  We consider separable nonconvex optimization problems under affine
constraints. For these problems, the Shapley-Folkman theorem provides an upper
bound on the duality gap as a function of the nonconvexity of the objective
functions, but does not provide a systematic way to construct primal solutions
satisfying that bound. In this work, we develop a two-stage approach to do so.
The first stage approximates the optimal dual value with a large set of primal
feasible solutions. In the second stage, this set is trimmed down to a primal
solution by computing (approximate) Caratheodory representations. The main
computational requirement of our method is tractability of the Fenchel
conjugates of the component functions and their (sub)gradients. When the
function domains are convex, the method recovers the classical duality gap
bounds obtained via Shapley-Folkman. When the function domains are nonconvex,
the method also recovers classical duality gap bounds from the literature,
based on a more general notion of nonconvexity.
