Relaxed Lagrangian duality in convex infinite optimization: reducibility and strong duality

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 H of finite subsets of T, a suitable Lagrangian-Haar dual problem. We obtain necessary and sufficient conditions for H-reducibility, that is, equivalence to some subproblem obtained by replacing the whole index set T by some element of H. Special attention is addressed to linear optimization, infinite and semi-infinite, and to convex problems with a countable family of constraints. Results on zero H-duality gap and on H-(stable) strong duality are provided. Examples are given along the paper to illustrate the meaning of the results.This research was supported by the Vietnam National University HoChiMinh City (VNU-HCM) [grant number B2021-28-03], and by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI), and European Regional Development Fund (ERDF) (Project PGC2018-097960-B-C22)