Stable strong duality for cone-constrained set-valued optimization problems: A perturbation approach

Abstract

In this paper we consider  a general  set-valued otimimization problem of the model

Winf  F(x),  s.t. x in X

where $F\colon X\rightrightarrows Y\cup\{+\infty_{Y}\}$  is a proper mapping. The problem is then embed  into a parametric problem and can be express as 

 

(P)  $\mathop{\winf}\bigcup\limits_{x\in X}\Phi(x,0_Z)$

 

where $\Phi\colon X\times Z\rightrightarrows      Y^\bullet$ is   a proper set-valued {\it perturbation mapping} such that $ \Phi(x,0_Z) =  F(x) $.  A representation of the epigraph of the conjugate mapping $\Phi^\ast$ is established (Theorem 1)  and it is used as the basic tool for establishing a general stable strong duality for the problem (P) (Theorem 3).  As  applications, the mentioned general strong duality result is then applied to a cone-constrained set-valued optimization problem (CSP) to derive  three dual problems for (CSP):  The Lagrange dual problem and two forms of Fenchel-Lagrange dual problems for (CSP). Consequently, three stable strong duality results for the three primal-dual pairs of problems are derived (Theorems 4 and 5),  among them, one is  entirely  new while the others extend some known ones in the literature.