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On Non-Monotone Variational Inequalities

Sina Arefizadeh, Angelia Nedić

TL;DR

This work addresses non-monotone variational inequalities by deriving sufficient conditions for the existence of solutions through inverse mapping theory and degree theory. It develops results for both unconstrained problems with differentiable mappings and constrained problems via the natural and normal mappings, Clark's inverse mapping theorem, and degree-theoretic arguments, including near-$\xi$-monotone mappings. The paper also establishes the existence of Minty solutions and proves subsequential convergence of the Korpelevich and Popov methods under Minty-solution assumptions, extending classical VI theory beyond monotone cases. Overall, it provides rigorous existence conditions and convergent algorithms for non-monotone VIs with practical implications for applications where monotonicity fails.

Abstract

In this paper, we provide some sufficient conditions for the existence of solutions to non-monotone Variational Inequalities (VIs) based on inverse mapping theory and degree theory. We have obtained several applicable sufficient conditions for this problem and have introduced a sufficient condition for the existence of a Minty solution. We have shown that the Korpelevich and Popov methods converge to a solution of a non-monotone VI, provided that a Minty solution exists.

On Non-Monotone Variational Inequalities

TL;DR

This work addresses non-monotone variational inequalities by deriving sufficient conditions for the existence of solutions through inverse mapping theory and degree theory. It develops results for both unconstrained problems with differentiable mappings and constrained problems via the natural and normal mappings, Clark's inverse mapping theorem, and degree-theoretic arguments, including near--monotone mappings. The paper also establishes the existence of Minty solutions and proves subsequential convergence of the Korpelevich and Popov methods under Minty-solution assumptions, extending classical VI theory beyond monotone cases. Overall, it provides rigorous existence conditions and convergent algorithms for non-monotone VIs with practical implications for applications where monotonicity fails.

Abstract

In this paper, we provide some sufficient conditions for the existence of solutions to non-monotone Variational Inequalities (VIs) based on inverse mapping theory and degree theory. We have obtained several applicable sufficient conditions for this problem and have introduced a sufficient condition for the existence of a Minty solution. We have shown that the Korpelevich and Popov methods converge to a solution of a non-monotone VI, provided that a Minty solution exists.

Paper Structure

This paper contains 9 sections, 23 theorems, 69 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notions and Terminology
  3. Main Results
  4. Unconstrained VI
  5. Constrained VI
  6. Inverse Mapping Based Approach
  7. Degree Theoretic Approach
  8. Algorithms
  9. Conclusions

Key Result

Lemma 1

Let $K \subseteq \mathbb{R}^m$ be a non-empty closed set and let $F:K \to\mathbb{R}^m$ be a mapping. The following statements hold:

Theorems & Definitions (46)

  • Definition 1: Variational Inequality Problem facchinei2003finite
  • Definition 2: Minty Solution Concepts crespi2005existence
  • Definition 3: Monotone Mappings C2
  • Lemma 1: Lemma 2.2 - Minty's Lemma C2
  • proof
  • Theorem 1: Inverse Mapping Theorem lebl2018introduction
  • Theorem 2
  • proof
  • Lemma 2: Weak Coupling Condition
  • proof
  • ...and 36 more