On the Construction of Gap Functions for Variational Inequalities via Conjugate Duality

Abstract

In this paper, we deal with the construction of gap functions for variational inequalities by using an approach which bases on the conjugate duality. Under certain assumptions we also investigate a further class of gap functions for the variational inequality problem, the so-called dual gap functions.

Conjugate duality in vector optimization and some applications to the vector variational inequality

“…If t ∈ N(x), there exists μ > 0 such that (F (x) − t)μ = 0. Therefore it holds F (x) = t. Consequently, the gap function for the variational inequality (VI) becomes γ VI F (x) = F (x) T x + max y∈K (−F (x) T y) = max y∈K F (x) T (x − y), which coincides with Auslender's gap function (see [3,4]). …”

“…On the other hand, the duality results investigated in Section 3 allow us to introduce some new gap functions for (VVI). Let us mention that such a similar approach has been used for scalar variational inequalities in [3]. We remark that x ∈ K is a solution to the problem (VVI) if and only if 0 is a minimal point of the set {F (x) T (y − x) | y ∈ K}.…”

“…The approach we consider here extends to the vector case the results in [2,3] where different gap functions have been constructed via conjugate duality for variational inequalities and for equilibrium problems, respectively.…”

Merit functions: a bridge between optimization and equilibria

“…A further merit function for (VI) can be obtained by means of the complementarity reformulation (4), which in turn can be associated with the merit function (1). Further examples of merit functions associated with a complementarity problem can be found in [35].…”

“…is defined as sup When the feasible set C is explicitly defined by convex constraints as in (3), the value of the Auslender gap function p at a given point x coincides (see [1,53]) with the opposite of the optimal value of the Fenchel dual of the problem…”