Characterization and recovery of optimal distributions in Wasserstein expectation problems with non-convex quadratic functions via single SDP

Abstract We consider the minimization of the expectation of piecewise (not necessarily convex) quadratic function over Wasserstein balls. This expectation problem often appears as a key sub-problem of distributionally robust optimization problems. We present a computationally accessible semidefinite program (SDP)-based characterization for the optimal distribution of this problem, requiring only the optimal solution of a single SDP. We show that strong duality holds between the expectation problem and its SDP dual problem. We then provide a constructive characterization of the associated optimal distributions and prove that they can be explicitly recovered from an optimal solution of the dual of the dual SDP. This result enables the direct computation of the optimal distributions of the expectation problem. Furthermore, we demonstrate through a numerical study on a distributionally robust mean-risk portfolio optimization problem using simulated data that the worst-case distributions can be computed efficiently and utilized to obtain probabilistic interpretation of worst-case solutions.