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Canonical Dual Approach for Minimizing a Nonconvex Quadratic Function over a Sphere

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Advances in Global Optimization

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 95))

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Abstract

In this paper, we study global optimal solutions of minimizing a nonconvex quadratic function subject to a sphere constraint. The main challenge is to solve the problem when it has multiple global solutions on the boundary of the sphere, which is called hard case. By canonical duality theory, a concave maximization problem is formulated, which is one-dimensional and without duality gaps to the primal problem. Then sufficient and necessary conditions are provided to identify whether the problem is in the hard case or not. A perturbation method and associated algorithms are proposed to solve hard-case problems. Theoretical results and methods are verified by numerical examples.

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Acknowledgements

This research is supported by US Air Force Office of Scientific Research under the grant AFOSR FA9550-10-1-0487, as well as by a grant from the Australian Government under the Collaborative Research Networks (CRN) program. The main results of this paper have been announced at the 3rd World Congress of Global Optimization, July 9–11, 2013, the Yellow Mountains, China.

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Authors and Affiliations

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

    Yi Chen & David Y. Gao

Authors
  1. Yi Chen
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  2. David Y. Gao
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Correspondence to Yi Chen .

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Editors and Affiliations

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

    David Gao

  2. School Science, Inform., Tech. & Engg., Federation University Australia, Ballarat, Victoria, Australia

    Ning Ruan

  3. Dept. of Mathematical Sciences, Tsinghua University, Beijing, China

    Wenxun Xing

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Chen, Y., Gao, D.Y. (2015). Canonical Dual Approach for Minimizing a Nonconvex Quadratic Function over a Sphere. In: Gao, D., Ruan, N., Xing, W. (eds) Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-08377-3_16

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