# Finite Convergence Analysis and Weak Sharp Solutions for Variational Inequalities

## TL;DR

This paper investigates the weak sharpness of solution sets in variational inequalities and establishes finite convergence properties for several iterative algorithms, providing theoretical insights and convergence estimates.

## Contribution

It offers new characterizations of weak sharpness for VIP solutions without primal or dual gap functions and proves finite convergence for multiple algorithms.

## Key findings

- Characterizations of weak sharpness without gap functions
- Finite convergence of proximal point, exact proximal point, and gradient projection methods
- Estimate on the number of iterations for convergence

## Abstract

In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of the sequence generated by some algorithm for finding the solutions of VIP. In particular, we give some characterizations of weak sharpness of the solution set of VIP without considering the primal or dual gap function. We establish an abstract result on the finite convergence property for a sequence generated by some iterative methods. We then apply such abstract result to discuss the finite termination property of the sequence generated by proximal point method, exact proximal point method and gradient projection method. We also give an estimate on the number of iterates by which the sequence converges to a solution of the VIP.