## 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**

**Abstract:**

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**

**Abstract:**

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**

**Abstract:**

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 × **R** → **P**(Z,K) such that

*F(x)=\sup*_{x\ast} R(x^{a}st,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.

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As an illustration, we study the dual representation of non complete preference orders.