Relaxed lagrangian duality in convex infinite optimization : reducibility and strong duality
- Authors: Dinh, Nguyen , Goberna, Miguel , López-Cerdá, Marco , Volle, Michel
- Date: 2023
- Type: Text , Journal article
- Relation: Optimization Vol. 72, no. 1 (2023), p. 189-214
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- Description: We associate with each convex optimization problem, posed on some locally convex space, with infinitely many constraints indexed by the set T, and a given non-empty family (Formula presented.) of finite subsets of T, a suitable Lagrangian-Haar dual problem. We obtain necessary and sufficient conditions for (Formula presented.) -reducibility, that is, equivalence to some subproblem obtained by replacing the whole index set T by some element of (Formula presented.). Special attention is addressed to linear optimization, infinite and semi-infinite, and to convex problems with a countable family of constraints. Results on zero (Formula presented.) -duality gap and on (Formula presented.) -(stable) strong duality are provided. Examples are given along the paper to illustrate the meaning of the results. © 2022 Informa UK Limited, trading as Taylor & Francis Group.
A new tour on the subdifferential of the Supremum function
- Authors: Hantoute, Abderrahim , López-Cerdá, Marco
- Date: 2023
- Type: Text , Conference paper
- Relation: International Meeting on Functional Analysis and Continuous Optimization, IMFACO 2022, Elche, Spain, 16-17 June 2022, Functional Analysis and Continuous Optimization In Honour of Juan Carlos Ferrando's 65th Birthday, Elche, Spain, June 16–17, 2022 Vol. 424, p. 167-194
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- Description: This chapter is a survey presenting various characterizations of the subdifferential of the pointwise supremum of convex functions, as well as some featured applications. We gathered here the main outcomes we obtained in a series of recent papers, dealing with different models, assumptions and scenarios. Starting by the maximum generality framework, we move after to particular contexts in which some continuity and compactness assumptions are either imposed or inforced via processes of compactification of the index set and regularization of the data functions. Some relevant applications of the general results are presented, in particular to derive rules for the subdifferential of the sum, and for convexifying a general (unconstrained) optimization problem. The last section gives some specific constraint qualifications for the convex optimization problem with an arbitrary set of constraints, and also contains different sets of KKT-type optimality conditions appealing to the subdifferential of the supremum function. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
New tour on the subdifferential of supremum via finite sums and suprema
- Authors: Hantoute, Abderrahim , López-Cerdá, Marco
- Date: 2022
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 81-106
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- Description: This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family. © 2021, The Author(s).