Epi/Hypo-Convergence of Bifunctions on General Domains and Approximations of Quasi-Variational Problems

Set-Convergence and Its Application: A Tutorial

“…Set-convergence is also central to the analysis of saddle point and minsup problems. For bivariate functions, the simultaneous epi-convergence in one variable and hypo-convergence in the other is referred to as epi/hypo-convergence and is a fundamental concept in the study of convergence of saddle points; see the pioneering work in [6] and the recent paper [15]. An adjustment to epi/hypo-convergence under the name lopsided convergence places a firm emphasis on the first variable over the second one and thus becomes a key tool in analyzing approximations of minsup problems [5,31].…”

“…smoothing of complementarity problems). For a smooth15 mapping F :R n → R n , the complementarity problem x ≥ 0, F(x) ≥ 0, F (x), x = 0 corresponds to the generalized equation 0 ∈ F (x) + N C (x), with C = R n+ . This, in turn, is equivalent to the normal map condition…”

Variational approximations of a dual pair of mathematical programming problems

“…First, consider A, A k ⊂ X, φ k : A k → R, and φ : A → R. Definition 2.2. (epi-convergence [7], inside epi-convergence [6]) {φ k } k is called epi-convergent to φ, denoted by φ k e → φ or φ = e-lim k φ k if the following conditions are satisfied (a) for all…”

“…In [7], the above Definition 2.2 for epi-convergence together with a modification for finite-valued bifunctions Φ : A × B → R of the concept of misup-lop-convergence defined in [2] (for Φ : X × Y → R ∪ {+∞} ∪ {−∞}) in order to apply more effectively to practical problems involving finite-valued bifunctions. The weaker notion of inside epi-convergence was recently proposed in [6] with effective applications in approximations. A notion symmetric to epiconvergence (and strongly concerned when maximization is considered) is:…”

“…Definition 2.4. (epi/hypo convergence [5], inside epi/hypo-convergence [6]) Bifunctions Φ k , are called epi/hypo convergent (e/h-convergent) to a bifunction Φ if (a) for all x k j ∈ A k j → x ∈ A and y ∈ B, there exist y k j ∈ B k j → y such that liminf j Φ k j (x k j , y k j ) ≥ Φ(x, y), and for all…”