Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations

This paper studies the so-called biquadratic optimization over unit spheres ... , subject to ... . The authors show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, they present various approximation methods based on sem...

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Bibliographic Details
Published in:SIAM journal on optimization Vol. 20; no. 3; pp. 1286 - 1310
Main Authors: Ling, Chen, Nie, Jiawang, Qi, Liqun, Ye, Yinyu
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
Subjects:
Algorithms
Approximation
Eigenvalues
Hierarchies
Mathematical analysis
Mathematical models
Mathematical problems
Optimization
Optimization algorithms
Programming
Semidefinite programming
Studies
ISSN:1052-6234, 1095-7189
Online Access:Get full text
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Summary:This paper studies the so-called biquadratic optimization over unit spheres ... , subject to ... . The authors show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, they present various approximation methods based on semidefinite programming (SDP) relaxations.Their theoretical results are as follows: For general biquadratic forms, they develop a ... -approximation algorithm under a slightly weaker approximation notion; for biquadratic forms that are square-free, we give a relative approximation bound ... when min ... is a constant, we present two polynomial time approximation schemes (PTASs) which are based on sum of squares (SOS) relaxation hierarchy and grid sampling of the standard simplex. For practical computational purposes, we propose the first order SOS relaxation, a convex quadratic SDP relaxation, and a simple minimum eigenvalue method and show their error bounds. Some illustrative numerical examples are also included.(ProQuest: ... denotes formulae/symbols omitted.)
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ISSN:1052-6234
1095-7189
DOI:10.1137/080729104