A reconsideration of quasimonotone variational inequality problems

TL;DR

This work addresses variational inequality problems $VI(C,F)$ in real Hilbert spaces where $F$ is quasimonotone and $L$-Lipschitz. It adopts a self-adaptive Tseng's extragradient algorithm and proves strong convergence to the solution set $S$, along with sublinear and $Q$-linear convergence rates, while relaxing prior finiteness assumptions on the auxiliary set $A$. The authors also show how the convergence behavior depends on a relaxed condition $(A5')$ with parameter $\alpha\in[0,1]$, yielding sublinear rates for $\alpha>0$ and linear convergence for $\alpha=0$. Numerical experiments validate the algorithm’s effectiveness and demonstrate improved performance over existing methods, highlighting practical implementability for quasimonotone VI problems.

Abstract

This paper is based on Tseng's exgradient algorithm for solving variational inequality problems in real Hilbert spaces. Under the assumptions that the cost operator is quasimonotone and Lipschitz continuous, we establish the strong convergence, sublinear convergence, and Q-linear convergence of the algorithm. The results of this paper provide new insights into quasimonotone variational inequality problems, extending and enriching existing results in the literature. Finally, we conduct numerical experiments to illustrate the effectiveness and implementability of our proposed condition and algorithm.

Figures (3)

• Figure 1: Original signal and recovered signal for different $M$, $N$ and $K$.
• Figure 2: The relationship between the Ratio and the number of iterations.
• Figure 3: Comparison of Error and number of iterations.

Theorems & Definitions (22)

• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Remark 3.1
• Example 3.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3