Extragradient methods with complexity guarantees for hierarchical variational inequalities

TL;DR

This paper introduces extragradient methods with proven convergence rates for hierarchical variational inequalities, covering a broad class of problems like bilevel optimization and equilibrium constraints.

Contribution

It develops new extragradient algorithms with complexity guarantees for hierarchical variational inequalities, improving upon existing methods in generality and efficiency.

Findings

Established convergence rates using gap functions.

Provided worst-case iteration complexity bounds.

Achieved weak convergence under geometric weak sharpness.

Abstract

In the framework of a real Hilbert space we consider the problem of approaching solutions to a class of hierarchical variational inequality problems, subsuming several other problem classes including certain mathematical programs under equilibrium constraints, constrained min-max problems, hierarchical game problems, optimal control under VI constraints, and simple bilevel optimization problems. For this general problem formulation, we establish rates of convergence in terms of suitably constructed gap functions, measuring feasibility gaps and optimality gaps. We present worst-case iteration complexity results on both levels of the variational problem, as well as weak convergence under a geometric weak sharpness condition on the lower level solution set. Our results match and improve the state of the art in terms of their iteration complexity and the generality of the problem formulation.