New glimpses on convex infinite optimization duality

Abstract

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 $(\Delta )$, 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 $\inf (P)=\max (D)$, $\inf (P)=\max (Q),$ and $\inf (P)=\max (\Delta )$ (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing $\min (P)=\sup (Q)$ (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.