Sectional convexity of epigraphs of conjugate mappings with applications to robust vector duality

TL;DR

This paper investigates the sectional convexity properties of epigraphs of conjugate mappings in robust vector optimization, providing new duality results and extending existing scalar problem theories.

Contribution

It introduces the concept of $k$-sectional convexity of epigraphs of conjugate mappings and develops representations that lead to robust duality theorems.

Findings

Epigraphs of conjugate mappings are $k$-sectionally convex.

Representation of epigraphs via $k$-sectionally convex hulls.

New robust duality and Farkas lemmas for vector problems.

Abstract

This paper concerns the robust vector problems \begin{equation*} \mathrm{(RVP)}\ \ {\rm Wmin}\left\{ F(x): x\in C,\; G_u(x)\in -S,\;\forall u\in\mathcal{U}\right\}, \end{equation*} where $X, Y, Z$ are locally convex Hausdorff topological vector spaces, $K$ is a closed and convex cone in $Y$ with nonempty interior, and $S$ is a closed, convex cone in $Z$, $\mathcal{U}$ is an \textit{uncertainty set}, $F\colon X\rightarrow {Y}^\bullet,$ $G_u\colon X\rightarrow Z^\bullet$ are proper mappings for all $ u \in \mathcal{U}$, and $\emptyset \ne C\subset X$. Let $ A:=C\cap \left(\bigcap_{u\in\mathcal{U}}G_u^{-1}(-S)\right)$ and $I_A : X \to Y^\bullet $ be the indicator map defined by $I_A(x) = 0_Y $ if $x \in A$ and $I_A(x) = + \infty_Y$ if $ x \not\in A$. It is well-known that the epigraph of the conjugate mapping $(F+I_A)^\ast$, in general, is not a convex set. We show that, however, it is "$k$-sectionally convex" in the sense that each section form by the intersection of epi$(F+I_A)^\ast$ and any translation of a "specific $k$-direction-subspace" is a convex subset, for any $k$ taking from int$\,K$. The key results of the paper are the representations of the epigraph of the conjugate mapping $(F+I_A)^\ast$ via the closure of the $k$-sectionally convex hull of a union of epigraphs of conjugate mappings of mappings from a family involving the data of the problem (RVP). The results are then given rise to stable robust vector/convex vector Farkas lemmas which, in turn, are used to establish new results on robust strong stable duality results for (RVP). It is shown at the end of the paper that, when specifying the result to some concrete classes of scalar robust problems (i.e., when $Y = \mathbb{R}$), our results cover and extend several corresponding known ones in the literature.