The topology of theρ-hausdorff distance

Abstract

Abstract. In this paper we extend the theory of strong uniform continuity and strong uniform convergence, developed in the setting of metric spaces in [13,14], to the uniform space setting, where again the notion of shields plays a key role. Further, we display appropriate bornological/variational modifications of classical properties of Alexandroff [1] and of Bartle [7] for nets of continuous functions, that combined with pointwise convergence, yield continuity of the limit for functions between metric spaces.2000 AMS Classification: Primary 40A30, 54C35; Secondary 54E15, 54C08

Approximations and solution estimates in optimization

“…For ε, δ ≥ 0, the set of near-optimal near-feasible solutions of the problem min{f 0 (x) : f (x) ≤ 0, x ∈ X}, which of course is equivalent to 2 …”

“…This notion of convergence is the only natural choice for minimization problems as it guarantees the convergence of optimal solutions and optimal values of approximate problems to those of a limiting problem. Quantification of the distance between epi-graphs, which then leads to a quantification of epi-convergence, is placed on a firm footing in [4,2,5] with the development of the Attouch-Wets (aw) distance; see also [9,11,12,10]. We follow these lines and especially those of [6,7] that utilize such quantification as the basis for solution estimates in minimization problems.…”

Weak topologies for the closed subsets of a metrizable space

“…Only recently has a completely acceptable replacement (at least in the convex case) for the Hausdorff metric been investigated: the metrizable topology of uniform convergence of (d(-, A")) to d(-, A) on bounded subsets of X. Given x0 £ X, a local base for this topology [Be2,BDC,AP,ALW] In the setting of convex analysis, this topology reduces to the Hausdorff metric topology for closed and bounded convex sets [BL1], is stable with respect to duality [Be3,Pe], and is well suited for approximation and optimization. In view of its seminal study in [AW], we call this the Attouch-Wets topology xav/d, although it has been often called the bounded Hausdorff topology [AP, Pe].…”

Weak Topologies for the Closed Subsets of a Metrizable Space

“…Only recently has a completely acceptable replacement (at least in the convex case) for the Hausdorff metric been investigated: the metrizable topology of uniform convergence of (d(-, A")) to d(-, A) on bounded subsets of X. Given x0 £ X, a local base for this topology [Be2,BDC,AP,ALW] In the setting of convex analysis, this topology reduces to the Hausdorff metric topology for closed and bounded convex sets [BL1], is stable with respect to duality [Be3,Pe], and is well suited for approximation and optimization. In view of its seminal study in [AW], we call this the Attouch-Wets topology xav/d, although it has been often called the bounded Hausdorff topology [AP, Pe].…”