Prederivatives of convex set-valued maps and applications to set optimization problems

Abstract

International audienceWe investigate the existence of several kinds of prederivatives for set-valued mappings enjoying convex properties. Then, we established both necessary and sufficient optimality conditions, involving such prederivatives, for set optimization problems

Prederivatives of gamma paraconvex set-valued maps and Pareto optimality conditions for set optimization problems

“…Theorem 3.1 extends [ 10 , Theorem 3.3] from convex set-valued mappings to γ -paraconvex set-valued mappings. Recall [ 16 ] that G is said to be open at if is a neighborhood of ȳ for every neighborhood U of x̄ .…”

“…In contrast with [ 10 , Theorem 3.8], Theorem 3.2 has some improvements. Firstly, we extend Y from finite dimensional spaces to general Banach spaces.…”

“…In [ 10 , Theorem 3.8], Gaydu, Geoffroy and Marcelin proved the following result. Let Y be a finite dimensional Banach space, be a nonempty closed convex cone, be a C -convex set-valued mapping, .…”

“…In [ 10 ], Gaydu, Geoffroy and Marcelin established necessary condition and sufficient conditions for the weak minimizers and the strong minimizers of the optimization problem (SP). It is well known that each strong minimizer is a Pareto minimizer and each Pareto minimizer is a weak minimizer, but the converses are not true.…”

“…In the later publication of Pang [ 7 , 8 ], and Gaydu, Geoffroy and Jean-Alexis [ 9 ], some notions of prederivatives were posed and further studied. In 2016, Gaydu, Geoffroy and Marcelin [ 10 ] studied the existence of some kinds of prederivatives of convex set-valued mappings and established necessary and sufficient optimality conditions for the weak minimizers and the strong minimizers of set optimization problems. γ -paraconvex set-valued mappings are an extension of convex set-valued mappings, and were studied by some researchers [ 11 , 12 ].…”