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Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

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Abstract

A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, in order to get rid of the sign constraints. We study the first- and second-order optimality conditions of the original StBQP and the reformulated bi-quartic problem over the product of two Euclidean spheres. Furthermore, we discuss the one-to-one correspondence between the global/local solutions of StBQP and the global/local solutions of the reformulation. We introduce a continuously differentiable penalty function. Based upon this, the original problem is converted into the problem of locating an unconstrained global minimizer of a (specially structured) polynomial of degree eight.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vienna, Vienna, Austria

    Immanuel M. Bomze

  2. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Chen Ling

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Liqun Qi

  4. Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China

    Xinzhen Zhang

Authors
  1. Immanuel M. Bomze
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  2. Chen Ling
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  3. Liqun Qi
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  4. Xinzhen Zhang
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Corresponding author

Correspondence to Immanuel M. Bomze.

Additional information

Chen Ling’s work is supported by the National Natural Science Foundation of China (Grant No. 10871168 and 10971187), and the Zhejiang Provincial National Science Foundation of China (Grant No. Y6100366).

Liqun Qi’s work is supported by the Hong Kong Research Grant Council (Projects: PolyU 5019/09P and PolyU 5018/08P).

Xinzhen Zhang’s work is supported by Chinese NSF Grants 10771120.

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Bomze, I.M., Ling, C., Qi, L. et al. Standard bi-quadratic optimization problems and unconstrained polynomial reformulations. J Glob Optim 52, 663–687 (2012). https://doi.org/10.1007/s10898-011-9710-5

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  • DOI: https://doi.org/10.1007/s10898-011-9710-5

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