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- W2036726769 abstract "Given a compact parameter set $mathbf{Y}subsetmathbb{R}^p$, we consider polynomial optimization problems $(mathbf{P}_{mathbf{y}}$) on $mathbb{R}^n$ whose description depends on the parameter $mathbf{y}inmathbf{Y}$. We assume that one can compute all moments of some probability measure $varphi$ on $mathbf{Y}$, absolutely continuous with respect to the Lebesgue measure (e.g., $mathbf{Y}$ 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 $mathbf{x}^*(mathbf{y})$ of $mathbf{P}_{mathbf{y}}$, as $mathbf{y}inmathbf{Y}$. 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 $mathbf{y}mapsto x^*_k(mathbf{y})$ of the kth 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 $varphi({mathbf{y}:x^*_k(mathbf{y})=1}$, i.e., the probability that in an optimal solution $mathbf{x}^*(mathbf{y})$, the coordinate $x^*_k(mathbf{y})$ takes the value 1. 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." @default.
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- W2036726769 date "2010-01-01" @default.
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- W2036726769 title "A “Joint+Marginal” Approach to Parametric Polynomial Optimization" @default.
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- W2036726769 doi "https://doi.org/10.1137/090759240" @default.
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