Strong Duality in Risk-Constrained Nonconvex Functional Programming

Abstract

We show that a wide class of risk-constrained nonconvex functional optimization problems exhibit strong duality, regardless of nonconvexity. We develop two novel results under distinct sets of assumptions, establishing strong duality over both decomposable policies (matching and extending prior work in the risk neutral case) and nondecomposable policies with structure (e.g., continuity or smoothness), including certain universal finite-dimensional (fixed depth/width) neural network parametrizations as special cases (improving established results in the risk-neutral setting as well). Our results hold for constrained models featuring arbitrary convex risk functionals on $L_p,p\in[1,\infty]$. We further discuss reductions and generalizations of our base model, and establish its necessity/minimality by presenting explicitly constructed counterexamples, rigorously refuting the manifestation of strong duality for certain, potentially simpler optimization models. Lastly, we discuss applications in wireless systems resource allocation, and supervised constrained learning. Our core proof technique appears to be new and relies on a non-trivial application of the Kingman-Robertson generalization of Lyapunov's convexity theorem for vector measures taking values in infinite-dimensional topological spaces.