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Approximation Bounds for Trilinear and Biquadratic Optimization Problems Over Nonconvex Constraints

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Abstract

This paper presents new approximation bounds for trilinear and biquadratic optimization problems over nonconvex constraints. We first consider the partial semidefinite relaxation of the original problem, and show that there is a bounded approximation solution to it. This will be achieved by determining the diameters of certain convex bodies. We then show that there is also a bounded approximation solution to the original problem via extracting the approximation solution of its semidefinite relaxation. Under some conditions, the approximation bounds obtained in this paper improve those in the literature.

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Acknowledgments

The authors would like to thank Professor Giannessi for his helpful comments. The second author’s work was supported by the NSFC Grant 11271206, Doctoral fund of Chinese Ministry of Education (Grant No. 20120031110024). The third author’s work was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 501909, 502510, 502111, and 501212).

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin , 300071, People’s Republic of China

    Yuning Yang & Qingzhi Yang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, People’s Republic of China

    Liqun Qi

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  1. Yuning Yang
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  2. Qingzhi Yang
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  3. Liqun Qi
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Correspondence to Yuning Yang.

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Yang, Y., Yang, Q. & Qi, L. Approximation Bounds for Trilinear and Biquadratic Optimization Problems Over Nonconvex Constraints. J Optim Theory Appl 163, 841–858 (2014). https://doi.org/10.1007/s10957-014-0538-2

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  • DOI: https://doi.org/10.1007/s10957-014-0538-2

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