This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in Dinh et al. (Set Valued Var Anal, 2019). It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal-dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdifferential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to obtain conditions guaranteeing the existence of best approximate solutions to inconsistent convex inequality systems.