Weak Sharp Solutions of Variational Inequalities in Hilbert Spaces

Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities

“…They characterized the weak sharp solutions for variational inequalities by minimum principle sufficiency property under the condition that the underlying mapping is pseudomonotone + and continuous on the compact polyhedral set. Wu and Wu [8] extended the results of Marcotte and Zhu [7] under a different assumption. They also introduced the concept of maximum principle sufficiency property for variational inequalities.…”

“…We consider the following condition which is the extension of a similar condition considered by Wu and Wu [8].…”

“…, d for all x ∈ K and all d ∈ X , then Condition A is same as condition (3) in [8]. Note that A(x) is nonempty for all x ∈ K * and since K * is assumed to be nonempty, B(y) is also nonempty for all y ∈ K .…”

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

“…Zhang et al [16] have also obtained relevant results by studying the dual gap function for a variational inequality. Wu and Wu [13] have further presented several equivalent conditions for the weak sharpness of C * in terms of an error bound of G in a Hilbert space. Hu and Song [11] have extended the results of weak sharpness for the solutions of VIP(C, F) under the conditions that F is pseudomonotone and continuous and G is Gâteaux differentiable and Lipschitz in a Banach space.…”

“…We note that several results of [8,11,13,15] have been obtained by utilizing the dual gap function G while g is seldom used. Although G is convex, it is more complicated to compute its values than that of g since for a fixed point x, G(x) is the maximum of a nonlinear program usually while g(x) is that of a linear program.…”