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.
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17 December 2019
The authors would like to correct the errors caused by a wrong equation (A=B+z), at the end of the proof of Lemma 2.1 in the original article, which affects this lemma, two subsequent examples, as well as other statements spread along the paper.
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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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Dinh, N., Goberna, M.A. & Volle, M. Primal–Dual Optimization Conditions for the Robust Sum of Functions with Applications. Appl Math Optim 80, 643–664 (2019). https://doi.org/10.1007/s00245-019-09596-9
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DOI: https://doi.org/10.1007/s00245-019-09596-9
Keywords
- Robust sum function
- Duality
- Optimality conditions
- Existence of optimal solutions
- Inconsistent convex inequality systems best approximation