Abstract
We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvex quadratic programs.
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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(2), 171–206 (1998)
Anstreicher, K., Brixius, N., Goux, J.-P., Linderoth, J.: Solving large quadratic assignment problems on computational grids. Math. Program. 91(3, Ser. B), 563–588 (2002). ISMP 2000, Part 1 (Atlanta, GA)
Braun, S., Mitchell, J.E.: A semidefinite programming heuristic for quadratic programming problems with complementarity constraints. Comput. Optim. Appl. 31(1), 5–29 (2005)
Burer, S., Vandenbussche, D.: Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16(3), 726–750 (2006)
Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Manuscript, Department of Management Sciences, University of Iowa, Iowa City, IA, USA, June 2005. Revised April 2006 and June 2006. Math. Program. (to appear)
De Angelis, P., Pardalos, P., Toraldo, G.: Quadratic programming with box constraints. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P. (eds.) Developments in Global Optimization, pp. 73–94 (1997)
Giannessi, F., Tomasin, E.: Nonconvex quadratic programs, linear complementarity problems, and integer linear programs. In: Fifth Conference on Optimization Techniques, Rome, 1973, Part I. Lecture Notes in Comput. Sci., vol. 3, pp. 437–449. Springer, Berlin (1973)
Gould, N.I.M., Toint, P.L.: Numerical methods for large-scale non-convex quadratic programming. In: Trends in Industrial and Applied Mathematics, Amritsar, 2001. Appl. Optim., vol. 72, pp. 149–179. Kluwer Acad., Dordrecht (2002)
Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subject to box constraints. Naval Res. Logist. 40(3), 373–392 (1993)
Helmberg, C.: Fixing variables in semidefinite relaxations. SIAM J. Matrix Anal. Appl. 21(3), 952–969 (2000) (Electronic)
ILOG, Inc. ILOG CPLEX 9.0, User Manual (2003)
Kozlov, M.K., Tarasov, S.P., Khachiyan, L.G.: Polynomial solvability of convex quadratic programming. Dokl. Akad. Nauk SSSR 248(5), 1049–1051 (1979)
Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190 (1991)
Pardalos, P.: Global optimization algorithms for linearly constrained indefinite quadratic problems. Comput. Math. Appl. 21, 87–97 (1991)
Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1(1), 15–22 (1991)
Vandenbussche, D., Nemhauser, G.: A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program. 102(3), 531–557 (2005)
Vandenbussche, D., Nemhauser, G.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3), 559–575 (2005)
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S. Burer was supported in part by NSF Grants CCR-0203426 and CCF-0545514.
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Burer, S., Vandenbussche, D. Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound. Comput Optim Appl 43, 181–195 (2009). https://doi.org/10.1007/s10589-007-9137-6
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DOI: https://doi.org/10.1007/s10589-007-9137-6