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Necessary Optimality Conditions and New Optimization Methods for Cubic Polynomial Optimization Problems with Mixed Variables

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Abstract

Multivariate cubic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot of practical applications in real world. In this paper, some necessary local optimality conditions and some necessary global optimality conditions for cubic polynomial optimization problems with mixed variables are established. Then some local optimization methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization methods for cubic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local optimality conditions and necessary global optimality conditions. A global optimization method is proposed for cubic polynomial optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical examples are also given to illustrate that these approaches are very efficient.

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Authors and Affiliations

  1. School of Science, Information Technology and Engineering, University of Ballarat, Ballarat, Victoria, 3353, Australia

    Z. Y. Wu & J. Tian

  2. Department of Mathematics, Yibin University, Yibin, 644000, China

    J. Quan

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    G. Q. Li

Authors
  1. Z. Y. Wu
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  2. J. Quan
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  3. G. Q. Li
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  4. J. Tian
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Correspondence to Z. Y. Wu.

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Communicated by Xiaoqi Yang.

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Wu, Z.Y., Quan, J., Li, G.Q. et al. Necessary Optimality Conditions and New Optimization Methods for Cubic Polynomial Optimization Problems with Mixed Variables. J Optim Theory Appl 153, 408–435 (2012). https://doi.org/10.1007/s10957-011-9961-9

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  • DOI: https://doi.org/10.1007/s10957-011-9961-9

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