Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems

Wenling Zhao,Hongxiang Zhang,Ruyu Wang

doi:10.3390/mca22020036

Abstract

We establish the notion of augmented weak sharpness of solution sets for the variational inequality problems which can be abbreviated to VIPs. This notion of augmented weak sharpness is an extension of the weak sharpness of the solution set of monotone variational inequality, and it overcomes the defect of the solution set not satisfying the weak sharpness in many cases. Under the condition of the solution set being augmented weak sharp, we present a necessary and sufficient condition for finite convergence for feasible solution sequence of VIP. The result is an extension of published results, and the augmented weak sharpness also provides weaker sufficient conditions for the finite convergence of many optimization algorithms.

Highlights

The variational inequality problem (VIP) is one of classical mathematical problems
Rockafellar [1], Polyak [2], and Ferris [3] successively put forward the weak sharp minima and strong non-degeneracy of the solution sets of mathematical programming problems, and prove that any one of them is the sufficient condition for finite convergence of the proximal point algorithm [1,4,5,6] and some important iterative algorithms [7,8,9,10,11,12]
This paper presents the notion of augmented weak sharpness of solution sets for VIP

Summary

Introduction

The variational inequality problem (VIP) is one of classical mathematical problems. Many models of problems in engineering and physics are constructed by partial differential equations with some suitable boundary conditions and primal conditions and are described by different kinds of variational inequality problems. Rockafellar [1], Polyak [2], and Ferris [3] successively put forward the weak sharp minima and strong non-degeneracy of the solution sets of mathematical programming problems, and prove that any one of them is the sufficient condition for finite convergence of the proximal point algorithm [1,4,5,6] and some important iterative algorithms [7,8,9,10,11,12]. In order to give finite convergence of a feasible solution sequence under more relaxed conditions, we establish the notion of augmented weak sharpness relative to a feasible solution sequence for a variational inequality problem. We give a necessary and sufficient condition for the finite convergence of a feasible solution sequence, under the feasible solution set of VIP is augmented weak sharp. We say that {xk} ⊂ Rn converges finitely to C if there exists k0 such that xk ∈ C for all k≥k0

The Augmented Weak Sharpness of Solution Sets

Finite Convergence

Conclusions