Exact Algorithms for Global Optimization of Mixed-Integer Nonlinear Programs

  • Mohit Tawarmalani
  • Nikolaos V. Sahinidis
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 62)


This chapter presents recent advances in the development and application of global optimization algorithms for solving mixed-integer nonlinear programs (MINLPs). It is demonstrated that practically relevant nonlinear programs can be solved to global optimality in a completely automated fashion when carefully chosen relaxation schemes, branching strategies, and domain reduction techniques are are used in conjunction with branch and bound to enhance its performance. In particular, this chapter presents a) applications of the convex extensions theory for constructing tight relaxations, b) unifying ideas behind domain reduction schemes, c) linear outer-approximation schemes with proven convergence guarantees, and d) branching schemes for factorable nonlinear programs. The chapter concludes with computational results on some benchmark mixed-integer nonlinear problems. New solutions are reported for four of these problems.


Global Optimization Exact Algorithm Nonlinear Program Global Optimization Algorithm Convex Envelope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adjiman, C.S., Dallwig, S., Floudas, C.A., and Neumaier, A. (1998). A Global Optimization Method, aBB, for General Twice-Differentiable Constrained NLPs–I. Theoretical Advances. Computers Chemical Engineering, 22: 1137–1158.CrossRefGoogle Scholar
  2. Al-Khayyal, F.A. and Falk, J.E. (1983). Jointly Constrained Biconvex Programming. Mathematics of Operations Research, 8: 273–286.MathSciNetMATHCrossRefGoogle Scholar
  3. Andersen, D.E. and Andersen, K.D. (1995). Presolving in linear programming. Mathematical Programming, 71A: 221–245.MathSciNetMATHGoogle Scholar
  4. Balas, E. and Mazzola, J. (1984). Nonlinear 0–1 Programming: I. Linearization Techniques. Mathematical programming, 30: 1–21.MathSciNetMATHCrossRefGoogle Scholar
  5. Ben-Tal, A., Eiger, G., and Gershovitz, V. (1994). Global Minimization by Reducing the Duality Gap. Mathematical Programming, 63: 193–212.MathSciNetMATHCrossRefGoogle Scholar
  6. Bienstock, D. (1996). Computational Study of a Family of Mixed-Integer Quadratic Programming Problems. Mathematical Programmming, 74: 121–140.MathSciNetMATHGoogle Scholar
  7. Borchers, B. and Mitchell, J.E. (1994). An Improved Branch and Bound for Mixed Integer Nonlinear Programs. Comput. Oper. Res., 21. 359–367.MathSciNetMATHCrossRefGoogle Scholar
  8. Borchers, B. and Mitchell, J.E. (1997). A Computational Comparison of Branch and Bound and Outer Approximation Algorithms for 0–1 Mixed Integer Nonlinear Programs. Comput. Oper. Res., 24: 699–701.MathSciNetMATHCrossRefGoogle Scholar
  9. Burkard, R.E., Hamacher, H., and Rote, G. (1992). Sandwich Approximation of Univariate Convex Functions with an Application to Separable Convex Programming. Naval Research Logistics, 38: 911–924.MathSciNetGoogle Scholar
  10. Bussieck, M.R., Drud, A.S., and Meeraus, A. (2001). MINLPLib—A Collection of Test Models for Mixed-Integer Nonlinear Programming. Technical report, GAMS Development Corporation, Washington, D.C.Google Scholar
  11. Cheung, B. K-S., Langevin, A., and Delmaire, H. (1997). Coupling Genetic Algorithm with a Grid Search Method to Solve Mixed Integer Nonlinear Programming Problems. Comput. Math. Appl., 34: 13–23.MathSciNetMATHCrossRefGoogle Scholar
  12. Dakin, R.J. (1965). A Tree Search Algorithm for Mixed Integer Programming Problems. Computer Journal, 8: 250–255.MathSciNetMATHCrossRefGoogle Scholar
  13. Duran, M.A. and Grossmann, I.E. (1986). An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs Mathematical Programming, 36: 307–339.MathSciNetMATHCrossRefGoogle Scholar
  14. Epperly, T.G.W. and Swaney, R.E. (1996). Branch and Bound for Global NLP: New Bounding LP. In Grossmann, I. E., editor, Global Optimization in Engineering Design, chapter 1, pages 1–36. Kluwer Academic Publishers Inc., Dordrecht.Google Scholar
  15. Falk, J.E. and Soland, R.M. (1969). An Algorithm for Separable Non- convex Programming Problems. Management Science, 15: 550–569.MathSciNetMATHCrossRefGoogle Scholar
  16. Glover, F. and Woolsey, E. (1974). Converting a 0–1 Polynomial Programming Problem to a 0–1 Linear Program. Operations Research, 22: 180–182.MATHCrossRefGoogle Scholar
  17. Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley Publication Co., Reading, MA.Google Scholar
  18. Gupta, O.K. and Ravindran, A. (1985). Branch and Bound Experiments in Convex Nonlinear Integer Programming. Management Science, 31: 1533–1546.MathSciNetMATHCrossRefGoogle Scholar
  19. Hamed, A.S.E. and McCormick, G.P. (1993). Calculation of Bounds on Variables Satisfying Nonlinear Inequality Constraints. Journal of Global Optimization, 3: 25–47.MathSciNetMATHCrossRefGoogle Scholar
  20. Hansen, E.R. (1992). Global optimization using interval analysis. Pure and Applied Mathematics. Marcel Dekker, New York.Google Scholar
  21. Hansen, P., Jaumard, B., and Lu, S.-H. (1991). An Analytic Approach to Global Optimization. Mathematical Programming, 52: 227–254.MathSciNetMATHCrossRefGoogle Scholar
  22. Hillestad, R.J. and Jacobsen, S.E. (1980). Reverse Convex Programming. Applied Mathematics and Optimization, 6: 63–78.MathSciNetMATHCrossRefGoogle Scholar
  23. Hoffman, K. (1981). A Method for Globally Minimizing Concave Functions Over Convex Sets. Mathematical Programming, 20: 22–32.MathSciNetMATHCrossRefGoogle Scholar
  24. Horst, R., Thoai, N.V., and Tuy, H. (1989). On an Outer-Approximation in Global Optimization. Optimization, 20: 255–264.MathSciNetMATHCrossRefGoogle Scholar
  25. Horst, R. and Thy, H. (1996). Global Optimization: Deterministic Approaches. Springer Verlag, Berlin, Third edition.MATHGoogle Scholar
  26. Kearfott, R.B. (1996). Rigorous Global Search: Continuous Problems, volume 13 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht.Google Scholar
  27. Lamar, B.W. (1993). An Improved Branch and Bound Algorithm for Minimum Concave Cost Network Flow Problems. Journal of Global Optimization, 3: 261–287.MathSciNetMATHCrossRefGoogle Scholar
  28. Land, A.H. and Doig, A.G. (1960). An Automatic Method for Solving Discrete Programming Problems. Econometrica, 28: 497–520.MathSciNetMATHCrossRefGoogle Scholar
  29. Lazimy, R. (1982). Mixed-Integer Quadratic Programming. Mathematical Programmming, 22: 332–349.MathSciNetMATHCrossRefGoogle Scholar
  30. Lazimy, R. (1985). Improved Algorithm for Mixed-Integer Quadratic Programs and a Computational Study. Mathematical Programmming, 32: 100–113.MathSciNetMATHCrossRefGoogle Scholar
  31. Lee, E.K. and Mitchell, J. (1997). Computational experience in Nonlinear Mixed Integer Programming. In Zimmermann, U., editor, Operations Research Proceedings, Selected papers of the Symposium, SOR ‘96, Braunschweig, Germany, pages 95–100. Springer, Berlin.Google Scholar
  32. Li, H-L. and Chou, C-T. (1994). A Global Approach for Nonlinear Mixed Discrete Programming in Design Optimization. Engineering Optimization, 22: 109–122.CrossRefGoogle Scholar
  33. Mangasarian, O.L. and McLinden, L. (1985). Simple Bounds for Solutions of Monotone Complementarity Problems and Convex Programs. Mathematical Programming, 32: 32–40.MathSciNetMATHCrossRefGoogle Scholar
  34. Martelli, G. (1962). Jemmy Twitcher - A Life on the Fourth Earl of Sandwich, 1782–1792. Jonathan Cape, London.Google Scholar
  35. McBride, R.D. and Yormark, J.S. (1994). An Implicit Enumeration Algorithm for Quadratic Integer Programming. Management Science, 26: 282–296.MathSciNetCrossRefGoogle Scholar
  36. McCormick, G.P. (1976). Computability of Global Solutions to Factorable Nonconvex Programs: Part I–Convex Underestimating Problems. Mathematical Programming, 10: 147–175.MathSciNetMATHCrossRefGoogle Scholar
  37. McCormick, G.P. (1982). Nonlinear Programming: Theory, Algorithms and Applications. John Wiley and Sons.Google Scholar
  38. Nemhauser, G.L. and Wolsey, L.A. (1988). Integer and Combinatorial Optimization. Wiley Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons.MATHGoogle Scholar
  39. Rinnooy Kan, A.H.G. and Timmer, G.T. (1987a). Stochastic global optimization methods. I: Clustering methods. Mathematical Programming, 39: 27–56.MathSciNetMATHCrossRefGoogle Scholar
  40. Rinnooy Kan, A.H.G. and Timmer, G.T. (1987b). Stochastic global optimization methods. II: Multi level methods. Mathematical Programming, 39: 57–78.MathSciNetMATHCrossRefGoogle Scholar
  41. Rockafellar, R.T. and Wets, R.J.-B. (1998). Variational Analysis. A Series of Comprehensive Studies in Mathematics. Springer, Berlin.MATHCrossRefGoogle Scholar
  42. Rote, G. (1992). The Convergence Rate of the Sandwich Algorithm for Approximating Convex Functions. Computing, 48: 337–361.MathSciNetMATHCrossRefGoogle Scholar
  43. 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
  44. Ryoo, H.S. and Sahinidis, N.V. (1996). A Branch-and-Reduce Approach to Global Optimization. Journal of Global Optimization, 8: 107–139.MathSciNetMATHCrossRefGoogle Scholar
  45. Sahinidis, N.V. and Tawarmalani, M. (2000). Applications of Global Optimization to Process and Molecular Design. Computers é. Chemical Engineering, 24: 2157–2169.CrossRefGoogle Scholar
  46. Savelsbergh, M.W.P. (1994). Preprocessing and Probing for Mixed Inte- ger Programming Problems. ORSA J. on Computing, 6: 445–454.MathSciNetMATHCrossRefGoogle Scholar
  47. Schoen, F. (1991). Stochastic Techniques for Global Optimization: A Survey of Recent Advances. Journal of Global Optimization, 1: 207–228.MathSciNetMATHCrossRefGoogle Scholar
  48. 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.MathSciNetMATHCrossRefGoogle Scholar
  49. Sherali, H.D. (1997). Convex Envelopes of Multilinear Functions over a Unit Hypercube and over Special Discrete Sets. Acta Mathematica Vietnamica, 22: 245–270.MathSciNetMATHGoogle Scholar
  50. Sherali, H.D. and Adams, W.P. (1990). A Hierarchy of Relaxations between the Continuous and Convex Hull Representations for Zero-One Programming Problems. SIAM Journal of Discrete Mathematics, 3: 411–430.MathSciNetMATHCrossRefGoogle Scholar
  51. Tawarmalani, M., Ahmed, S., and Sahinidis, N.V. ((submitted 1999)). Global Optimization of 0–1 Hyperbolic Programs. Journal of Global Optimization. http://archimedes.scs.uiuc.edu/papers/fractional.pdf.Google Scholar
  52. Tawarmalani, M., Ahmed, S., and Sahinidis, N.V. ((submitted 2001)). Reformulations of Rational Functions of 0–1 Variables. Optimization and Engineering. http://archimedes.scs,uiuc.edu/papers/rational.pdf.Google Scholar
  53. Tawarmalani, M. and Sahinidis, N.V. (online). BARON on the Web. http://archimedes.scs.uiuc.edu/cgi/run.pl.Google Scholar
  54. Tawarmalani, M. and Sahinidis, N.V. (2001). Semidefinite Relaxations of Fractional Programs via Novel Techniques for Constructing Convex Envelopes of Nonlinear Functions. Journal of Global Optimization, 20: 137–158.MathSciNetMATHCrossRefGoogle Scholar
  55. Tawarmalani, M. and Sahinidis, N.V. ((submitted 1999)). Global Optimization of Mixed Integer Nonlinear Programs. A Theoretical and Computational Study. Mathematical Programming. http://archimedes.scs.uiuc.edu/papers/comp.pdf.Google Scholar
  56. Tawarmalani, M. and Sahinidis, N.V. ((submitted 2000)). Convex Extensions and Convex Envelopes of l.s.c. Functions. Mathematical Programming. Can be downloaded from http://archimedes.scs.uiuc.edu/papers/extensions.pdf.Google Scholar
  57. Thakur, L.S. (1990). Domain Contraction in Nonlinear Programming: Minimizing a Quadratic Concave Function over a Polyhedron. Mathematics of Operations Research, 16: 390–407.MathSciNetCrossRefGoogle Scholar
  58. Törn, A. and Zilinskas, A. (1989). Global Optimization. Lecture Notes in Computer Science, Vol. 350. Springer-Verlag, Berlin.Google Scholar
  59. Tuy, H. (1964). Concave Programming Under Linear Constraints. Doklady Akademic Nauk, 159: 32–35.Google Scholar
  60. Tuy, H. (1985). Concave Programming Under Linear Constraints with Special Structure. Optimization, 16: 335–352.MathSciNetMATHCrossRefGoogle Scholar
  61. Tuy, H. (1987). Global Optimization of a Difference of two Convex Functions. Mathematical Programming Study, 30: 150–182.MATHCrossRefGoogle Scholar
  62. Tuy, H., Thieu, T.V., and Thai, N.Q. (1985). A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set. Mathematics of Operations Research, 10: 498–514.MathSciNetMATHCrossRefGoogle Scholar
  63. Visweswaran, V. and Floudas, C.A. (1993). New Properties and Computational Improvement of the GOP Algorithm for Problems with Quadratic Objective Functions and Constraints. Journal of Global Optimization, 3: 439–462.MathSciNetMATHCrossRefGoogle Scholar
  64. Wu, S-J. and Chow, P-T. (1995). Genetic Algorithms for Nonlinear Mixed Discrete-Integer Optimization Problems Via Meta-Genetic Parameter Optimization. Engineering Optimization, 24: 137–159.CrossRefGoogle Scholar
  65. Zabinsky, Z.B. (1998). Stochastic methods for practical global optimization. Journal of Global Optimization, 13: 433–444.MathSciNetMATHCrossRefGoogle Scholar
  66. Zamora, J.M. and Grossmann, I.E. (1999). A Branch and Contract Algorithm for problems with Concave Univariate, Bilinear and Linear Fractional Terms. Journal of Global Optimization, 14: 217–249.MathSciNetMATHCrossRefGoogle Scholar
  67. Zhang, C. and Wang, H-P. (1993). Mixed-Discrete Nonlinear Optimization with Simulated Annealing. Engineering Optimization, 21: 277–291.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Mohit Tawarmalani
    • 1
  • Nikolaos V. Sahinidis
    • 2
  1. 1.Krannert School of ManagementPurdue UniversityWest LafayetteUSA
  2. 2.Department of Chemical EngineeringUniversity of Illinois at Urbana-ChampaignUSA

Personalised recommendations