Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

Abstract

Artículo de publicación ISIThis paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in linear semi-infinite optimization under perturbations of all coefficients. With this aim in mind, the paper establishes as a key tool a basic result on finite-valued convex functions in the -dimensional Euclidean space. Specifically, this result provides an upper limit characterization of the boundary of the subdifferential of such a convex function. When applied to the supremum function associated with our constraint system, this characterization allows us to derive an upper estimate for the aimed calmness modulus in linear semi-infinite optimization under the uniqueness of nominal optimal solution.Fondecyt 111001

Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

“…The first case was already dealt in [1] for a convex function (without differentiability assumptions), and the new results provided in this paper constitute its (nonconvex) differentiable counterpart. The second type of outer limits is the one which constitutes a key ingredient in the estimations of the calmness modulus of feasible set mappings associated with right-hand-side perturbations of a nominal system (recall Theorem 2.1) under differentiability/convexity assumptions.…”

“…Remark 3.2 Theorem 3.1 in [1] shows that condition (ii) in the previous theorem also holds as equality in the case when function g is convex, without differentiability assumptions, in which case, ∂g represents the usual subdifferential of convex analysis. …”

“…In relation to this problem, the immediate antecedents are gathered in [1][2][3], where the assumption of the uniqueness of nominal optimal solution is essential. To this respect, we advance that an exact formula for the aimed modulus is obtained in Sect.…”

Recent contributions to linear semi-infinite optimization: an update

“…In [33], the main result (Theorem 3.1) gives the following characterization of the boundary of the subdi¤erential set of a convex function f :…”

Recent contributions to linear semi-infinite optimization

“…In [28], the main result (Theorem 3.1) gives the following characterization of the boundary of the subdi¤erential set of a convex function f :…”