Skip to main content
SpringerLink
Log in

Global optimization of multi-parametric MILP problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we present a novel global optimisation approach for the general solution of multi-parametric mixed integer linear programs (mp-MILPs). We describe an optimisation procedure which iterates between a (master) mixed integer nonlinear program and a (slave) multi-parametric program. Moreover, we explain how to overcome the presence of bilinearities, responsible for the non-convexity of the multi-parametric program, in two classes of mp-MILPs, with (i) varying parameters in the objective function and (ii) simultaneous presence of varying parameters in the objective function and the right-hand side of the constraints. Examples are provided to illustrate the solution steps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acevedo J., Pistikopoulos E.N.: A multiparametric programming approach for linear process engineering problems under uncertainty. Ind. Eng. Chem. Res. 36, 717–728 (1997)

    Article  Google Scholar 

  2. Acevedo J., Salgueiro M.: An efficient algorithm for convex multiparametric nonlinear programming problems. Ind. Eng. Chem. Res. 42(23), 5883–5890 (2003)

    Article  Google Scholar 

  3. Adjiman C.S., Androulakis I.P., Floudas C.A.: A global optimization method, αbb, for general twice-differentiable constrained nlps. ii. implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179 (1998)

    Article  Google Scholar 

  4. Adjiman C.S., Dallwig S., Floudas C.A., Neumaier A.: A global optimization method, αbb, for general twice-differentiable constrained nlps. i. theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  5. Armacost R.L.: Computational experience in sensitivity analysis for nonlinear programming. Math. Program. 6, 301–326 (1974)

    Article  Google Scholar 

  6. Armacost, R.L.: Sensitivity analysis in parametric nonlinear programming. PhD thesis, School of Engineering and Applied Science, George Washington University, Washington, USA (1976)

  7. Arnold V.I.: Catastrophe Theory. Springer-Verlag, Berlin (1984)

    Google Scholar 

  8. Baotić M., Christophersen F.J., Morari M.: Constrained optimal control of hybrid systems with linear performance index. IEEE Trans. Autom. Control 51(12), 1903–1919 (2006)

    Article  Google Scholar 

  9. Bazaraa M.S., Shetty C.M.: Nonlinear Programming—Theory and Algorithms. Wiley, New York (1979)

    Google Scholar 

  10. Bemporad A., Filippi C.: Suboptimal explicit receding horizon control via approximate multiparametric quadratic programming. J. Optim. Theory Appl. 117(1), 9–38 (2003)

    Article  Google Scholar 

  11. Bemporad A., Morari M., Dua V., Pistikopoulos E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002)

    Article  Google Scholar 

  12. Bemporad A., Borrelli F., Morari M.: Min–max control of constrained uncertain discrete-time linear systems. IEEE Trans. Autom. Control 48(9), 1600–1606 (2003)

    Article  Google Scholar 

  13. Biegler L.T., Grossman I.E., Westerberg A.W.: Systematic Methods of Chemical Process Design. Prentice Hall, New Jersey (1997)

    Google Scholar 

  14. Borrelli F., Bemporad A., Morari M.: Geometric algorithm for multiparametric linear programming. J. Optim. Theory Appl. 118(3), 515–540 (2003)

    Article  Google Scholar 

  15. Borrelli F., Baotić M., Bemporad A., Morari M.: Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica 41, 1709–1721 (2005)

    Article  Google Scholar 

  16. Dua, V.: Parametric programming techniques for process engineering problems under uncertainty. PhD thesis. Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine London, London, UK (2000)

  17. Dua V., Pistikopoulos E.N.: Algorithms for the solution of multiparametric mixed-integer nonlinear optimization problems. Ind. Eng. Chem. Res 38, 3976–3987 (1999)

    Article  Google Scholar 

  18. Dua V., Pistikopoulos E.N.: An algorithm for the solution of multiparametric mixed integer linear programming problems. Ann. Oper. Res. 99, 123–139 (2000)

    Article  Google Scholar 

  19. Dua P., Pistikopoulos E.N.: Modelling and control of drug delivery systems. Comput. Chem. Eng. 29(11–12), 2290–2296 (2005)

    Article  Google Scholar 

  20. Dua V., Bozinis N.A., Pistikopoulos E.N.: A multiparametric programming approach for mixed-integer quadratic engineering problems. Comput. Chem. Eng. 26, 715–733 (2002)

    Article  Google Scholar 

  21. Dua V., Papalexandri K.P., Pistikopoulos E.N.: Global optimization issues in multiparametric continuous and mixed-integer optimization problems. J. Glob. Optim. 30, 59–89 (2004)

    Article  Google Scholar 

  22. Dua P., Doyle F.J., Pistikopoulos E.N.: Model-based blood glucose control for type 1 diabetes via parametric programming. IEEE Trans. Biomed. Eng. 53(8), 1478–1491 (2006)

    Article  Google Scholar 

  23. Faísca, N.P., Saraiva, P.M., Rustem, B., Pistikopoulos, E.N.: A multi-parametric programming approach for multi-level hierarchical and decentralised optimisation problems. Comput. Manag. Sci. (published electronically) (2007)

  24. Faísca N.P., Dua V., Saraiva P.M., Rustem B., Pistikopoulos E.N.: Parametric global optimisation for bilevel programming. J. Glob. Optim. 38(4), 609–623 (2007)

    Article  Google Scholar 

  25. Faísca N.P., Kouramas K.I., Saraiva P.M., Rustem B., Pistikopoulos E.N.: A multi-parametric programming approach for constrained dynamic programming problems. Optim. Lett. 2(2), 267–280 (2008)

    Article  Google Scholar 

  26. Fiacco A.V.: Sensitivity analysis for nonlinear programming using penalty methods. Math. Program. 10, 287–311 (1976)

    Article  Google Scholar 

  27. Fiacco A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)

    Google Scholar 

  28. Fiacco A.V., Kyparisis J.: Computable bounds on parametric solutions of convex problems. Math. Program. 40, 213–221 (1988)

    Article  Google Scholar 

  29. Filippi C.: An algorithm for approximate multiparametric linear programming. J. Optim. Theory Appl. 120(1), 73–95 (2004)

    Article  Google Scholar 

  30. Floudas C.A.: Deterministic Global Optimization. Kluwer, Dordrecht (2000)

    Google Scholar 

  31. Gal T.: Rim multiparametric linear programming. Manag. Sci. 21(5), 567–575 (1975)

    Article  Google Scholar 

  32. Gal T., Nedoma J.: Multiparametric linear programming. Math. Program. Stud. 18, 406–422 (1972)

    Google Scholar 

  33. Gass S.I., Saaty T.L.: Parametric objective function (part 2). J. Oper. Res. Soc. Am. 3(4), 395–401 (1954)

    Google Scholar 

  34. Kosmidis, V.S.: Multiparametric analysis of mixed integer linear programming problems. Master’s thesis, Imperial College London, University of London, London, UK (1999)

  35. Kouramas, K.I., Faísca, N.P., Rustem, B., Pistikopoulos, E.N.: Design of robust multi-parametric controllers for constrained multi-stage optimization problems. Automatica (submitted)(2008)

  36. Kyparisis J., Fiacco A.V.: Generalized convexity and concavity of the optimal value function in nonlinear programming. Math. Program. 39, 285–304 (1987)

    Article  Google Scholar 

  37. Li Z., Ierapetritou M.G.: A new methodology for the general mixed-integer linear programming (milp) problems. Ind. Eng. Chem. Res. 46, 5141–5151 (2007)

    Article  Google Scholar 

  38. Li Z., Ierapetritou M.G.: Process scheduling under uncertainty using multiparametric programming. AIChE J 53(12), 3183–3203 (2007)

    Article  Google Scholar 

  39. Lundberg B.N., Poore A.B.: Numerical continuation and singularity detection methods for parametric nonlinear programming. SIAM J. Optim. 3(1), 134–154 (1993)

    Article  Google Scholar 

  40. Mangasarian O.L., Fromovitz S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)

    Article  Google Scholar 

  41. McCormick, G.P.: Optimality criteria in nonlinear programming. In: SIAM-AMS Proceedings 1975, vol. 9, p. 27 (1976)

  42. Morari M., Barić M.: Rcent developments in the control of constrained hybrid systems. Comput. Chem. Eng. 30(10–12), 1619–1631 (2006)

    Article  Google Scholar 

  43. Pertsinidis, A.: On the parametric optimization of mathematical programs with binary variables and its application in the chemical engineering process synthesis. PhD thesis, Department of Chemical Engineering, Carnegie-Mellon University, Pittsburg, USA (1992)

  44. Pertsinidis A., Grossman I.E., McRae G.J.: Parametric optimization of milp programs and a framework for the parametric optimization of minlps. Comput. Chem. Eng. 22, S205–S212 (1998)

    Article  Google Scholar 

  45. Pistikopoulos E.N., Dua V., Bozinis N.A., Bemporad A., Morari M.: On-line optimization via off-line parametric optimization tools. Comput. Chem. Eng. 24, 183–188 (2000)

    Article  Google Scholar 

  46. Pistikopoulos E.N., Georgiadis M.C., Dua V.: Multi-Parametric Programming: Theory, Algorithms, and Applications, vol. 1. Wiley-VCH, Weinheim (2007)

    Google Scholar 

  47. Pistikopoulos E.N., Georgiadis M.C., Dua V.: Multi-Parametric Model-Based Control: Theory and Applications, vol. 2. Wiley-VCH, Weinheim (2007)

    Google Scholar 

  48. Pistikopoulos, E.N., Faísca, N.P., Georgiadis, M.C.: Advanced optimization techniques in the supply chain management of manufacturing industries. In: PulPaper 2007. Finish Paper Engineers’ Association, Helsinki, Finland (2007)

  49. Poore A.B., Tiahrt C.A.: Bifurcation problems in nonlinear parametric programming. Math. Program. 39, 189–205 (1987)

    Article  Google Scholar 

  50. Ryu J.H., Pistikopoulos E.N.: A novel approach to scheduling of zero-wait batch processes under processing time variations. Comput. Chem. Eng. 31(3), 101–106 (2007)

    Article  Google Scholar 

  51. Ryu J.H., Dua V., Pistikopoulos E.N.: A bilevel programming framework for enterprise-wide process networks under uncertainty. Comput. Chem. Eng. 28, 1121–1129 (2004)

    Article  Google Scholar 

  52. Saaty T.L., Gass S.I.: Parametric objective function (part 1). J. Oper. Res. Soc. Am. 2(3), 316–319 (1954)

    Google Scholar 

  53. Sahinidis N.V.: BARON: User’s Manual Version 4.0. Department of Chemical Engineering, University of Illinois, Urbana-Campaign (2000)

    Google Scholar 

  54. Sakizlis, V., Dua, V., Perkins, J.D., Pistikopoulos E.N.: The explicit control law for hybrid systems via parametric programming. In: Proceedings of the American Control Conference, Anchorage, AK, pp. 674–679 (2002)

  55. Sakizlis V., Perkins J.D., Pistikopoulos E.N.: Parametric controllers in simultaneous process and control design optimization. Ind. Eng. Chem. Res. 42, 4545–4563 (2003)

    Article  Google Scholar 

  56. Sakizlis V., Perkins J.D., Pistikopoulos E.N.: Recent advances in optimization-based simulataneous process and control design. Comput. Chem. Eng. 28, 2069–2086 (2004)

    Article  Google Scholar 

  57. Sakizlis V., Kakalis M.P., Dua V., Perkins J.D., Pistikopoulos E.N.: Design of robust controllers via parametric programming. Automatica 40, 189–201 (2004)

    Article  Google Scholar 

  58. Sakizlis V., Kakalis N.M.P., Dua V., Perkins J.D., Pistikopoulos E.N.: Design of robust model-based controllers via parametric programming. Automatica 40(2), 189–201 (2004)

    Article  Google Scholar 

  59. Sakizlis V., Perkins J.D., Pistikopoulos E.N.: Explicit solutions to optimal control problems for constrained continuous-time linear systems. IEE Proc. Control Theory Appl. 152(4), 443–452 (2005)

    Article  Google Scholar 

  60. Sakizlis, V., Kouramas, K.I., Faisca, N.P., Pistikopoulos, E.N.: Towards the design of parametric model predictive controllers for non-linear constrained systems. In: Assessment and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol. 358. Springer, Berlin (2007)

  61. Smith E.M.B., Pantelides C.C.: A symbolic reformulation: spatial branch-and-bound algorithm for the global optimisation of nonconvex minlps. Comput. Chem. Eng. 23, 457–478 (1999)

    Article  Google Scholar 

  62. Tøndel P., Johansen T.A., Bemporad A.: An algorithm for multi-parameffic quadratic programming and explicit mpc solutions. Automatica 39(3), 489–497 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Process Systems Engineering, Imperial College London, London, SW7 2AZ, UK

    N. P. Faísca, B. Rustem & E. N. Pistikopoulos

  2. EDF Trading Markets Limited, 71 High Holborn, London, WC1V 6ED, UK

    V. D. Kosmidis

Authors
  1. N. P. Faísca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. D. Kosmidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Rustem
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. N. Pistikopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. N. Pistikopoulos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Faísca, N.P., Kosmidis, V.D., Rustem, B. et al. Global optimization of multi-parametric MILP problems. J Glob Optim 45, 131–151 (2009). https://doi.org/10.1007/s10898-008-9398-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-008-9398-3

Keywords

Search

Navigation