New Classes of Non-monotone Variational Inequality Problems Solvable via Proximal Gradient on Smooth Gap Functions

TL;DR

The paper addresses solving non-monotone variational inequalities by reformulating them with a smooth gap function $g_{\lambda}$ and applying proximal-gradient methods. It establishes a novel link between uniform error bounds and level-set error bounds, enabling local linear convergence of PG on the gap function under level-set error bounds, gradient-Lipschitzness, and suitable initialization. It also introduces an initialization-free homotopy continuation method for affine VIs to guarantee global convergence by path following, and demonstrates practical performance across Nash equilibria, traffic equilibrium, and GAN applications. This work extends the gap-function PG framework to broad non-monotone VIP classes, providing verifiable conditions and scalable algorithms with global convergence guarantees for challenging non-monotone settings.

Abstract

In this paper, we study the local linear convergence behavior of proximal-gradient (PG) descent algorithm on a parameterized gap-function reformulation of a smooth but non-monotone variational inequality problem (VIP). The aim is to solve the non-monotone VI problem without assuming the existence of a Minty-type solution. We first introduce and study various error bound conditions for the gap functions in relation to the VI model. In particular, we show that uniform type error bounds imply level-set type error bounds for composite optimization, revealing a key hierarchical structure there. As a result, local linear convergence is established under some easy-verifiable conditions induced by level-set error bounds, the gradient Lipschitz condition and a suitable initialization condition. Furthermore, for non-monotone affine VIs we present a homotopy continuation scheme that achieves global convergence by dynamically tracing a solution path. Our numerical experiments show the efficacy of the proposed approach, leading to the solutions of a broad class of non-monotone VI problems resulting from the need to compute Nash equilibria, traffic controls, and the GAN models.

Figures (11)

• Figure 1: $g_{\lambda}(x)$ and $\nabla g_{\lambda}(x)$ of Example \ref{['exp:nvip_1']}
• Figure 2: The relationships among the notions of the level-set error bounds
• Figure 3: The relationships among the notions of the level-set error bounds on $[\phi^*\leq\phi <\phi^*+\nu]$
• Figure 4: From the properties of (VIP) to the linear convergence of PG for solving (S-VIP)
• Figure 5: Left: Convergence of Algorithm \ref{['alg:PGOPg']} on Example \ref{['exp:nvip_1']}; Right: Solution generated by Algorithm \ref{['alg:PGOPg']} on Example \ref{['exp:nvip_1']}

Theorems & Definitions (31)

• Definition 1
• Example 1.1
• Example 1.2
• Definition 2
• Proposition 1
• Lemma 2.1: Generalized descent inequality
• Definition 3: Level-set proximal error bound ($\ell$-PEB)
• Theorem 1
• Definition 4: Uniform error bounds
• Definition 5: Level-set error bounds