Exact Solution of Uncertain Convex Optimization Problems

This paper proposes a novel approach for the solution of a wide class of convex programs characterized by the presence of bounded stochastic uncertainty. The data of the problem is assumed to depend polynomially on a vector of uncertain parameters q isin R^d, uniformly distributed in a box, and the solution should minimize the expected value of the cost function with respect to q. The proposed methodology is based on a combination of low-order quadrature formulae, which allow for the construction of a cubature rule with high degree of exactness and low number of nodes. The algorithm is shown to depend polynomially on the problem dimension d. A specific application to uncertain least-squares problems, along with a numerical example, concludes the paper.