Existence and uniqueness of a solution for a minimization problem with a generic increasing function

Abstract

In this paper we study the existence and uniqueness of a solution for minimization problems with generic increasing functions in an ordered Banach space X. The standard approaches are not suitable in such a setting. We propose a new type of perturbation adjusted for the problem under consideration, prove the existence and point out sufficient conditions providing the uniqueness of a solution. These results are proved by assuming that the space X enjoys the following property: each decreasing norm-bounded sequence has a limit. We supply a counterexample, which shows that this property is essential and give a modification of obtained results for the space C(T), which does not possess this property.1991 Mathematics subject classification (Amer. Math. Soc): primary 49J27, 90C30.

Almost all normal sets are strictly normal

“…When we say that a certain property holds for a generic element of a complete metric space Y we mean that the set of points which have this property contains a Gg everywhere dense subset of Y. Such an approach, when a certain property is investigated for the whole space Y and not just for a single point in Y, has already been successfully applied in many areas of Analysis [1,2,3,4,5,11,12,13]. The first generic result in monotonic analysis was obtained in [11] where we showed that a generic increasing function defined on an ordered Banach space has a point of minimum.…”

Existence of solutions of minimization problems with an increasing cost function and porosity

“…The study of a generic existence of solutions in optimization has recently been a rapidly growing area of research (see [1,2,3,4,5,6,8,9,10,12,13,14,15] and the references mentioned there). Instead of considering the existence of solutions for a single cost function, we study it for a space of all such cost functions equipped with an appropriate complete uniformity and show that a solution exists for most of these functions.…”

“…The property (A) is well known in the theory of ordered Banach spaces (see, e.g., [7,10,11]). Recall that the cone X + has the property (A) if the space X is reflexive.…”