Abstract
We develop eight different mixed-integer convex programming reformulations of 0-1 hyperbolic programs. We obtain analytical results on the relative tightness of these formulations and propose a branch and bound algorithm for 0-1 hyperbolic programs. The main feature of the algorithm is that it reformulates the problem at every node of the search tree. We demonstrate that this algorithm has a superior convergence behavior than directly solving the relaxation derived at the root node. The algorithm is used to solve a discrete p-choice facility location problem for locating ten restaurants in the city of Edmonton.
Similar content being viewed by others
References
Agrawal, S.C. (1977), An alternative method of integer solutions to linear fractional functionals by a Bbranch and bound technique. Z. Angew. Math. Mech. 57: 52–53.
Al-Khayyal, F.A. and Falk, J.E. (1983), Jointly constrained biconvex programming. Mathematics of Operations Research 8: 273–286.
Arora, S.R., K. Swarup, K. and Puri, M.C. (1977), The set covering problem with linear fractional functional. Indian Journal of Pure and Applied Mathematics 8: 578–588.
Blum, M., Floyd, R.W., Pratt, V., Rivest R.L. and Tarjan, R.E. (1973), Time bounds for selection. Journal of Computer and System Sciences 7: 448–461.
Brook, A., Kendrick, D. and Meeraus, A. (1988), GAMS–A User's Guide. Scientific Press, Redwood City, CA.
Charnes, A. and Cooper, W.W. (1962), Programming with linear fractional functionals. Naval Research Logistics Quarterly 9: 181–186.
CPLEX. (1997), CPLEX 6.0 User's Manual. ILOG CPLEX Division, Incline Village, NV.
Dorneich, M.C. and Sahinidis, N.V. (1995), Global optimization algorithms for chip layout and compaction. Engineering Optimization 25: 131–154.
Falk, J.E. and Polocsay, S.W. (1994), Image space analysis of generalized fractional programs. Journal of Global Optimization 4: 63–88.
Ghildyal, V. and Sahinidis, N.V. (2001), Solving global optimization problems with BARON. In: Migdalas, A., Pardalos, P., Varbrand, P. and Holmqvist, K. (eds.), From Local to Global Optimization. A Workshop on the Occasion of the 70th Birthday of Professor Hoang Tuy, Kluwer Academic Publishers, Boston, MA.
Ghosh, A. and McLafferty, S. (1987), Location Strategies for Retail and Service Firms. Lexington Books, Massachusetts.
Ghosh, A., McLafferty, S. and Craig, S. (1995), Multifacility retail networks. In: Drezner Z. (ed.), Facility Location: A Survey of Applications and Methods, Springer, New York, pp. 301–330.
Gilmore, P.C. and Gomory, R.E. (1963), A linear programming approach to the cutting stock problem – Part II. Operations Research 11: 52–53.
Granot, D. and Granot, F. (1976), On solving fractional (0 ? 1) programs by implicit enumeration. INFOR 14: 241–249.
Granot, D. and Granot, F. (1977), On integer and mixed integer fractional programming problems. Annals of Discrete Mathematics 1: 221–231.
Grunspan, M. and Thomas, M.E. (1973), Hyperbolic integer programming. Naval Research Logistics Quarterly 20: 341–356.
Gutierrez, R.A. and Sahinidis, N.V. (1996), A branch-and-bound approach for machine selection in just-in-time manufacturing systems. International J. Production Research 34: 797–818.
Hammer, P.L. and Rudeanu, S. (1968), Boolean Methods in Operations Research and Related Areas. Springer, New York.
Hansen, P., de Aragao, M.V.P. and Ribeiro, C.C. (1991), Hyperbolic 0 ? 1 programming and query optimization in information retrieval. Mathematical Programming 52: 255–263.
Hansen, P., Jaumard, B. and Mathon, V. (1993), Constrained nonlinear 0-1 programming. ORSA Journal of Computing 5: 87–119.
Haque, M.A. and Ahmed, S. (1998), p-Choice facility location in discrete space. in preparation.
Hashizume, S., Fukushima, M., Katoh, N. and Ibaraki, T. (1987), Approximation algorithms for combinatorial fractional programming problems. Mathematical Programming 37: 255–267.
Hiriart-Urruty, J. and Lemaréchal, C. (1993), Convex Analysis and Minimization Algorithms I. Springer, Berlin.
Li, H. (1994), A global approach for general 0 ? 1 fractional programming. European Journal of Operational Research 73: 590–596.
Liu, M.L., Sahinidis N.V. and Shectman, J.P. (1996), Planning of chemical process networks via global concave minimization. In: Grossmann I.E. (ed.), Global Optimization in Engineering Design. Kluwer Academic Publishers, Boston, MA. Chapt. 7, pp. 195–230.
McCormick, G.P. (1982), Nonlinear Programming: Theory, Algorithms and Applications. John Wiley and Sons, New York.
Megiddo, N. (1979), Combinatorial optimization with rational objective functions. Mathematics of Operations Research 4: 414–424.
Murtagh, B.A. and Saunders, M.A. (1995), MINOS 5.4 User's Guide. Technical Report SOL 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, CA.
Nakanishi, M. and Cooper, L.G. (1974), Parameter estimate for multiplicative interactive choice models: least squares approach. Journal of Marketing Research 11: 303–311.
OSL. (1995), Optimization subroutine library guide and reference release 2.1. International Business Machines Corporation, Kingston, NY, fifth edition.
Quesada, I. and Grossmann, I.E. (1995), A global optimization algorithm for linear fractional and bilinear programs. Journal of Global Optimization 6: 39–76.
Robillard, P. (1971), (0, 1) Hyperbolic programming problems. Naval Research Logistics Quarterly 18: 47–57.
Ryoo, H.S. and Sahinidis, N.V. (1995), Global optimization of nonconvex NLPs and MINLPs with applications in process design. Computers & Chemical Engineering 19: 551–566.
Ryoo, H.S. and Sahinidis, N.V. (1996), A branch-and-reduce approach to global optimization. Journal of Global Optimization 8: 107–139.
Sahinidis, N.V. (1996), BARON: A general purpose global optimization software package. Journal of Global Optimization 8: 201–205.
Sahinidis, N.V. and Tawarmalani, M. (2000), Applications of global optimization to process and molecular design. Computers & Chemical Engineering 24: 2157–2169.
Saipe, A.L. (1975), Solving a (0, 1) hyperbolic program by branch and bound. Naval Research Logistics Quarterly 22: 497–515.
Schaible, S. (1995), Fractional Programming. In: Horst, R. and Pardalos, P.M. (eds.) Handbook of Global Optimization. Kluwer Academic Publishers, Norwell, Massachusetts. pp. 495–608.
Schaible, S. (1996), Fractional programming with sums of ratios. working paper 96-04, A.G. Anderson Graduate School of Management, University of California, Riverside.
Shectman, J.P. and Sahinidis, N.V. (1998), A finite algorithm for global minimization of separable concave programs. Journal of Global Optimization 12: 1–36.
Stancu-Minasian, I.M. (1997), Fractional Programming. Kluwer Academic Publishers, Dordrecht.
Tawarmalani, M., Ahmed S. and Sahinidis, N.V. (submitted 2001), Product disaggregation in global optimation and an application to rational programs Optimization and Engineering.
Tawarmalani, M. and N.V. Sahinidis, N.V. (accepted 2002), Convex extensions and convex envelopes of l.s.c. functions. Mathematical Programming.
VanAntwerp, J.G., Braatz, R.D. and Sahinidis, N.V. (1999), Globally optimal robust control. Journal of Process Control pp. 375–383.
Williams, H.P. (1974), Experiments in the formulation of integer programming problems. Mathematical Programming Study 2: 180–197.
Wu, T. (1997), A Note on a global approach for general 0-1 fractional programming. European Journal of Operational Research 101 220–223.
Author information
Authors and Affiliations
Additional information
The research was supported in part by NSF awards DMII 95-02722 and BES 98-73586 to NVS.
Rights and permissions
About this article
Cite this article
Tawarmalani, M., Ahmed, S. & Sahinidis*, N.V. Global Optimization of 0-1 Hyperbolic Programs. Journal of Global Optimization 24, 385–416 (2002). https://doi.org/10.1023/A:1021279918708
Issue Date:
DOI: https://doi.org/10.1023/A:1021279918708