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    A "joint+marginal" approach to parametric polynomial optimization

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    Given a compact parameter set Ysubset R^p, we consider polynomial optimization problems (Py) on R^n whose description depends on the parameter yinY. 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 Py. 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 ymapsto 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 xk, 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 L1(phi) (resp. almost uniform) convergence to the optimal value function.

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    Description

    Title : A "joint+marginal" approach to parametric polynomial optimization
    Author(s) : Jean B Lasserre
    Abstract : Given a compact parameter set Ysubset R^p, we consider polynomial optimization problems (Py) on R^n whose description depends on the parameter yinY. 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 Py. 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 ymapsto 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 xk, 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 L1(phi) (resp. almost uniform) convergence to the optimal value function.
    Subject : unspecified
    Area : Other
    Language : English
    Year : 2009

    Affiliations LAAS-CNRS, Toulouse, France
    Journal : SIAM Journal on Optimization
    Volume : 20
    Issue : 4
    Pages : 1995
    Url : http://arxiv.org/abs/0905.2497

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    Jean's Peer Evaluation activity

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    • Jean Lasserre, Principal Research Fellow, LAAS-CNRS, Toulouse, France.

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