A "joint+marginal" approach to parametric polynomial optimization

Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $\phi$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $\phi$ 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 $P_y$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their $\phi$-mean. In addition, using this knowledge on moments, the measurable function $y\mapsto x^*_k(y)$ of the $k$-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable $x_k$, one may approximate as closely as desired its persistency $\phi(\{y:x^*_k(y)=1\})$, i.e. the probability that in an optimal solution $x^*(y)$, the coordinate $x^*_k(y)$ 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 $L_1(\phi)$ (resp. almost uniform) convergence to the optimal value function.