Sum of Squares Method for Sensor Network Localization
This paper has been withdrawn by the author due to its publication
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arXiv subjects
Jiawang Nie
Publications and source records attributed to Jiawang Nie.
Semidefinite Representation of the $k$-Ellipse
The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.
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Semidefnite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization
In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x \mid x^* A_k x \ge 1, x\in\mathbb{F}^n, k=0,1,...,m\}$; and (2) $\max \{x^* C x \mid x^* A_k x \le 1, x\in\mathbb{F}^n, k=0,1,...,m\}$. If \emph{one} of $A_k$'s is indefinite while others and $C$ are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by $O(m^2)$ when $\mathbb{F}$ is the real line $\mathbb{R}$, and by $O(m)$ when $\mathbb{F}$ is the complex plane $\mathbb{C}$. This result is an extension of the recent work of Luo {\em et al.} \cite{LSTZ}. For (2), we show that the same ratio is bounded from below by $O(1/\log m)$ for both the real and complex case, whenever all but one of $A_k$'s are positive semidefinite while $C$ can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal {\em et al.} \cite{BNR02}. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.
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An elementary and constructive solution to Hilbert's 17th Problem for matrices
We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring $\mathbb R[x_1,...,x_m]$. The result is that if $A$ is postive semidefinite for all substitutions $(x_1,...,x_m) \in \mathbb R^m$, then $A$ can be expressed as a sum of squares of symmetric matrices with entries in $\mathbb R(x_1,...,x_m)$. Moreover, our proof is constructive and gives explicit representations modulo the scalar case.
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Minimizing Polynomials Over Semialgebraic Sets
This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in $\re^m$ defined by finitely many polynomial equations and inequalities, using the Karush-Kuhn-Tucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper \cite{njw_grad}, which considers minimizing polynomials on algebraic sets, i.e., sets in $\re^m$ defined by finitely many polynomial equations. Most of the theorems and conclusions in \cite{njw_grad} generalize to semialgebraic sets, even in the case where the semialgebraic set is not compact. We discuss the method in some special cases, namely, when the semialgebraic set is contained in the nonnegative orthant $\re^n_+$ or in box constraints $[a,b]_n$. These constraints make the computations more efficient.
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