Journal of Global Optimization

, Volume 22, Issue 1–4, pp 17–37

Branch-and-bound approaches to standard quadratic optimization problems

  • Immanuel M. Bomze


This paper explores several possibilities for applying branch-and-bound techniques to a central problem class in quadratic programming, the so-called Standard Quadratic Problems (StQPs), which consist of finding a (global) minimizer of a quadratic form over the standard simplex. Since a crucial part of the procedures is based on efficient local optimization, different procedures to obtain local solutions are discussed, and a new class of ascent directions is proposed, for which a convergence result is established. Main emphasis is laid upon a d.c.-based branch-and-bound algorithm, and various strategies for obtaining an efficient d.c. decomposition are discussed.

D.C. decomposition Semidefinite relaxation Replicator dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. An, L. T. H. and Tao, P. D. Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J. Global Optimiz. 11: 253–285, 1997.Google Scholar
  2. An, L. T. H. and Tao, P. D. A branch and bound method via d. c. optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems. J. Global Optimiz. 13: 171–206, 1998.Google Scholar
  3. Bazaraa, M. S. and Shetty, C. M. Nonlinear programming - theory and algorithms. Wiley, New York, 1979.Google Scholar
  4. Bomze, I. M. On standard quadratic optimization problems. J. Global Optimiz. 13: 369–387, 1998.Google Scholar
  5. Bomze, I. M., Budinich, M., Pardalos, P. M. and Pelillo, M. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization suppl. Vol. A:1–74. Kluwer, Dordrecht, 1999.Google Scholar
  6. Bomze, I. M., Budinich, M., Pelillo, M. and Rossi, C. Annealed replication: a new heuristic for the maximum clique problem. To appear in: Discrete Applied Math., 2001.Google Scholar
  7. Bomze, I. M., Dür, M., de Klerk, E., Quist, A. J., Roos, C. and Terlaky, T. On copositive programming and standard quadratic optimization problems. J. Global Optimiz. 18: 301–320, 2000.Google Scholar
  8. Bomze, I. M. and Stix, V. Genetical engineering via negative fitness: evolutionary dynamics for global optimization. Annals of O.R. 89: 279–318, 1999.Google Scholar
  9. Cegielski, A. The Polyak subgradient projection method in matrix games. Discuss. Math. 13: 155–166, 1993.Google Scholar
  10. Dür, M. A Note on Local and Global Optimality Conditions in D.C.-Programming. Research Report No. 56, Dept. of Statistics, Vienna Univ. Econ., 1999.Google Scholar
  11. Hansen, P., Jaumard, B., Ruiz, M. and Xiong, J. Global minimization of indefinite quadratic functions subject to box constraints. Nav. Res. Logist. 40: 373–392, 1993.Google Scholar
  12. Horst, R. On generalized bisection of n-simplices. Math. of Comput. 66: 691–698, 1997.Google Scholar
  13. Horst, R., Pardalos, P. M. and Thoai, V. N. Introduction to Global Optimization. Kluwer, Dordrecht, 1995. BRANCH-AND-BOUND FOR STANDARD QUADRATIC OPTIMIZATION 37Google Scholar
  14. Horst, R. and Thoai, V. N. Modification, implementation and comparison of three algorithms for globally solving linearly constrained concave minimization problems. Computing 42: 271–289, 1989.Google Scholar
  15. Horst, R. and Thoai, V. N. A new algorithm for solving the general quadratic programming problem. Comput. Optim. Appl. 5: 39–48, 1996.Google Scholar
  16. Horst, R., Thoai, V. N. and de Vries, J. On geometry and convergence of a class of simplicial covers. Optimization 25: 53–64, 1992.Google Scholar
  17. Horst, R. and Tuy, H. Global Optimization. Springer, Heidelberg, 1993.Google Scholar
  18. Johnson, D. S. and Trick, M. A. (editors). Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26. American Mathematical Society, Providence, RI, 1996.Google Scholar
  19. Kuznetsova, A. and Strekalovsky, A. On solving the maximum clique problem. J. Global Optimiz. 21: 265–288, 2001.Google Scholar
  20. Massaro, A., Pelillo, M. and Bomze, I. M. A complementary pivoting approach to the maximum weight clique problem. To appear in: SIAM J. Optimiz., 2001.Google Scholar
  21. Murty, K. G. and Kabadi, S. N. Some NP-complete problems in quadratic and linear programming. Math. Programming 39: 117–129, 1987.Google Scholar
  22. Nowak, I. A new semidefinite programming bound for indefinite quadratic forms over a simplex. J. Global Optimiz. 14: 357–364, 1999.Google Scholar
  23. Phong, T. Q., An, L. T. H. and Tao, P. D. On globally solving linearly constrained indefinite quadratic minimization problems by decomposition branch and bound method. RAIRO, Rech. Oper. 30: 31–49, 1996.Google Scholar
  24. Quist, A. J., de Klerk, E., Roos, C. and Terlaky, T. Copositive relaxation for general quadratic programming. Optimization Methods and Software 9: 185–209, 1998.Google Scholar
  25. Raber, U. A simplicial branch-and-bound method for solving nonconvex all-quadratic programs. J. Global Optimiz. 13: 417–432, 1998.Google Scholar
  26. Renegar, J. A mathematical view of interior-point methods in convex optimization. Forthcoming, SIAM, Philadelphia, PA, 2001.Google Scholar
  27. Stix, V. Global optimization of standard quadratic problems including parallel approaches. Ph.D. thesis, Univ. Vienna, 2000.Google Scholar
  28. Stix, V. Target-oriented branch-and-bound method for global optimization. Preprint, Univ. Vienna, 2001.Google Scholar
  29. Weibull, J. W. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  1. 1.Department of Statistics and Decision Support SystemsUniversity of ViennaViennaAustria

Personalised recommendations