Skip to main content
SpringerLink
Log in

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Khayyal F.A., Larsen C., Van Voorhis T.: A relaxation method for nonconvex quadratically constrained quadratic programs. J. Glob. Optim. 6, 215–230 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. An L.T.H., Tao P.D.: Solving a class of linearly constrained indefinite quadratic problems by D.C. algorithms. J. Glob. Optim. 11, 253–285 (1997)

    Article  MATH  Google Scholar 

  4. Anstreicher K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43, 471–484 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anstreicher, K.M.: On Convex Relaxations for Quadratically Constrained Quadratic Programming. Tech. rep., Department of Management Sciences, University of Iowa (2010). Avaialable at http://www.optimization-online.org/DB_FILE/2010/08/2699.pdf

  6. Anstreicher K.M., Burer S.: D.C. versus copositive bounds for standard QP. J. Glob. Optim. 33, 299–312 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Anstreicher K.M., Wolkowicz H.: On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. Appl. 22, 41 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Audet C., Hansen P., Jaumard B., Savard G.: A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Math. Program. 87, 131–152 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Beck A., Eldar Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17, 844–860 (2007)

    Article  MathSciNet  Google Scholar 

  10. Ben-Tal, A.: Conic and Robust Optimization. Lecture Notes, Universita di Roma La Sapienzia, Rome, Italy (2002). Available at http://ie.technion.ac.il/Home/Users/morbt/rom.pdf

  11. Bertsekas D.P., Nedić A., Ozdaglar A.E.: Convex Analysis and Optimization. Athena Scientific Belmont, Massachusetts (2003)

    MATH  Google Scholar 

  12. Bomze I.M., Dür M., de Klerk E., Roos C., Quist A.J., Terlaky T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18, 301–320 (2000)

    Article  MATH  Google Scholar 

  13. Burer S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Burer, S., Saxena, A.: Old Wine in New Bottle: The MILP Road to MIQCP. Tech. rep., Department of Management Sciences, University of Iowa (2009). Available at http://www.optimization-online.org/DB_FILE/2009/07/2338.pdf

  15. 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 

  16. Fu M., Luo Z.Q., Ye Y.: Approximation algorithms for quadratic programming. J. Comb. Optim. 2, 29–50 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujie T., Kojima M.: Semidefinite programming relaxation for nonconvex quadratic programs. J. Glob. Optim. 10, 367–380 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goemans M.X., Williamson D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 1. 21 (2011). http://cvxr.com/cvx

  20. Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  21. Kellerer H., Pferschy U., Pisinger D.: Knapsack Problems. Springer, Berlin (2004)

    MATH  Google Scholar 

  22. Kim S., Kojima M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26, 143–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. de Klerk E., Pasechnik D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. de Klerk E., Pasechnik D.V.: A linear programming reformulation of the standard quadratic optimization problem. J. Glob. Optim. 37, 75–84 (2007)

    Article  MATH  Google Scholar 

  25. Kojima M., Tunçel L.: Cones of matrices and successive convex relaxations of nonconvex sets. SIAM J. Optim. 10, 750–778 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 

  27. McCormick G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nesterov Y.: Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9, 141–160 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nesterov Y., Nemirovsky A.: Interior-Point Polynomial Methods in Convex Programming. SIAM, Philadelphia (1994)

    Google Scholar 

  30. Nesterov Y., Todd M.J., Ye Y.: Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math. Program. 84, 227–267 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. thesis, California Institute of Technology (2000)

  32. Povh J., Rendl F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18, 223–241 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Povh J., Rendl F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6, 231–241 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Raber U.: A simplicial branch-and-bound method for solving nonconvex all-quadratic programs. J. Glob. Optim. 13, 417–432 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sahinidis N.V.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8, 201–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. (2011). doi:10.1007/s10107-010-0340-3

  37. Sherali H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnamica 22, 245–270 (1997)

    MathSciNet  MATH  Google Scholar 

  38. Sherali H.D., Adams W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  39. Sherali H.D., Tuncbilek C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7, 1–31 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Tseng P.: Further results on approximating nonconvex quadratic optimization by semidefinite programming relaxation. SIAM J. Optim. 14, 268 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vanderbussche D., Nemhauser G.L.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102, 371–405 (2005)

    Article  MathSciNet  Google Scholar 

  44. Ye Y.: Approximating global quadratic optimization with convex quadratic constraints. J. Glob. Optim. 15, 1–17 (1999)

    Article  MATH  Google Scholar 

  45. Ye Y.: Approximating quadratic programming with bound and quadratic constraints. Math. Program. 84, 219–226 (1999)

    MathSciNet  MATH  Google Scholar 

  46. Ye Y., Zhang S.: New results on quadratic minimization. SIAM J. Optim. 14, 245–267 (2004)

    Article  MathSciNet  Google Scholar 

  47. Zhang S.: Quadratic maximization and semidefinite relaxation. Math. Program. 87, 453–465 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zheng, X.J., Sun, X.L., Li, D.: Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations. J. Glob. Optim. (2011). doi:10.1007/s10898-010-9630-9

Download references

Author information

Authors and Affiliations

  1. Department of Management Science, School of Management, Fudan University, 670 Guoshun Road, Shanghai, 200433, People’s Republic of China

    Xiao Jin Zheng & Xiao Ling Sun

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Duan Li

Authors
  1. Xiao Jin Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao Ling Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Duan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Duan Li.

Additional information

Dedicated to the memory of Professor Paul Tseng. This work was supported by National Natural Science Foundation of China under grants 10971034 and 70832002, by NSFC/RGC Joint Research Project under grant 71061160506, and by Research Grants Council of Hong Kong under grant 414207. The authors are grateful to the valuable comments and suggestions from two anonymous reviewers, which help improve the paper significantly.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, X.J., Sun, X.L. & Li, D. Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation. Math. Program. 129, 301–329 (2011). https://doi.org/10.1007/s10107-011-0466-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-011-0466-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation