  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.
