Journal of Global Optimization

, Volume 24, Issue 4, pp 385–416 | Cite as

Global Optimization of 0-1 Hyperbolic Programs

  • Mohit Tawarmalani
  • Shabbir Ahmed
  • Nikolaos V. Sahinidis*

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.

Fractional programming Convex extensions Facility location 

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References

  1. 1.
    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.Google Scholar
  2. 2.
    Al-Khayyal, F.A. and Falk, J.E. (1983), Jointly constrained biconvex programming. Mathematics of Operations Research 8: 273–286.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    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.Google Scholar
  5. 5.
    Brook, A., Kendrick, D. and Meeraus, A. (1988), GAMS–A User's Guide. Scientific Press, Redwood City, CA.Google Scholar
  6. 6.
    Charnes, A. and Cooper, W.W. (1962), Programming with linear fractional functionals. Naval Research Logistics Quarterly 9: 181–186.Google Scholar
  7. 7.
    CPLEX. (1997), CPLEX 6.0 User's Manual. ILOG CPLEX Division, Incline Village, NV.Google Scholar
  8. 8.
    Dorneich, M.C. and Sahinidis, N.V. (1995), Global optimization algorithms for chip layout and compaction. Engineering Optimization 25: 131–154.Google Scholar
  9. 9.
    Falk, J.E. and Polocsay, S.W. (1994), Image space analysis of generalized fractional programs. Journal of Global Optimization 4: 63–88.Google Scholar
  10. 10.
    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.Google Scholar
  11. 11.
    Ghosh, A. and McLafferty, S. (1987), Location Strategies for Retail and Service Firms. Lexington Books, Massachusetts.Google Scholar
  12. 12.
    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.Google Scholar
  13. 13.
    Gilmore, P.C. and Gomory, R.E. (1963), A linear programming approach to the cutting stock problem – Part II. Operations Research 11: 52–53.Google Scholar
  14. 14.
    Granot, D. and Granot, F. (1976), On solving fractional (0 ? 1) programs by implicit enumeration. INFOR 14: 241–249.Google Scholar
  15. 15.
    Granot, D. and Granot, F. (1977), On integer and mixed integer fractional programming problems. Annals of Discrete Mathematics 1: 221–231.Google Scholar
  16. 16.
    Grunspan, M. and Thomas, M.E. (1973), Hyperbolic integer programming. Naval Research Logistics Quarterly 20: 341–356.Google Scholar
  17. 17.
    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.Google Scholar
  18. 18.
    Hammer, P.L. and Rudeanu, S. (1968), Boolean Methods in Operations Research and Related Areas. Springer, New York.Google Scholar
  19. 19.
    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.Google Scholar
  20. 20.
    Hansen, P., Jaumard, B. and Mathon, V. (1993), Constrained nonlinear 0-1 programming. ORSA Journal of Computing 5: 87–119.Google Scholar
  21. 21.
    Haque, M.A. and Ahmed, S. (1998), p-Choice facility location in discrete space. in preparation.Google Scholar
  22. 22.
    Hashizume, S., Fukushima, M., Katoh, N. and Ibaraki, T. (1987), Approximation algorithms for combinatorial fractional programming problems. Mathematical Programming 37: 255–267.Google Scholar
  23. 23.
    Hiriart-Urruty, J. and Lemaréchal, C. (1993), Convex Analysis and Minimization Algorithms I. Springer, Berlin.Google Scholar
  24. 24.
    Li, H. (1994), A global approach for general 0 ? 1 fractional programming. European Journal of Operational Research 73: 590–596.Google Scholar
  25. 25.
    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.Google Scholar
  26. 26.
    McCormick, G.P. (1982), Nonlinear Programming: Theory, Algorithms and Applications. John Wiley and Sons, New York.Google Scholar
  27. 27.
    Megiddo, N. (1979), Combinatorial optimization with rational objective functions. Mathematics of Operations Research 4: 414–424.Google Scholar
  28. 28.
    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.Google Scholar
  29. 29.
    Nakanishi, M. and Cooper, L.G. (1974), Parameter estimate for multiplicative interactive choice models: least squares approach. Journal of Marketing Research 11: 303–311.Google Scholar
  30. 30.
    OSL. (1995), Optimization subroutine library guide and reference release 2.1. International Business Machines Corporation, Kingston, NY, fifth edition.Google Scholar
  31. 31.
    Quesada, I. and Grossmann, I.E. (1995), A global optimization algorithm for linear fractional and bilinear programs. Journal of Global Optimization 6: 39–76.Google Scholar
  32. 32.
    Robillard, P. (1971), (0, 1) Hyperbolic programming problems. Naval Research Logistics Quarterly 18: 47–57.Google Scholar
  33. 33.
    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.Google Scholar
  34. 34.
    Ryoo, H.S. and Sahinidis, N.V. (1996), A branch-and-reduce approach to global optimization. Journal of Global Optimization 8: 107–139.Google Scholar
  35. 35.
    Sahinidis, N.V. (1996), BARON: A general purpose global optimization software package. Journal of Global Optimization 8: 201–205.Google Scholar
  36. 36.
    Sahinidis, N.V. and Tawarmalani, M. (2000), Applications of global optimization to process and molecular design. Computers & Chemical Engineering 24: 2157–2169.Google Scholar
  37. 37.
    Saipe, A.L. (1975), Solving a (0, 1) hyperbolic program by branch and bound. Naval Research Logistics Quarterly 22: 497–515.Google Scholar
  38. 38.
    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.Google Scholar
  39. 39.
    Schaible, S. (1996), Fractional programming with sums of ratios. working paper 96-04, A.G. Anderson Graduate School of Management, University of California, Riverside.Google Scholar
  40. 40.
    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.Google Scholar
  41. 41.
    Stancu-Minasian, I.M. (1997), Fractional Programming. Kluwer Academic Publishers, Dordrecht.Google Scholar
  42. 42.
    Tawarmalani, M., Ahmed S. and Sahinidis, N.V. (submitted 2001), Product disaggregation in global optimation and an application to rational programs Optimization and Engineering.Google Scholar
  43. 43.
    Tawarmalani, M. and N.V. Sahinidis, N.V. (accepted 2002), Convex extensions and convex envelopes of l.s.c. functions. Mathematical Programming.Google Scholar
  44. 44.
    VanAntwerp, J.G., Braatz, R.D. and Sahinidis, N.V. (1999), Globally optimal robust control. Journal of Process Control pp. 375–383.Google Scholar
  45. 45.
    Williams, H.P. (1974), Experiments in the formulation of integer programming problems. Mathematical Programming Study 2: 180–197.Google Scholar
  46. 46.
    Wu, T. (1997), A Note on a global approach for general 0-1 fractional programming. European Journal of Operational Research 101 220–223.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Mohit Tawarmalani
    • 1
  • Shabbir Ahmed
    • 2
  • Nikolaos V. Sahinidis*
    • 3
  1. 1.Krannert School of ManagementPurdue UniversityWest LafayetteUSA
  2. 2.School of Industrial & Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of Chemical EngineeringThe University of Illinois at Urbana-ChampaignUrbanaUSA

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