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April  2008, 4(2): 271-285. doi: 10.3934/jimo.2008.4.271

New approach to global minimization of normal multivariate polynomial based on tensor

Zhong Wan 1, and  Chunhua Yang 2,

1. 

School of Mathematics Sciences and Computing Technology, Central South University, Hunan Changsha, 410083, China
2. 

School of Information Science and Engineering, Central South University, Changsha, 410083, China

Received  January 2007 Revised  September 2007 Published  April 2008

In this paper, we first present a concise representation of multivariate polynomial, based on which we deduce the calculation formulae of its derivatives using tensor. Then, we propose a solution method to determine a global descent direction for the minimization of general normal polynomial. At a local and non-global maximizer or saddle point, we could use this method to get a global descent direction of the objective function. By using the global descent direction, we can transform an $n$-dimensional optimization problem into a one-dimensional one. Based on some efficient algorithms for one dimensional global optimization, we develop an algorithm to compute the global minimizer of normal multivariate polynomial. Numerical examples show that the proposed algorithm is promising.
Keywords: polynomial optimization, representation of polynomials, global minimization., global descent direction, tensor.
Mathematics Subject Classification: Primary: 90C22,90C26; Secondary: 49M20,49N1.
Citation: Zhong Wan, Chunhua Yang. New approach to global minimization of normal multivariate polynomial based on tensor. Journal of Industrial and Management Optimization, 2008, 4 (2) : 271-285. doi: 10.3934/jimo.2008.4.271
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