Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
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.
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.
Bazaraa, M. S. and Shetty, C. M. Nonlinear programming - theory and algorithms. Wiley, New York, 1979.
Bomze, I. M. On standard quadratic optimization problems. J. Global Optimiz. 13: 369–387, 1998.
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.
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.
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.
Bomze, I. M. and Stix, V. Genetical engineering via negative fitness: evolutionary dynamics for global optimization. Annals of O.R. 89: 279–318, 1999.
Cegielski, A. The Polyak subgradient projection method in matrix games. Discuss. Math. 13: 155–166, 1993.
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.
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.
Horst, R. On generalized bisection of n-simplices. Math. of Comput. 66: 691–698, 1997.
Horst, R., Pardalos, P. M. and Thoai, V. N. Introduction to Global Optimization. Kluwer, Dordrecht, 1995. BRANCH-AND-BOUND FOR STANDARD QUADRATIC OPTIMIZATION 37
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.
Horst, R. and Thoai, V. N. A new algorithm for solving the general quadratic programming problem. Comput. Optim. Appl. 5: 39–48, 1996.
Horst, R., Thoai, V. N. and de Vries, J. On geometry and convergence of a class of simplicial covers. Optimization 25: 53–64, 1992.
Horst, R. and Tuy, H. Global Optimization. Springer, Heidelberg, 1993.
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.
Kuznetsova, A. and Strekalovsky, A. On solving the maximum clique problem. J. Global Optimiz. 21: 265–288, 2001.
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.
Murty, K. G. and Kabadi, S. N. Some NP-complete problems in quadratic and linear programming. Math. Programming 39: 117–129, 1987.
Nowak, I. A new semidefinite programming bound for indefinite quadratic forms over a simplex. J. Global Optimiz. 14: 357–364, 1999.
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.
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.
Raber, U. A simplicial branch-and-bound method for solving nonconvex all-quadratic programs. J. Global Optimiz. 13: 417–432, 1998.
Renegar, J. A mathematical view of interior-point methods in convex optimization. Forthcoming, SIAM, Philadelphia, PA, 2001.
Stix, V. Global optimization of standard quadratic problems including parallel approaches. Ph.D. thesis, Univ. Vienna, 2000.
Stix, V. Target-oriented branch-and-bound method for global optimization. Preprint, Univ. Vienna, 2001.
Weibull, J. W. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bomze, I.M. Branch-and-bound approaches to standard quadratic optimization problems. Journal of Global Optimization 22, 17–37 (2002). https://doi.org/10.1023/A:1013886408463
Issue Date:
DOI: https://doi.org/10.1023/A:1013886408463