Weak convergence of projection algorithm with momentum terms and new step size rule for quasimonotone variational inequalities
Gourav Kumar, Santanu Soe, V. Vetrivel
TL;DR
We address variational inequality problems for a quasimonotone, $L$-Lipschitz operator $\mathscr{A}$ on a closed convex set $\mathscr{C}$. We propose a simple projection method augmented with two momentum terms and a nondecreasing step-size rule, requiring only one projection and one $\mathscr{A}$ evaluation per iteration. Under standard assumptions, we prove weak convergence to the Minty solution set $\mathscr{S}_{\mathscr{M}}$ and validate performance with numerical experiments, including a signal-recovery application, demonstrating practical efficiency gains. The results suggest the method is a competitive tool for high-dimensional VIPs and motivate future extensions to broader problem classes such as mixed VIPs and equilibrium problems.
Abstract
This article analyses the simple projection method proposed by Izuchukwu et al. [8, Algorithm 3.2] for solving variational inequality problems by incorporating momentum terms. A new step size strategy is also introduced, in which the step size sequence increases after a finite number of iterations. Under the assumptions that the underlying operator is quasimonotone and Lipschitz continuous, we establish weak convergence of the proposed method. The effectiveness and efficiency of the algorithm are demonstrated through numerical experiments and are compared with existing approaches from the literature. Finally, we apply the proposed algorithm to a signal recovery problem.