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A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications

Masakazu Muramatsu (muramatu***at***cs.uec.ac.jp)
Hayato Waki (waki***at***imi.kyushu-u.ac.jp)
Levent Tuncel (ltuncel***at***math.uwaterloo.ca)

Abstract: We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^¥ast - ¥epsilon)$ and from above by $f^¥ast + ¥epsilon (n+1)$, where $f^¥ast$ is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

Keywords:

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Nonlinear Optimization (Quadratic Programming )

Citation:

Download: [PDF]

Entry Submitted: 03/29/2013
Entry Accepted: 03/29/2013
Entry Last Modified: 03/29/2013

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