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Convex Robust Sum Optimization Problems with Conic and Set Constraints: Duality and Optimality Conditions Revisited

Abstract

This paper deals with optimization problems consisting in the minimization of a robust sum of infinitely many functions under a conic and a set constraints. Through suitable perturbation functions, the problem is embedded into a family of linearly perturbed problems which have an associated qualifying set which is contained in the Cartesian product of the dual space to the decision space by the real line. The strong duality is characterized in terms of the weak-star closed convexity of the intersection of the qualifying set with the vertical axis. Each strong duality theorem allows to characterize the linear perturbations of the objective function which are consequence of the conic and set constraints, and these Farkas-type lemmas provide the desired optimality conditions for the robust sum constrained problem. This scheme is developed for different perturbation functions providing Lagrange, Fenchel-Lagrange and other types of dual problems, and also for particular types of robust sums as the supremum function or the sequential robust sum.

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Acknowledgements

The authors are grateful to the anonymous referees for reading carefully our manuscript and for their valuable comments and detailed suggestions which helped us to improve considerably the quality of the paper. The authors are also grateful to Professor Miguel A. Goberna for private communication and for his help during the process of realizing this paper. The work of the first author is funded by the Vietnam National University HoChiMinh city (VNU-HCM) under the grant number B2021-28-03.

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Correspondence to Nguyen Dinh.

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To Professor Miguel A. Goberna on the occasion of his 70th anniversary

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Dinh, N., Long, D.H. & Volle, M. Convex Robust Sum Optimization Problems with Conic and Set Constraints: Duality and Optimality Conditions Revisited. Set-Valued Var. Anal (2022). https://doi.org/10.1007/s11228-022-00646-z

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  • DOI: https://doi.org/10.1007/s11228-022-00646-z

Keywords

  • Robust sums of families of functions
  • Robust sum optimization problems
  • Stable Farkas lemmas
  • Stable strong duality
  • Sup-functions

Mathematics Subject Classification (2010)

  • 90C46
  • 49N15
  • 65F20