Let X be a Banach space and Z a nonempty closed subset of X. Let J :Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈Z{J (z)+ ‖x − z‖}, which is denoted by (x, J )-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈X for which ...

In this paper, we deal with a class of reflected backward stochastic differential equations associated to the subdifferential operator of a lower semi-continuous convex function driven by Teugels martingales associated with Lévy process. We obtain the existence and uniqueness of solutions to these equations by means of the penalization method. As its application, we give a probabilistic interpr...

To address the plurality of interpretations of the subjective notion of risk, we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity. Our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets. This representation covers most instruments related to ...

Lower semicontinuity properties of multiple integrals u ∈W (Ω;R) 7→ ∫ Ω f(x, u(x), · · · ,∇u(x)) dx are studied when f may grow linearly with respect to the highest-order derivative, ∇ku, and admissible W k,1(Ω;Rd) sequences converge strongly in W k−1,1(Ω;Rd). It is shown that under certain continuity assumptions on f, convexity, 1-quasiconvexity or k-polyconvexity of ξ 7→ f(x0, u(x0), · · · ,∇...

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the pr...

Let f : X ! IR f+1g be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdiierential @f is quasimonotone. As an immediate consequence, we get that f is convex if and only if @f is monotone.

Suppose that (X,A) is a measurable space and Y is a metrizable, Souslin space. Let Au denote the universal completion of A. For x ∈ X, let f(x, ·) be the lower semicontinuous hull of f(x, ·). If f : X × Y → R is (Au ⊗ B(Y ),B(R))-measurable, then f is (Au ⊗ B(Y ),B(R))-measurable.

It is known that the subdifferential of a lower semicontinuous convex function f over a Banach space X determines this function up to an additive constant in the sense that another function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c for some real constant c. Recently, Thibault and Zagrodny introduced a large cl...

We will present the problem of the Central Limit Theorem for random upper semicontinuous functions. Let us begin by providing the context for that problem. We need to present the space in which we will work, and specify the notions of random element of that space, convergence and the operations used to average random elements. Then we will make some historical remarks on limit theorems in this ...