Convex Robust Sum Optimization Problems with Conic and Set Constraints: Duality and Optimality Conditions Revisited

This paper deals with optimization problems consisting in the minimization of a robust sum of infinitely many functions under a conic and a set constraints. Through suitable perturbation functions, the problem is embedded into a family of linearly perturbed problems which have an associated qualifying set which is contained in the Cartesian product of the dual space to the decision space by the real line. The strong duality is characterized in terms of the weak-star closed convexity of the intersection of the qualifying set with the vertical axis. Each strong duality theorem allows to characterize the linear perturbations of the objective function which are consequence of the conic and set constraints, and these Farkas-type lemmas provide the desired optimality conditions for the robust sum constrained problem. This scheme is developed for different perturbation functions providing Lagrange, Fenchel-Lagrange and other types of dual problems, and also for particular types of robust sums as the supremum function or the sequential robust sum.