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http://hdl.handle.net/10397/4765
| 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 |
| Appears in Collections: | Journal/Magazine Article |
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| Qi_Global_minimization_normal.pdf | 360.14 kB | Adobe PDF | View/Open |
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