New glimpses on convex infinite optimization duality
- Authors: Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2015
- Type: Text , Journal article
- Relation: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas Vol. 109, no. 2 (2015), p. 431-450
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- Description: Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that (P)=max(D), (P)=max(Q), and (P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing (P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described. © 2014, Springer-Verlag Italia.
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.