Minimizing worst-case polynomial expectation over Wasserstein balls: convergent SDP hierarchy, exact SDP relaxation and optimal distribution extraction

This paper investigates a novel worst-case polynomial expectation problem, designed to minimize the maximum expected values over Wasserstein balls. We present a convergent hierarchy of semi-definite programming (SDP) relaxations for the minimax expectation problem, and the extraction of optimal distribution via SDP, under suitable conditions. This is done by first establishing a finite reduction theorem for the minimax expectation problem. Furthermore, we demonstrate that for non-convex quadratic functions with ball supports, the optimal distribution can be constructed by solving a single SDP. We also identify a class of convex polynomials (possibly of degree greater than two) for which we show that the SDP hierarchy achieves exact SDP relaxation and provides an alternative construction of optimal distribution via a single SDP. Finally, the methodologies are illustrated through numerical examples and an application to bounding the expected payoffs across multiple options.