A concept of inner prederivative for set-valued mappings and its applications

Abstract

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

An extension of the Rådström cancellation theorem to cornets

“…Let A, B, C ∈ P W (S) such that B is covered by the sum of a topologically bounded subset of S and W and C is a topologically closed m-convex subset of S for some m ≥ 2. Assume that A + B ⊆ C + B (20) holds. Then A ⊆ C.…”

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,[44][45][46]48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52][53]…”

An extension of the Rådström Cancellation Theorem to Cornets

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,44,45,47,48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52]…”

“…Let A, B, C ∈ P W (S) such that B is covered by the sum of a topologically bounded subset of S and W and C is a topologically closed m-convex subset of S for some m ≥ 2. Assume that(20) A + B ⊆ C + B holds. Then A ⊆ C.…”