Strong Duality in Risk-Constrained Nonconvex Functional Programming

Abstract: We show that risk-constrained functional optimization problems with general integrable nonconvex instantaneous reward/constraint functions exhibit strong duality, regardless of nonconvexity. We consider risk constraints featuring convex and positively homogeneous risk measures admitting dual representations with bounded risk envelopes, generalizing expectations. Popular risk measures supported within our setting include the conditional value-at-risk (CVaR), the mean-absolute deviation (MAD, including the non-monotone case), certain distributionally robust representations and more generally all real-valued coherent risk measures on the space $L_1$. We highlight the usefulness of our results by further discussing various generalizations of our base model, extensions for risk measures supported on $L_{p>1}$, implications in the context of mean-risk tradeoff models, as well as more specific applications in wireless systems resource allocation, and supervised constrained learning. Our core proof technique appears to be new and relies on risk conjugate duality in tandem with J. J. Uhl's weak extension of A. A. Lyapunov's convexity theorem for vector measures taking values in general infinite-dimensional Banach spaces.