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    Principal Research Fellow

    LAAS-CNRS, Toulouse, France

    Convexity in semi-algebraic geometry and polynomial optimization

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    We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.

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    Description

    Title : Convexity in semi-algebraic geometry and polynomial optimization
    Author(s) : Jean B Lasserre
    Abstract : We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.
    Keywords : phrases, anr grant nt05 3 41612, basic semi algebraic sets, convex, convex polynomials, french, jensen inequality, research partially supported, semide nite programming, sets, sums squares

    Subject : unspecified
    Area : Other
    Language : English
    Year : 2008

    Affiliations LAAS-CNRS, Toulouse, France
    Journal : SIAM Journal on Optimization
    Volume : 19
    Issue : 4
    Pages : 21
    Url : http://arxiv.org/abs/0806.3784

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    Jean's Peer Evaluation activity

    Downloads 10
    Views 73
    Collected by 1
    • Jean Lasserre, Principal Research Fellow, LAAS-CNRS, Toulouse, France.

    Jean has...

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    Collected 8
    • Jean Lasserre, Principal Research Fellow, LAAS-CNRS, Toulouse, France.
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