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A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming

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Abstract

In this paper, we design an eigenvalue decomposition based branch-and-bound algorithm for finding global solutions of quadratically constrained quadratic programming (QCQP) problems. The hardness of nonconvex QCQP problems roots in the nonconvex components of quadratic terms, which are represented by the negative eigenvalues and the corresponding eigenvectors in the eigenvalue decomposition. For certain types of QCQP problems, only very few eigenvectors, defined as sensitive-eigenvectors, determine the relaxation gaps. We propose a semidefinite relaxation based branch-and-bound algorithm to solve QCQP. The proposed algorithm, which branches on the directions of the sensitive-eigenvectors, is very efficient for solving certain types of QCQP problems.

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Notes

  1. The terms \(s_i\) and \(t_i\) are computed for all \(i\in \{1,\ldots ,r\}\), rather than only for \(i\in \mathcal {A}_{a}\). The computational cost is very low, even when r is large.

  2. The solvers BARON, COUENNE and SCIP have provided interfaces to control their error tolerances.

  3. As discussed in [31], the complex Gaussian distribution is a reasonable assumption.

  4. The problem size \((n,m)=(4,8)\) is a typical setting, as discussed in [31].

References

  1. Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1, 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. An, L.T.H., Tao, P.D.: A branch and bound method via D.C. optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems. J. Glob. Optim. 13, 171–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, X., Sahinidis, N.V.: Polyhedral relaxations for nonconvex, quadratically constrained quadratic programs. Optim. Method Softw. 24, 485–504 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programs: a review and comparisons. Math. Program. 129, 129–157 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Tal, A., Teboulle, M.: Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72, 51–63 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Buchheim, C., Wiegele, A.: Semidefinite relaxations for non-convex quadratic mixed-integer programming. Math. Program. 141, 435–452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113, 259–282 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151, 89–116 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cambini, R., Sodini, C.: Decomposition methods for solving nonconvex quadratic programs via branch and bound. J. Glob. Optim. 33, 316–336 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4, 33–52 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. He, S., Luo, Z.-Q., Nie, J., Zhang, S.: Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization. SIAM J. Optim. 19, 503–523 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Method Softw. 15, 201–224 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lemarechal, C., Oustry, F.: SDP relaxations in combinatorial optimization from a Lagrangian point of view. In: Hadjisavvas, N., Pardalos, P. (eds.) Proceedings of Advances in Convex Analysis and Global Optimization, pp. 119–134. Kluwer, Amsterdam (2001)

    Chapter  Google Scholar 

  14. Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103, 251–282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo, Z.-Q., Sidiropoulos, N.D., Tseng, P., Zhang, S.: Approximation bounds for quadratic optimization with homogeneous quadratic constraints. SIAM J. Optim. 18, 1–28 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luo, Z.-Q., Ma, W.-K., So, A.M.-C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems: from its practical deployments and scope of applicability to key theoretical results. IEEE Signal Process. Mag. 27, 20–34 (2010)

    Article  Google Scholar 

  18. Lu, C., Deng, Z., Jin, Q.: An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints. J. Global Optim. 67, 475–493 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Matskani, E., Sidiropoulos, N.D., Luo, Z.-Q., Tassiulas, L.: Convex approximation techniques for joint multiuser downlink beamforming and admission control. IEEE Trans. Wireless Commun. 7, 2682–2693 (2008)

    Article  Google Scholar 

  20. Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Global Optim. 57, 3–50 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Global Optim. 59, 503–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mitchell, J.E., Pang, J.S., Yu, B.: Convex quadratic relaxations of nonconvex quadratically constrained quadratic programs. Optim. Method Softw. 29, 120–136 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pardalos, P.M., Rodgers, G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45, 131–144 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim. 1, 15–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77, 273–299 (1997)

    MathSciNet  MATH  Google Scholar 

  26. Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8, 201–205 (1996)

  27. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 130, 359–413 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124, 383–411 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3, 411–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sherali, H.D., Liberti, L.: Reformulation-Linearization methods for global optimization. Technical report

  31. Sidiropoulos, N.D., Davidson, T.N., Luo, Z.-Q.: Transmit beamforming for physical layer multicasting. IEEE Trans. Signal Process. 54, 2239–2251 (2006)

    Article  MATH  Google Scholar 

  32. Sojoudi, S., Lavaei, J.: Exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure. SIAM J. Optim. 24, 1746–1778 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Method Softw. 11, 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vandenbussche, D., Nemhauser, G.L.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102, 559–575 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)

  38. Wang, S., Xia, Y.: On the ball-constrained weighted maximin dispersion problem. SIAM J. Optim. 26, 1565–1588 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14, 245–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the two anonymous reviewers, whose invaluable comments have significantly improved the quality of this paper.

Author information

Authors and Affiliations

  1. School of Economics and Management, North China Electric Power University, Beijing, 102206, China

    Cheng Lu

  2. School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing, 100190, China

    Zhibin Deng

  3. Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, 310023, China

    Jing Zhou

  4. Departmemt of Mathematics, China University of Mining and Technology, Beijing, 100083, China

    Xiaoling Guo

Authors
  1. Cheng Lu
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  2. Zhibin Deng
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  3. Jing Zhou
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  4. Xiaoling Guo
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Corresponding author

Correspondence to Zhibin Deng.

Additional information

Lu’s research has been supported by National Natural Science Foundation of China Grant Nos. 11701177 and 11771243, and Fundamental Research Funds for the Central Universities Grant Nos. 2017MS058 and 2018ZD14. Deng’s research has been supported by National Natural Science Foundation of China Grant No. 11501543. Zhou’s research has been supported by National Natural Science Foundation of China Grant No. 11701512, and the Zhejiang Provincial Natural Science Foundation of China Grant No. LQ16A010010. Guo’s research has been supported by National Natural Science Foundation of China Grant No. 11801557, and Fundamental Research Funds for the Central Universities Grant No. 800015LF.

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Lu, C., Deng, Z., Zhou, J. et al. A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming. J Glob Optim 73, 371–388 (2019). https://doi.org/10.1007/s10898-018-0726-y

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  • DOI: https://doi.org/10.1007/s10898-018-0726-y

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