New Representations of Epigraphs of Conjugate Mappings and Lagrange, Fenchel-Lagrange Duality for Vector Optimization Problems

In this paper we concern the vector problem of the model: \begin{align*} ({\rm VP})\quad\qquad &\rm{WInf} \{F(x): x\in C,\; G(x)\in -S\}. \end{align*} where $X, Y, Z$ are locally convex Hausdorff topological vector spaces, $F\colon X\rightarrow Y\cup\{+\infty_{Y}\}$ and $G\colon X\rightarrow Z\cup\{+\infty_{Z}\}$ are proper mappings, $C$ is a nonempty convex subset of $X$, and $S$ is a non-empty closed, convex cone in $Z$. Several new presentations of epigraphs of composite conjugate mappings associated to (VP) are established under variant qualifying conditions. The significance of these representations is twofold: Firstly, they play a key role in establish new kinds of vector Farkas lemmas which serve as tools in the study of vector optimization problems; secondly, they pay the way to define Lagrange dual problem and two new kinds of Fenchel-Lagrange dual problems for the vector problem (VP). Strong and stable strong duality results corresponding to these three mentioned dual problems of (VP) are established with the help of new Farkas-type results just obtained from the representations. It is shown that in the special case where $Y = \mathbb{R}$, the Lagrange and Fenchel-Lagrange dual problems for (VP), go back to Lagrange dual problem, and Fenchel-Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some setting, extend the corresponding ones for scalar problems in the literature.