- A "joint+marginal" approach to parametric polynomial optimization Jean B. Lasserre(lasserrelaas.fr) Abstract: 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 $y\in Y$. We assume that one can compute all moments of some probability measure $\varphi$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $\varphi$ 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 $P_y$, as $y\in Y$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their $\varphi$-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 $\varphi(\{y:x^*_k(y)=1\}$, 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 $L_1(\varphi)$ (resp. $\varphi$-almost uniform) convergence to the optimal value function. Keywords: Parametric and polynomial optimization; semidefinite relaxations Category 1: Global Optimization Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Category 3: Global Optimization (Theory ) Citation: To appear in SIAM J. Optim. Download: [PDF]Entry Submitted: 01/13/2010Entry Accepted: 01/13/2010Entry Last Modified: 01/13/2010Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.