A "Joint+Marginal" Approach to Parametric Polynomial Optimization

Abstract. Given a compact parameter set Y ⊂ Rp, we consider polynomial optimization problems (Py) on Rn whose description depends on the parameter y ∈ Y. We assume that one can compute all moments of some probability measure φ on Y, absolutely continuous with respect to the Lebesgue measure (e.g. Y is a box or a simplex and φ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions x∗(y) of Py, as y ∈ Y. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their φ-mean. In addition, using this knowledge on moments, the measurable function y #→ xk(y) of the k-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable xk, one may approximate as closely as desired its persistency φ({y : xk(y) = 1}, i.e. the probability that in an optimal solution x∗(y), the coordinate xk(y) takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with L1(φ) (resp. φ-almost uniform) convergence to the optimal value function.