Mathematical Programming

, Volume 103, Issue 2, pp 225–249 | Cite as

A polyhedral branch-and-cut approach to global optimization

  • Mohit Tawarmalani
  • Nikolaos V. Sahinidis


A variety of nonlinear, including semidefinite, relaxations have been developed in recent years for nonconvex optimization problems. Their potential can be realized only if they can be solved with sufficient speed and reliability. Unfortunately, state-of-the-art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software.

In this paper, we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch-and-cut approach. Our algorithm exploits convexity, either identified automatically or supplied through a suitable modeling language construct, in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems. We prove that, if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions, our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function. The convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited.

Root-node relaxations are computed for 87 problems from globallib and minlplib, and detailed computational results are presented for globally solving 26 of these problems with BARON 7.2, which implements the proposed techniques. The use of cutting planes for these problems reduces root-node relaxation gaps by up to 100% and expedites the solution process, often by several orders of magnitude.


Mixed-integer nonlinear programming Outer approximation Convexification Factorable programming Convexity identification 


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  1. 1.
    Audet, C., Hansen, P., Jaumard, B., Savard, G. : A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Math. Prog. 87, 131–152 (2000)Google Scholar
  2. 2.
    Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum Press, 1988Google Scholar
  3. 3.
    Böröczky K.Jr., Reitzner,M.: Approximation of smooth convex bodies by random circumscribed polytopes. Annals of Applied Probability 14, 239–273 (2004)CrossRefGoogle Scholar
  4. 4.
    Bussieck, M.R.: MINLP World. 2002
  5. 5.
    Chinneck, J.W.: Discovering the characteristics of mathematical programming via sampling. Optimization Methods and Software 17, 319–352 (2002)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Prog. 36, 307–339 (1986)Google Scholar
  7. 7.
    Fourer, R., Moré, J., Munson, T., Sarich, J.: Next-generation servers for optimization as an internet resource. Available at 2004
  8. 8.
    Griewank, A.: Evaluating derivatives. Principles and Techniques of Algorithmic Differentiation, vol 19 of Frontiers in Applied Mathematics. SIAM, Philadelphia, PA, 2000Google Scholar
  9. 9.
    Griffith, R.E., Stewart, R.A.: A nonlinear programming technique for the optimization of continuous processing systems. Management Science 7, 379–392 (1961)Google Scholar
  10. 10.
    Gruber, P.M.: Asymptotic estimates for best and stepwise approximation of convex bodies II. Forum Mathematicum 5, 521–538 (1993)Google Scholar
  11. 11.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin, 1993Google Scholar
  12. 12.
    Kelley, J.E.: The cutting plane method for solving convex programs. Journal of the SIAM 8, 703–712 (1960)Google Scholar
  13. 13.
    Maheshwari, C., Neumaier, A., Schichl, H.: Convexity and concavity detection. Available at 2003
  14. 14.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems. Math. Prog. 10, 147–175 (1976)CrossRefGoogle Scholar
  15. 15.
    Meeraus, A.: GLOBAL World. 2002
  16. 16.
    Nowak, I.: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Habilitation thesis Humboldt-Universität zu Berlin, Germany, 2004Google Scholar
  17. 17.
    Rote, G.: The convergence rate of the sandwich algorithm for approximating convex functions. Computing 48, 337–361 (1992)Google Scholar
  18. 18.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel techniques for constructing convex envelopes of nonlinear functions. Journal of Global Optimization 20, 137–158 (2001)CrossRefGoogle Scholar
  19. 19.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and convex envelopes of l.s.c. functions. Mathematical Programming 93, 247–263 (2002)CrossRefGoogle Scholar
  20. 20.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Dordrecht, 2002Google Scholar
  21. 21.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Prog. 99, 563–591 (2004)CrossRefGoogle Scholar
  22. 22.
    Vandenbussche, D.: Polyhedral Approaches to Solving Nonconvex Quadratic Programs. PhD thesis, Georgia Institute of Technology, Department of Indystrial and Systems Engineering, Atlanta, GA, 2003Google Scholar
  23. 23.
    Zamora, J.M., Grossmann, I.E.: A global MINLP optimization algorithm for the synthesis of heat exchanger networks with no stream splits. Computers & Chemical Engineering 22, 367–384 (1998)Google Scholar
  24. 24.
    Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. Journal of Global Optimization 14, 217–249 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Krannert School of ManagementPurdue UniversityWest LafayetteUSA
  2. 2.Department of Chemical and Biomolecular EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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