Inexact Direct-Search Methods for Bilevel Optimization Problems

TL;DR

This paper introduces novel direct search algorithms for bilevel optimization problems that use a fixed accuracy oracle, providing convergence guarantees and complexity bounds, and includes preliminary numerical validation.

Contribution

It presents the first direct search schemes for bilevel problems with a fixed accuracy oracle, including adaptations of mesh adaptive schemes and convergence analysis.

Findings

Convergence guarantees to approximate stationary points.

Complexity bounds in the smooth case.

Preliminary numerical results demonstrate effectiveness.

Abstract

In this work, we introduce new direct search schemes for the solution of bilevel optimization (BO) problems. Our methods rely on a fixed accuracy black box oracle for the lower-level problem, and deal both with smooth and potentially nonsmooth true objectives. We thus analyze for the first time in the literature direct search schemes in these settings, giving convergence guarantees to approximate stationary points, as well as complexity bounds in the smooth case. We also propose the first adaptation of mesh adaptive direct search schemes for BO. Some preliminary numerical results on a standard set of bilevel optimization problems show the effectiveness of our new approaches.