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SIAM J. on Optimization / Volume 22 / Issue 1

SIAM J. Optim. 22, pp. 87-107 (21 pages)

Maximum Block Improvement and Polynomial Optimization

Bilian Chen, Simai He, Zhening Li, and Shuzhong Zhang

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In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we accept only a block update that achieves the maximum improvement, hence the name of our new search method: maximum block improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proved. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.

© 2012 Society for Industrial and Applied Mathematics

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KEYWORDS

AMS Subject Headings

90C26, 90C30, 15A69, 49M27

PUBLICATION DATA

ISSN

1052-6234 (print)  
1095-7189 (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received May 18, 2011
Accepted October 07, 2011
Published online January 24, 2012
Digital Object Identifier
http://dx.doi.org/10.1137/110834524

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