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Title: Global minimization of normal quartic polynomials based on global descent directions
Authors: Qi, L 
Wan, Z
Yang, YF
Issue Date: 2004
Source: SIAM journal on optimization, 2004, v. 15, no. 1, p. 275-302
Abstract: A normal quartic polynomial is a quartic polynomial whose fourth degree term coefficient tensor is positive definite. Its minimization problem is one of the simplest cases of nonconvex global optimization, and has engineering applications. We call a direction a global descent direction of a function at a point if there is another point with a lower function value along this direction. For a normal quartic polynomial, we present a criterion to find a global descent direction at a noncritical point, a saddle point, or a local maximizer. We give sufficient conditions to judge whether a local minimizer is global and give a method for finding a global descent direction at a local, but not global, minimizer. We also give a formula at a critical point and a method at a noncritical point to find a one-dimensional global minimizer along a global descent direction. Based upon these, we propose a global descent algorithm for finding a global minimizer of a normal quartic polynomial when n = 2. For the case n ≥ 3, we propose an algorithm for finding an ε-global minimizer. At each iteration of a second algorithm, a system of constrained nonlinear equations is solved. Numerical tests show that these two algorithms are promising.
Keywords: Global optimization
Normal quartic polynomial
Tensor
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on optimization 
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/S1052623403420857
Rights: © 2004 Society for Industrial and Applied Mathematics
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