A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations

Mathematical Programming

Volumn 113, Issue 2, 2008, Pages 259-282

  Burer, Samuel ^a   Vandenbussche, Dieter ^b  

^a UNIVERSITY OF IOWA   (United States)

^b Axioma Inc   (United States)

Author keywords

Branch and bound; Lift and project relaxations; Nonconcave quadratic maximization; Nonconvex quadratic programming; Semidefinite programming

Indexed keywords

APPROXIMATION ALGORITHMS; BOUNDARY CONDITIONS; COMPUTATIONAL METHODS; FINITE ELEMENT METHOD; LINEAR PROGRAMMING; RELAXATION PROCESSES; SET THEORY;

BRANCH-AND-BOUND CONDITIONS; NONCONCAVE QUADRATIC MAXIMIZATION; NONCONVEX QUADRATIC PROGRAMMING; SEMIDEFINITE PROGRAMMING;

QUADRATIC PROGRAMMING;

EID: 38749116247     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-006-0080-6     Document Type: Article

References (28)

1. Absil, P.-A., Tits, A.: Newton-KKT interior-point methods for indefinite quadratic programming. Manuscript, Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA (2006). To appear in Computational Optimization and Applications
2. Anstreicher, K.M.: Combining RLT and SDP for nonconvex QCQP. Talk given at the Workshop on Integer Programming and Continuous Optimization, Chemnitz University of Technology, November 7-9 (2004)
3. Bomze, I.M., de Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24(2), 163-185 (2002). Dedicated to Professor Naum Z. Shor on his 65th birthday
4. Burer S. and Vandenbussche D. (2006). Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16(3): 726-750
5. Floudas, C., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 217-269. Kluwer Academic Publishers (1995)
6. Giannessi, F., Tomasin, E.: Nonconvex quadratic programs, linear complementarity problems, and integer linear programs. In: Fifth Conference on Optimization Techniques (Rome, 1973), Part I, pp. 437-449. Lecture Notes in Computer Science, vol. 3. Springer, Berlin Heidelberg New York (1973)
7. Globallib: http://www.gamsworld.org/global/globallib.htm
8. Goemans M. and Williamson D. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42: 1115-1145
9. Gould, N.I.M., Toint, P.L.: Numerical methods for large-scale non-convex quadratic. In: Trends in Industrial and Applied Mathematics (Amritsar, 2001), vol. 72 of Appl. Optim., pp. 149-179. Kluwer Academic Publishers, Dordrecht (2002)
10. Hansen P., Jaumard B., Ruiz M. and Xiong J. (1993). Global minimization of indefinite quadratic functions subject to box constraints. Naval Res. Logist. 40(3): 373-392
11. ILOG, Inc.: ILOG CPLEX 9.0, User Manual (2003).
12. Kojima M. and Tunçel L. (2000). Cones of matrices and successive convex relaxations of nonconvex sets. SIAM J. Optim. 10(3): 750-778
13. Kojima M. and Tunçel L. (2002). Some fundamental properties of successive convex relaxation methods on LCP and related problems. J. Global Optim. 24(3): 333-348
14. Kozlov M.K., Tarasov S.P. and Khachiyan L.G. (1979). Polynomial solvability of convex quadratic programming. Dokl. Akad. Nauk SSSR 248(5): 1049-1051
15. Lootsma F.A. and Pearson J.D. (1970). An indefinite-quadratic-programming model for a continuous-production problem. Philips Res. Rep. 25: 244-254
16. Lovász L. and Schrijver A. (1991). Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1: 166-190
17. Nesterov Y. (1998). Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9: 141-160
18. Pardalos P. (1991). Global optimization algorithms for linearly constrained indefinite quadratic problems. Comput. Math. Appl. 21: 87-97
19. Pardalos P.M. and Vavasis S.A. (1991). Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim. 1(1): 15-22
20. Sahinidis N.V. (1996). BARON a general purpose global optimization software package. J. Glob. Optim. 8: 201-205
21. Sherali H.D. and Fraticelli B.M.P. (2002). Enhancing RLTrelaxations via a new class of semidefinite cuts. J. Global Optim. 22: 233-261
22. Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Global Optim. 2(1), 101-112 (1992).
23. Conference on Computational Methods in Global Optimization, I (Princeton, NJ, 1991).
24. Sherali H.D. and Tuncbilek C.H. (1995). A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Global Optim. 7: 1-31
25. Skutella M. (2001). Convex quadratic and semidefinite programming relaxations in scheduling. J. ACM 48(2): 206-242
26. Vandenbussche D. and Nemhauser G. (2005). A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program. 102(3): 531-557
27. Vandenbussche D. and Nemhauser G. (2005). A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3): 559-575
28. Ye Y. (1999). Approximating quadratic programming with bound and quadratic constraints. Math. Program. 84(2, Ser. A): 219-226