  This paper associates a dual problem to the minimization of an arbitrary
linear perturbation of the robust sum function introduced in DOI
10.1007/s11228-019-00515-2. 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 subdiffential 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.
