On the Hypomonotone Class of Variational Inequalities

TL;DR

This paper investigates the extragradient algorithm's limitations when applied to hypomonotone operators in variational inequalities, showing it may diverge and providing theoretical and numerical evidence for this behavior.

Contribution

It characterizes hypomonotone linear operators via eigenvalues and demonstrates the divergence of the extragradient method in this setting.

Findings

Extragradient method diverges for certain hypomonotone operators

Eigenvalue-based characterization of hypomonotone linear operators

Numerical example illustrating divergence in practice

Abstract

This paper studies the behavior of the extragradient algorithm [Korpelevich, 1976] when applied to hypomonotone operators, a class of problems that extends beyond the classical monotone setting. To support the understanding of this variational inequality problem class, we focus on a subclass of hypomonotone linear operators, characterizing them based on their eigenvalues and providing concrete examples. While the extragradient method is widely recognized for its efficiency in solving variational inequalities involving monotone and Lipschitz continuous operators, we demonstrate that it does not guarantee convergence in the hypomonotone case. In particular, we construct a counterexample where the extragradient method diverges regardless of the step size. A numerical experiment is presented to support this result.