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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 familyof finite subsets of  T , a suitable Lagrangian-Haar dual problem. We obtain necessary and sufficient conditions for-reducibility, that is, equivalence to some subproblem obtained by replacing the whole index set  T  by some element of. Special attention is addressed to linear optimization, infinite and semi-infinite, and to convex problems with a countable family of constraints. Results on zero-duality gap and on-(stable) strong duality are provided. Examples are given along the paper to illustrate the meaning of the results.

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