A Relaxed-Projection Splitting Algorithm for Variational Inequalities in Hilbert Spaces

TL;DR

This paper presents a new relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces that simplifies computations by replacing projections with hyperplane projections and uses subgradient steps.

Contribution

The paper introduces a novel algorithm that replaces orthogonal projections with hyperplane projections and employs simple subgradient steps for variational inequalities.

Findings

Proven weak convergence of the algorithm under monotonicity assumptions.

Algorithm handles nonsmooth maximal monotone operators and nonlinear convex constraints.

Simplifies computations by avoiding complex subproblem solutions.

Abstract

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the structure of the problem. Assuming monotonicity of the individual operators and the existence of solutions, we prove that the generated sequence converges weakly to a solution.