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Complete Characterizations of Robust Strong Duality for Robust Vector Optimization Problems

Abstract

Consider the robust vector optimization problem of the model

$$\text{(RVP)}\qquad \text{WMin}\{F(x):~ x\in C,~ G_{u}(x)\in -S~ \forall u\in \mathcal{U}\} , $$

where X, Y, and Z are locally convex Hausdorff topological vector spaces, S is a nonempty convex cone in Z, CX is a nonempty subset, \(\mathcal {U}\) is an uncertainty set, and F : XY ∪{ + Y }, G u : XZ ∪{ + Z } for all \(u\in \mathcal {U}\). The Lagrangian robust dual problem and the weakly Lagrangian robust dual problem for (RVP) are proposed and then principles of robust strong duality for (RVP) associated to these two dual problems in a general setting (not necessarily convex) are established. The results are obtained based mainly on the robust vector Farkas-type results and the representations of epigraphs of conjugate mappings of vector-valued functions, which are the key tools of this paper. Next, we give characterizations of robust strong duality for (RVP) in the convex setting and also for vector optimization problems (i.e., when the uncertainty set is a singleton). To illustrate the ability of applications of the results just obtained, we show that when specifying our general results for (RVP) to robust scalar problems (i.e., when \(Y = \mathbb {R}\)), we get new principles of robust strong duality for general non-convex robust problems, and various characterizations of robust strong duality for convex problems which cover/extend known results in the literature. In addition, we consider a (scalar) robust infinite problem. It turns out that a further application of the general results obtained for scalar robust problems to this specific problem also leads to various forms of robust strong duality results for the problem in consideration, which are new or cover/extend the known ones published in the recent years.

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Notes

  1. The notion of positively K-lsc was used in [8] and in [27] as “star K-lower semicontinuous”.

  2. Several other types of orders can be defined on \(\mathcal {P}_{0}(Y^{\bullet })\) (see, e.g., [29], [28, p. 317]).

  3. By a “countable set” we mean a set that is finite or countably infinite.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments that significantly improved the quality of the paper.

Funding

This research was supported by the National Foundation for Science & Technology Development (NAFOSTED) of Vietnam, Project 101.01-2015.27, Generalizations of Farkas lemma with applications to optimization.

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Correspondence to Nguyen Dinh.

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Dedicated to the 70th birthday of Professor Michel Théra.

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Dinh, N., Long, D.H. Complete Characterizations of Robust Strong Duality for Robust Vector Optimization Problems. Vietnam J. Math. 46, 293–328 (2018). https://doi.org/10.1007/s10013-018-0283-1

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  • DOI: https://doi.org/10.1007/s10013-018-0283-1

Keywords

  • Robust vector optimization
  • Robust convex optimization
  • Vector optimization
  • Robust strong duality
  • Vector Farkas lemma

Mathematics Subject Classification (2010)

  • 90C25
  • 49N15
  • 90C31