Piecewise sum-of-squares convexity and Wasserstein distributionally robust optimization: Exact semi-definite programming reformulations with data-driven decision-making under uncertainty

This paper develops a numerically tractable framework for distributionally robust optimization (DRO) by extending the recently introduced concept of piecewise sum-of-squares (SOS)-convexity. We show that piecewise SOS-convex functions admit SOS certificates for non-negativity over broad classes of convex sets, thereby extending the numerical tractability of SOS-based convex optimization beyond polynomial settings. We apply this framework to infinite-dimensional Wasserstein worst-case expectation problems with piecewise SOS-convex loss functions, and prove that they admit exact finite-dimensional dual reformulations with semidefinite programming (SDP) representations. The approach accommodates general support sets defined by SOS-convex polynomial inequalities, including balls, ellipsoids, and polyhedral sets, and yields exact SDP reformulations for a range of DRO models arising in applications such as revenue estimation, portfolio optimization, and insurance risk assessment. Furthermore, we demonstrate that for certain important classes of problems, these SDP formulations can be further strengthened into specialized SDPs. In particular, we derive second-order cone programming representations for models with piecewise affine losses over ball uncertainty sets, and specialized SDP formulations in terms of problem data for piecewise quadratic losses over ellipsoidal supports. Finally, we illustrate the computational tractability of the proposed approach through a numerical experiment on a revenue estimation problem with higher-degree SOS-convex utility functions, and demonstrate the versatility of the scheme through a numerical example on insurance risk assessments, and a distributionally robust mean-CVaR portfolio optimization case study.