Exact Penalization and Optimality Conditions for Approximate Directional Minima

Abstract

In this paper, we study the concept of approximate directional efficiency for set-valued constrained and unconstrained optimization problems. In our work, we concerned with finding conditions under which the Clarke penalization technique can be applied, and we derive some optimality conditions via variational analysis tools such as limiting normal cones and its corresponding normal coderivative.

Cone enlargements and applications to vector optimization

“…Notice that no convexity is involved in these definitions since C can be a nonconvex cone. For instance, C (1)ε was used in [1] under the name of "conic ε−neighborhood" in the context of some continuity properties of cone-values multifunctions, while C (2)ε was introduced in [4] in order to deal with directional efficiency in vector optimization.…”

“…Certain types of conic enlargements are well known to be very useful in a wide range of applications from stability issues in variational analysis (see [1,2] and the references therein) to results concerning the density of proper solutions into the set of strong solutions in vector optimization (Arrow-Barankin-Blackwell type theorems; see, e.g., [3]). More recently, some conic enlargement were introduced in order to deal with some concepts of directional solutions (see [4]) and a study of several possible enlargements was done in [5] within the scope of presenting their applications to cone separation results and to vector optimization problems with variable ordering structure in the setting designed in [6].…”

Approximate efficiency in set-valued optimization with variable order

“…(ii) The second type enlargement was introduced and studied in [6] in relation with some directional vector optimization problems. It is known ([6, Proposition 2.9]) that C \ {0} ⊂ intC (2)ε , for all ε > 0.…”