Duality for the Robust Sum of Functions

In this paper we associate with an infinite family of real extended functions defined on a locally convex space a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems formed by the unconstrained minimization of its robust sum and the so-called optimistic dual. For such a dual pair, we characterize weak duality, zero duality gap, and strong duality, and their corresponding stable versions, in terms of multifunctions associated with the given family of functions and a given approximation parameter e ≥ 0 which is related to the e-subdifferential of the robust sum of the family. We also consider the particular case when all functions of the family are convex, assumption allowing to characterize the duality properties in terms of closedness conditions.