Abstract
In this paper several equivalent formulations for the quadratic binary programming problem are presented. Based on these formulations we describe four different kinds of strategies for estimating lower bounds of the objective function, which can be integrated into a branch and bound algorithm for solving the quadratic binary programming problem. We also give a theoretical explanation for forcing rules used to branch the variables efficiently, and explore several properties related to obtained subproblems. From the viewpoint of the number of subproblems solved, new strategies for estimating lower bounds are better than those used before. A variant of a depth-first branch and bound algorithm is described and its numerical performance is presented.
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P.L. De Angells, I.M. Bomze, and G. Toraldo, “Ellipsoidal approach to box-constrained quadratic problems,” Jouranl of Global Optimization, vol. 28, no. 1, pp. 1–15, 2004.
F. Barahona, M. Jünger, and G. Reinelt, “Experiments in quadratic 0–1 programming,” Mathematical Programming, vol. 44, pp. 127–137, 1989.
A. Beck and M. Teboulle, “Global optimality conditions for quadratic optimization problems with binary constraints,” SIAM Journal on Optimization, vol. 11, no. 1, pp. 179–188, 2000.
G. Gallo, P.L. Hammer, and B. Simeone, “Quadratic knapsack problems,” Mathematical Programming, vol. 12, pp. 132–149, 1980.
M.X. Goemans and D.P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” Journal of the Association for Computing Machinery, vol. 42, pp. 1115–1145, 1995.
P.L. Hammer and B. Simeone, “Order relations of variables in 0–1 programming,” Annals of Discrete Mathematics, vol. 31, pp. 83–112, 1987.
P. Hansen, “Methods of nonlinear 0–1 programming,” Annals of Discrete Mathematics, vol. 5, pp. 53–70, 1979.
R. Horst, P.M. Pardalos, and N.V. Thoai, “Introduction to Global Optimization,” 2nd edition, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
H.X. Huang, P.M. Pardalos, and O. Prokopyev, “Multi-quadratic binary programming,” Technical Report, University of Florida, 2004.
L.D. Iasemidis, P.M. Pardalos, J.C. Sackellares, and D.S. Shiau, “Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures,” Journal of Combinatorial Optimization, vol. 5, no. 1, pp. 9–26, 2001.
J.L. Klepeis, C.A. Floudas, D. Morikis, C.G. Tsokos, and J.D. Lambris, “Design of peptide analogues with improved activity using a novel de novo protein design approach,” Industrial & Engineering Chemistry Research, vol. 43, no. 14, pp. 3817–3826, 2004.
F. Körner, “An efficient branch and bound algorithm to solve the quadratic integer programming problem,” Computing, vol. 30, pp. 253–260, 1983.
J. Krarup and P.A. Pruzan, “Computer aided layout design,” Mathematical Programming Study, vol. 9, pp. 75–94, 1978.
P. Lewis, A.S. Goodman, and J.M. Miller, “Psudo-random number generator for the system/360,” IBM Systems Journal, vol. 8, no. 2, pp. 300–312, 1969.
R.D. McBride and J.S. Yormark, “An implicit enumeration algorithm for quadratic integer programming,” Management Science, vol. 26, no. 3, pp. 282–296, 1980.
K. Miettinen, Nonlinear Multiobjective Optimization. International Series in Operations Research and Management Science, vol. 12. Kluwer Academic Publishers, Norwell, Massachusetts, 1999.
Yu. E. Nesterov, “Quality of semidefinite relaxation for nonconvex quadratic optimization,” CORE Discussion Paper 9719, Belgium, March, 1997.
P.M. Pardalos, “Construction of test problems in quadratic bivalent programming,” ACM Transactions on Mathematical Software, vol. 17, no. 1, pp. 74–87, 1991.
P.M. Pardalos and S. Jha, “Complexity of uniqueness and local search in quadratic 0–1 programming,” Operations Research Letters, vol. 11, no. 2, pp. 119–123, 1992.
P.M. Pardalos and G.P. Rodgers, “Computational aspects of a branch and bound algorithm for quadratic zero-one programming,” Computing, vol. 45, pp. 131–144, 1990.
R.T. Rockafellar, Convex Analysis. Princeton University Press: Princeton, NJ, 1970.
Y. Ye, “Approximating quadratic programming with bound and quadratic constraints,” Mathematical Programming, vol. 84, no. 2, pp. 219–226, 1999.
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Huang, HX., Pardalos, P.M. & Prokopyev, O.A. Lower Bound Improvement and Forcing Rule for Quadratic Binary Programming. Comput Optim Applic 33, 187–208 (2006). https://doi.org/10.1007/s10589-005-3062-3
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DOI: https://doi.org/10.1007/s10589-005-3062-3