Distributionally robust chance-constrained standard quadratic optimization with Wasserstein ambiguity

Abstract

The standard quadratic optimization problem consists of minimizing a quadratic form over the standard simplex. It has highly relevant applications ranging from portfolio optimization to machine learning. However, if the quadratic form is neither convex nor concave, this problem is NP-hard. We study a distributionally robust chance-constrained variant whose ambiguity set is a Wasserstein ball around the sampled distribution and make four contributions. First, although the problem is nonconvex in the decision variable, we exploit its linear dependence on the uncertain matrix to reduce the problem to a mixed-integer quadratic optimization problem. Second, we show that a judicious choice of matrix norm---the entrywise maximum norm---makes this reformulation both exact and tractable, and we describe the induced worst-case distribution. Third, our formulation is, by construction, the minimization of the worst-case Value-at-Risk of the random quadratic form; this unifies it with existing frameworks for uncertain standard quadratic optimization problems, in particular under chance constraints, and we identify natural settings in which it collapses to a single deterministic instance. Finally, we report numerical experiments on positive-definite and indefinite instances which quantify how the model parameters drive the optimal value and solution time, and expose the computational cost of indefiniteness.