Invited Session Thu.3.H 2035

Thursday, 15:15 - 16:45 h, Room: H 2035

Cluster 24: Variational analysis [...]

Set-valued convex and quasiconvex duality


Chair: Andreas H. Hamel



Thursday, 15:15 - 15:40 h, Room: H 2035, Talk 1

Carola Schrage
Dini derivatives for vector- and set-valued functions

Coauthors: Giovanni Crespi, Andreas Hamel


We will introduce set-valued derivatives of Dini type for vector- and set-valued functions and provide basic calculus rules for these derivatives.
Using a solution concept for multicriteria optimality problems introduced by Heyde and Löhne in 2008, we will provide a variational principle of Minty type supplying necessary and sufficient optimality conditions and state a Fermat rule, a necessary condition under which a subset of the pre-image space is a solution to the given optimality problem.



Thursday, 15:45 - 16:10 h, Room: H 2035, Talk 2

Andreas H. Hamel
Lagrange duality in set optimization

Coauthor: Andreas Löhne


A Lagrange type duality theorem for set-valued optimization problems is presented. New features include set-valued Lagrangians and saddle set (rather than point) theorems based on infima and suprema in appropriate spaces of sets. An application to multivariate utility maximization is given.



Thursday, 16:15 - 16:40 h, Room: H 2035, Talk 3

Samuel Drapeau
Complete duality for convex and quasiconvex set-valued functions

Coauthors: Andreas Hamel, Michael Kupper


The Fenchel-Moreau theorem is a central result stating a one to one relation between l.s.c. convex functions and their conjugate. For quasiconvex functions,
a dual representation has been achieved by Penot, Volle. The complete duality, that is, the unique characterisation of the dual function, has been done by Cerreia-Voglio et al. on M-spaces and Drapeau, Kupper on lctvs. However, for vector valued convex functions problems appear for the existence of a Fenchel-Moreau theorem and the uniqueness is still open.
In this talk, we present a complete duality for quasiconvex and convex
set valued functions. More precisely, given a l.s.c. quasiconvex
function F: X → P(Z,K) where P(Z,K) is the
set of monotone subsets of a vector space Z with respect to a convex
cone K, then, there exists a function R: X\ast × RP(Z,K) such that
F(x)=\supx\ast R(xast,x\ast(x)).
Furthermore, R is uniquely determined if K\ (-K) ≠ ∅. For convex functions we provide a set-valued pendant to the Fenchel Moreau theorem.
As an illustration, we study the dual representation of non complete preference orders.


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