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On the Hypomonotone Class of Variational Inequalities

Khaled Alomar, Tatjana Chavdarova

TL;DR

This work investigates the extragradient method for variational inequalities when the governing operator is hypomonotone rather than monotone. It introduces a spectral view by focusing on normal linear operators and shows, via a constructed counterexample, that extragradient can diverge irrespective of the step size in the hypomonotone setting. The authors provide two simple hypomonotone examples and establish a negative-real-part eigenvalue criterion that guarantees hypomonotonicity without monotonicity, enriching the understanding of this broader VI class. A numerical experiment corroborates the divergence phenomenon, underscoring the need for new algorithms or regularization techniques to handle hypomonotone problems 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.

On the Hypomonotone Class of Variational Inequalities

TL;DR

This work investigates the extragradient method for variational inequalities when the governing operator is hypomonotone rather than monotone. It introduces a spectral view by focusing on normal linear operators and shows, via a constructed counterexample, that extragradient can diverge irrespective of the step size in the hypomonotone setting. The authors provide two simple hypomonotone examples and establish a negative-real-part eigenvalue criterion that guarantees hypomonotonicity without monotonicity, enriching the understanding of this broader VI class. A numerical experiment corroborates the divergence phenomenon, underscoring the need for new algorithms or regularization techniques to handle hypomonotone problems 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.

Paper Structure

This paper contains 18 sections, 4 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Works
  4. Monotonicity.
  5. Beyond monotonicity.
  6. Beyond monotonicity: cohypomonotonicity (or $\rho$-comonotonicity).
  7. Beyond monotonicity: hypomonotonicity.
  8. Preliminaries
  9. Hypomonotone Operators: Examples & Subclass Characterization
  10. Example 1: concave problem.
  11. Example 2: affine problem.
  12. Hypomonotone Subclass: Operators with Negative Eigenvalues
  13. Hypomonotonicity of ${\operatorname{T}}$.
  14. Non-monotonicity.
  15. Divergence of Extragradient on Hypomonotone VIs
  16. ...and 3 more sections

Key Result

Theorem 1

Any normal matrix $\mathbf{A}$---as per Def. def:normal---can be diagonalized by a unitary matrix $\mathbf{S}$, yielding: $\mathbf{A} = \mathbf{S}\mathbf{D} \mathbf{S}^\dagger \,,$ where $\mathbf{D}$ is a diagonal matrix of eigenvalues.

Figures (1)

  • Figure 1: Trajectory of the \ref{['eq:eg']} method (blue) on the \ref{['eq:numerical_exp']} problem; with $\gamma=0.5$. The gray arrows depict the vector field of the game. The \ref{['eq:eg']} method diverges for this example.

Theorems & Definitions (12)

  • Definition 1: Monotonicity
  • Definition 2: Hypomonotonicity
  • Definition 3: Lipschitz operator
  • Definition 4: Normal Matrix
  • Theorem 1: Spectral Theorem
  • Lemma 1: Normality of Matrix Representation
  • Theorem 2: Normal Spectral Characterization Theorem
  • Theorem 3: \ref{['eq:eg']} divergence on \ref{['eq:hypomonotone']}.
  • Remark 1
  • Definition 5: Operator Graph
  • ...and 2 more