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Complete Solutions and Extremality Criteria to Polynomial Optimization Problems

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Abstract

This paper presents a set of complete solutions to a class of polynomial optimization problems. By using the so-called sequential canonical dual transformation developed in the author’s recent book [Gao, D.Y. (2000), Duality Principles in Nonconvex Systems: Theory, Method and Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, xviii + 454 pp], the nonconvex polynomials in \(\mathbb{R}^n\) can be converted into an one-dimensional canonical dual optimization problem, which can be solved completely. Therefore, a set of complete solutions to the original problem is obtained. Both global minimizer and local extrema of certain special polynomials can be indentified by Gao-Strang’s gap function and triality theory. For general nonconvex polynomial minimization problems, a sufficient condition is proposed to identify global minimizer. Applications are illustrated by several examples.

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  1. Department of Mathematics, Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061, USA

    David Yang Gao

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  1. David Yang Gao
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Correspondence to David Yang Gao.

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Gao, D.Y. Complete Solutions and Extremality Criteria to Polynomial Optimization Problems. J Glob Optim 35, 131–143 (2006). https://doi.org/10.1007/s10898-005-3068-5

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  • DOI: https://doi.org/10.1007/s10898-005-3068-5

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