Gap Functions and Descent Methods for Minty Variational Inequality

Auxiliary problem principles for equilibria

“…Therefore, ψ α is a gap function for (EP) whenever S(f ) = M α (f ) (see Theorems 4.1 and 4.2). Moreover, unlike the gap function ϕ α of the previous section, it is always convex, provided that f α (x, ·) is convex and this property makes it attractive: in fact, it has been exploited in algorithmic frameworks for variational inequalities [6,21,23,26] and for more general EPs [24,27].…”

“…On the contrary, the class of problems (MEP α ) has been explicitly considered only for variational inequalities with α < 0 in [6,26], indirectly through gap functions with α > 0 in [27,28] and very recently in [29] to refine some existence results for EPs.…”

“…Case (b) with τ = 0, μ > 0 and α = 0 or α = μ was considered in [27]. Considering just variational inequalities, case (a) with μ = 0 and α ≤ 0 has been proved in [6,26], while case (b) with μ > 0 and α ≥ 0 in [28,Lemma 2.3]. To the best of our knowledge, up to now, cases (c) and (d) have not been considered in any framework.…”

A Recurrent Neural Network for Solving a Class of General Variational Inequalities

“…where D = N − M and r = q − p. All the stability results for the NN in (8) presented in Theorems 1-5 can be applied to the NNs in (20)- (22). For the latter two NNs, one point should be noted is that because ∇F (x)…”

“…Example 5: Consider a GLCP in (19) where F (x) = Mx + p by using the NN in (22). Let The problem has a unique solution x * = (−0.700, −0.600, 7.050, 5.000, 1.350)…”