Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality

Abstract

This paper concerns the robust vector problem

$$ \mathrm{(RVP)}\enskip \text{WMin}\left\{ F(x) : x\in C, G_{u}(x)\in -S,\forall u\in\mathcal{U}\right\}, $$

where X,Y,Z are locally convex Hausdorff topological vector spaces, K is a closed and convex cone in Y with a nonempty interior, and S is a closed, convex cone in Z, \(\mathcal {U}\) is an 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 CX. Let \( A:=C\cap \left (\bigcap _{u\in \mathcal {U}}G_{u}^{-1}(-S)\right )\) and IA : XY be the indicator map defined by IA(x) = 0Y if xA and \(I_{A}(x) = + \infty _{Y}\) if xA. It is well known that the epigraph of the conjugate mapping (F + IA), in general, is not a convex set. We show that, however, it is “k-sectionally convex” in the sense that each section formed by the intersection of epi(F + IA) and any translation of a “specific k-direction-subspace” is a convex subset, for any k taking from intK. The key results of the paper are the representations of the epigraph of the conjugate mapping (F + IA) 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 then give 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.

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Notes

  1. 1.

    It is also called a proper cone.

  2. 2.

    Note that, in the first part of the proof of [11, Theorem 4.2], no assumptions on the convexity or closedness of the mappings F and G are needed.

  3. 3.

    This would be an elementary result in the study of robust optimization problems. However, to the surprise of the authors, we could not find it in the references we had in hand and so we insert a short proof here.

  4. 4.

    For the sake of simplicity, we write \((x^{*}_{\alpha },r_{\alpha })_{\alpha \in D}\) for \(((x^{*}_{\alpha }, r_{\alpha }))_{\alpha \in D}\).

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Funding

This work is supported by the project 101.01-2018.310, NAFOSTED, Vietnam.

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Correspondence to Nguyen Dinh.

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Dedicated to Professor Hoang Tuy’s 90th birthday

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Dinh, N., Long, D.H. Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality. Acta Math Vietnam 45, 525–553 (2020). https://doi.org/10.1007/s40306-019-00349-y

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Keywords

  • Robust vector optimization
  • Robust convex optimization
  • Robust convex strong duality
  • Robust stable vector Farkas lemma
  • Sectionally convex sets
  • Sectionally closed sets

Mathematics Subject Classification (2010)

  • 90C25
  • 49N15
  • 90C31