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Applied Mathematics & Optimization

, Volume 80, Issue 3, pp 643–664 | Cite as

Primal–Dual Optimization Conditions for the Robust Sum of Functions with Applications

  • N. Dinh
  • M. A. GobernaEmail author
  • M. Volle
Article

Abstract

This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in Dinh et al. (Set Valued Var Anal, 2019). It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal–dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdifferential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to obtain conditions guaranteeing the existence of best approximate solutions to inconsistent convex inequality systems.

Keywords

Robust sum function Duality Optimality conditions Existence of optimal solutions Inconsistent convex inequality systems best approximation 

Mathematics Subject Classification

90C46 49N15 65F20 

Notes

Acknowledgements

The authors wish to thank two anonymous referees for their valuable comments which helped to improve the manuscript. This research was supported by the National Foundation for Science and Technology Development (NAFOSTED), Vietnam, Project 101.01-2018.310 Some topics on systems with uncertainty and robust optimization, and by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Project PGC2018-097960-B-C22.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.International University, Vietnam National University-HCMCHo Chi Minh CityVietnam
  2. 2.Department of MathematicsUniversity of AlicanteSan Vicente del RaspeigSpain
  3. 3.Avignon UniversityAvignonFrance

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