Characterization of weakly sharp solutions of a variational inequality by its primal gap function

Abstract

Our aim is to study weakly sharp solutions of a variational inequality in terms of its primal gap function g. We discuss sufficient conditions for the Lipschitz continuity and subdifferentiability of the primal gap function. Several sufficient conditions for the relevant mapping to be constant on the solutions have also been obtained. Based on these, we characterize the weak sharpness of the solutions of a variational inequality by g. Some finite convergence results of algorithms for solving variational inequality problems are also included.

Weak sharpness and finite convergence for mixed variational inequalities

“…Noticing that the pseudomonotonicity + of F implies that F is constant on X * (see [24,Proposition 2] We can also relax the continuity of the function F improving [23,Theorem 3.2].…”

“…Marcotte and Zhu [25] characterized weak sharpness of the solution set of a variational inequality in term of its dual gap function and studied finite convergence of sequences generated by some algorithms for solving variational inequalities under weak sharpness of the solution sets. Later, weak sharpness of solutions and its applications to the finite convergence property of methods for finding solutions of varitional inequalities have been investigated by many authors (see, e.g., [3,17,23,24,26,27,32,33,34] and references therein). Some authors extended and established the concept of weak sharp solutions to general variational inequalities, e.g., set-valued variational inequalities [2,31], variational-type inequalities [19], nonsmooth variational inequalities [4] and mixed variational inequalities [16].…”

Finite convergence analysis and weak sharp solutions for variational inequalities

“…Recently, Liu and Wu [5] gave an error bound in term of the primal gap function for the VIP defined by…”

“…We give some characterizations of weak sharpness of the solution set of VIP without using the dual gap or the primal gap function. We note that in [6] (respectively, in [5]), the authors gave an error bound in term of the dual gap function G (respectively, the primal gap function g). They therefore needed some more assumptions.…”

On weak sharp solutions in $(\rho , \mathbf{b}, \mathbf{d})$-variational inequalities

“…Based on the works of Burke and Ferris [3], Patriksson [11] and following Marcotte and Zhu [10], the concept of weak sharp solution associated with variational-type inequalities has attracted the attention of many researchers (see, for instance, Hu and Song [7], Liu and Wu [9], Zhu [17] and Jayswal and Singh [8]). Recently, by using gap-type functions, in accordance with Ferris and Mangasarian [5] and following Hiriart-Urruty and Lemaréchal [6], Alshahrani et al [1] studied the minimum and maximum principle sufficiency properties associated with nonsmooth variational inequalities.…”