A "joint+marginal" approach to parametric polynomial optimization
| dc.creator | Lasserre, Jean B. | |
| dc.date | 2009-05-15 | |
| dc.date.accessioned | 2026-07-07T13:15:35Z | |
| dc.date.available | 2026-07-07T13:15:35Z | |
| dc.description | 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 $ϕ$ 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 $P_y$. 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 $y\mapsto x^*_k(y)$ 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 $ϕ(\{y:x^*_k(y)=1\})$, 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 $L_1(ϕ)$ (resp. almost uniform) convergence to the optimal value function. | |
| dc.identifier | https://arxiv.org/abs/0905.2497 | |
| dc.identifier | http://arxiv.org/abs/0905.2497 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230515 | |
| dc.subject | Optimization and Control | |
| dc.subject | 65D15, 65K05, 46N10, 90C22 | |
| dc.title | A "joint+marginal" approach to parametric polynomial optimization | |
| dc.type | text |