Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints

  • Authors:

  • Xinzhen Zhang;Chen Ling;Liqun Qi

  • Affiliations:

  • Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong;School of Science, Hangzhou Dianzi University, Hangzhou, China 310018 and School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, China;Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2011

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Abstract

This paper studies the relationship between the so-called bi-quadratic optimization problem and its semidefinite programming (SDP) relaxation. It is shown that each r-bound approximation solution of the relaxed bi-linear SDP can be used to generate in randomized polynomial time an $${\mathcal{O}(r)}$$ -approximation solution of the original bi-quadratic optimization problem, where the constant in $${\mathcal{O}(r)}$$ does not involve the dimension of variables and the data of problems. For special cases of maximization model, we provide an approximation algorithm for the considered problems.