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Jiawang Nie

  1. Discriminants and Nonnegative Polynomials.

    Пт, 02/12/2010 - 16:01 — SomNambul
    Authors: Jiawang Nie
    Subjects: Optimization and Control
    Abstract

    For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in
    R^n of degrees at most d that are nonnegative on K. This paper studies the
    geometry of its boundary. When K=R^n and d is even, we show that its boundary
    lies on the irreducible hypersurface defined by the discriminant of a single
    polynomial. When K is a real algebraic variety, we show that P_d(K) lies on the
    hypersurface defined by the discriminant of several polynomials. When K is a
    general semialgebraic set, we show that P_d(K) lies on a union of hypersurfaces
    defined by the discriminantal equations.

  2. Regularization Methods for Sum of Squares Relaxations in Large Scale Polynomial Optimization.

    Втр, 09/22/2009 - 14:09 — SomNambul
    Authors: Jiawang Nie
    Subjects: Optimization and Control
    Abstract

    We study how to solve sum of squares (SOS) and Lasserre's relaxations for
    large scale polynomial optimization. When interior-point type methods are used,
    typically only small or moderately large problems could be solved. This paper
    proposes the regularization type methods which would solve significantly larger
    problems. We first describe these methods for general conic semidefinite
    optimization, and then apply them to solve large scale polynomial optimization.
    Their efficiency is demonstrated by extensive numerical computations.

  3. Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems.

    Втр, 08/25/2009 - 22:41 — SomNambul
    Authors: Lawrence Fialkow, Jiawang Nie
    Subjects: Functional Analysis
    Abstract

    We employ positivity of Riesz functionals to establish representing measures
    (or approximate representing measures) for truncated multivariate moment
    sequences. For a truncated moment sequence $y$, we show that $y$ lies in the
    closure of truncated moment sequences admitting representing measures supported
    in a prescribed closed set $K \subseteq \re^n$ if and only if the associated
    Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove
    that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure
    supported in $K$.

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