Subdifferentials of perturbed distance functions in Banach spaces

Abstract

Subdifferential, Fréchet subdifferential, Proximal subdifferential, Perturbed optimization problem, Well-posedness, 49J52, 46N10, 49K27,

A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

“…For instance, if f = δ {0 X } , then one gets the classical minimal time function (see [13][14][15]) T C : R n → R, T C (x) := inf{t ≥ 0 : (x − tC) ∩ = ∅}, that, when = {0 X } collapses to the gauge function. In [12] one finds two perturbations of the classical minimal time function, γ C f (introduced in [31] and motivated by a construction specific to differential inclusions) and γ C ( f + δ ), that contains as a special case the perturbed distance function introduced in [24]. The latter function has motivated us to introduce T C , f , where the function f and the set do not share the same variable anymore and can thus be split in the dual representations.…”

“…[13,19]), nonsmooth analysis (cf. [5,12,[14][15][16]19,24,31]), control theory and Hamilton-Jacobi partial differential equation (mentioned in [5]), best approximation problems (cf. [24]) and differential inclusions (cf.…”

“…In order to introduce the general nonlinear minmax location problems we propose a new perturbed minimal time function that generalizes the classical minimal time function, introduced over 4 decades ago and recently reconsidered by Mordukhovich and Nam in a series of papers (see, for instance, [14][15][16]19]) and the book [13], and several of its recent extensions (cf. [12,16,19,24,31]). The motivation to investigate such problems comes from both theoretical and practical reasons, as location type problems arise in various areas of research and real life, such as geometry, physics, economics or health management, applications from these fields being mentioned in our paper as possible interpretations of our results.…”

Subdifferential formulas for a class of non-convex infimal convolutions

“…Using Corollary 2.4 and the fact that the norm function is coercive with constant m = 1, it is easy to obtain the related results from [8,13], as well as the results from [14,Corollary 3.1] and [14,Corollary 3.2] without assuming the convexity of the set S therein. Note that it is not possible to apply [14,Theorem 3.1] to derive these results since the function f (x) := J(x) + δ(x; Ω) never satisfies a center-Lipschitz condition atx ∈ S 0 if Ω is a proper subset of X.…”