sets are shown to only be enriched through exploration steps in later iterations, and suitable stopping conditions can be devised. Finally, the results from a large experimental benchmark show that the proposed class of approaches far outperforms state-of-the-art methodologies.