Primal-Dual Proximal Splitting for Wasserstein Distributionally Robust Convex Semidefinite Optimization via SDP-Based Spectrahedral Projection

Abstract We present a primal-dual proximal splitting framework for solving a broad class of convex semidefinite optimization problems with distributionally robust constraints involving piecewise affine functions, spectrahedral support sets, and Wasserstein ambiguity, which arise in a broad range of applications. We develop a semidefinite programming (SDP)-based formula for computing the projection onto a spectrahedron enabling the integration of SDP techniques within a primal-dual splitting framework for solving these problems via finite-dimensional convex semidefinite optimization reformulations. By reformulating the distributionally robust constraints into a system of linear matrix inequalities, we express the original distributionally robust problem as a finite-dimensional convex problem which results in convex composite unconstrained optimization problems suitable for primal-dual projection algorithms. Finally, we demonstrate the computational tractability of the proposed framework through numerical experiments on two representative models: mean-risk portfolio optimization with transaction costs in finance, and binary classification in machine learning. The results reveal that the primal-dual algorithm effectively solves the models of moderate scale.