Journal of Global Optimization

, Volume 73, Issue 2, pp 371–388 | Cite as

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

  • Cheng Lu
  • Zhibin DengEmail author
  • Jing Zhou
  • Xiaoling Guo


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.


Quadratically constrained quadratic programming Semidefinite relaxation Branch-and-bound algorithm Global optimization 



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


  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1, 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bao, X., Sahinidis, N.V.: Polyhedral relaxations for nonconvex, quadratically constrained quadratic programs. Optim. Method Softw. 24, 485–504 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ben-Tal, A., Teboulle, M.: Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72, 51–63 (1996)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Buchheim, C., Wiegele, A.: Semidefinite relaxations for non-convex quadratic mixed-integer programming. Math. Program. 141, 435–452 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113, 259–282 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151, 89–116 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cambini, R., Sodini, C.: Decomposition methods for solving nonconvex quadratic programs via branch and bound. J. Glob. Optim. 33, 316–336 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4, 33–52 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Method Softw. 15, 201–224 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)CrossRefGoogle Scholar
  14. 14.
    Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103, 251–282 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 20.
    Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Global Optim. 57, 3–50 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Global Optim. 59, 503–526 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim. 1, 15–22 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77, 273–299 (1997)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8, 201–205 (1996)Google Scholar
  27. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sherali, H.D., Liberti, L.: Reformulation-Linearization methods for global optimization. Technical reportGoogle Scholar
  31. 31.
    Sidiropoulos, N.D., Davidson, T.N., Luo, Z.-Q.: Transmit beamforming for physical layer multicasting. IEEE Trans. Signal Process. 54, 2239–2251 (2006)CrossRefzbMATHGoogle Scholar
  32. 32.
    Sojoudi, S., Lavaei, J.: Exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure. SIAM J. Optim. 24, 1746–1778 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Method Softw. 11, 625–653 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Vandenbussche, D., Nemhauser, G.L.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102, 559–575 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 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)Google Scholar
  38. 38.
    Wang, S., Xia, Y.: On the ball-constrained weighted maximin dispersion problem. SIAM J. Optim. 26, 1565–1588 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14, 245–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Economics and ManagementNorth China Electric Power UniversityBeijingChina
  2. 2.School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge ManagementChinese Academy of SciencesBeijingChina
  3. 3.Department of Applied Mathematics, College of ScienceZhejiang University of TechnologyHangzhouChina
  4. 4.Departmemt of MathematicsChina University of Mining and TechnologyBeijingChina

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