Abstract
This paper focuses on formulas for the ε-subdifferential of the optimal value function of scalar and vector convex optimization problems. These formulas can be applied when the set of solutions of the problem is empty. In the scalar case, both unconstrained problems and problems with an inclusion constraint are considered. For the last ones, limiting results are derived, in such a way that no qualification conditions are required. The main mathematical tool is a limiting calculus rule for the ε-subdifferential of the sum of convex and lower semicontinuous functions defined on a (non necessarily reflexive) Banach space. In the vector case, unconstrained problems are studied and exact formulas are derived by linear scalarizations. These results are based on a concept of infimal set, the notion of cone proper set and an ε-subdifferential for convex vector functions due to Taa.
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The careful readings and insightful comments of the two anonymous referees are gratefully acknowledged.
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Dedicated to Professor R. Tyrrell Rockafellar on the occasion of his 85th birthday.
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This research was partially supported by the Mathematics Research Institute of the University of Valladolid (IMUVA). In addition, for the first author, it was partially supported by the Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology and Thai Nguyen University of Sciences (Vietnam). For the second author, it was partially supported by the Ministerio de Ciencia e Innovación (MCI), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PID2020-112491GB-I00 (MCI/AEI/FEDER, UE).
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An, D.T.V., Gutiérrez, C. Differential stability properties in convex scalar and vector optimization. Set-Valued Var. Anal 29, 893–914 (2021). https://doi.org/10.1007/s11228-021-00601-4
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DOI: https://doi.org/10.1007/s11228-021-00601-4
Keywords
- Differential stability
- ε-subdifferential
- Parametric convex programming
- Limiting calculus rule
- Optimal value function
- Approximate solution
- Vector optimization
- Infimal set
- Cone proper set
- Weak minimal solution