Journal of Global Optimization

, Volume 36, Issue 2, pp 161–189 | Cite as

An Exact Reformulation Algorithm for Large Nonconvex NLPs Involving Bilinear Terms

  • Leo Liberti
  • Constantinos C. Pantelides


Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not affect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an efficient graph-theoretical algorithm for effecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branch-and Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.


Bilinear Convex relaxation Global optimization NLP Reformulation-linearization technique RRLT constraints 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams W., Lassiter J., Sherali H. (1998), Persistency in 0–1 polynomial programming. Mathematics of Operations Research 2(23): 359–389Google Scholar
  2. Adhya N., Tawarmalani M., Sahinidis N. (1999), A Lagrangian approach to the pooling problem. Industrial and Engineering Chemistry Research 38, 1956–1972CrossRefGoogle Scholar
  3. Al-Khayyal F., Falk J. (1983), Jointly constrained biconvex programming. Mathematics of Operations Research 8, 273–286Google Scholar
  4. Audet C., Brimberg J., Hansen P., Le Digabel S., Mladenović N. (2004), Pooling problem: alternate formulations and solution methods. Management Science 50(6): 761–776CrossRefGoogle Scholar
  5. Ben-Tal A., Eiger G., Gershovitz V. (1994), Global minimization by reducing the duality gap. Mathematical Programming 63, 193–212CrossRefGoogle Scholar
  6. Duff I. (1981), On algorithms for obtaining a maximum transversal. ACM Transactions Mathematical Software 7, 315–330CrossRefGoogle Scholar
  7. Foulds L., Haughland D., Jornsten K. (1992), A bilinear approach to the pooling problem. Optimization 24, 165–180CrossRefGoogle Scholar
  8. Goodaire E., Parmenter M. (1998), Discrete Mathematics with Graph Theory. Prentice-Hall, LondonGoogle Scholar
  9. Harary F. (1971), Graph Theory, 2nd ed., Addison-Wesley, Reading, MAGoogle Scholar
  10. Hopcroft J.E., Karp R.M. (1973), An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2, 225–231CrossRefGoogle Scholar
  11. Korte B., Vygen J. (2000), Combinatorial Optimization, Theory and Algorithms. Springer-Verlag, BerlinGoogle Scholar
  12. Liberti L., Pantelides C. (2003), Convex envelopes of monomials of odd degree. Journal of Global Optimization 25, 157–168CrossRefGoogle Scholar
  13. McCormick G. (1976), Computability of global solutions to factorable nonconvex programs. Part I. Convex underestimating problems. Mathematical Programming 10, 146–175Google Scholar
  14. Pantelides C. (1988), The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9, 213–231CrossRefGoogle Scholar
  15. Pardalos P., Romeijn H. ed. (2002). Handbook of Global Optimization, Vol. 2. Kluwer Academic Publishers, DordrechtGoogle Scholar
  16. Quesada I., Grossmann I. (1995), Global optimization of bilinear process networks and multicomponent flows. Computers and Chemical Engineering 19(12): 1219–1242CrossRefGoogle Scholar
  17. Sahinidis N., Tawarmalani M. (2005), Accelerating branch-and-bound through a modeling language construct for relaxation-specific constraints. Journal of Global Optimization. 32(2): 259–280CrossRefGoogle Scholar
  18. Sherali, H. (2002), Tight relaxations for nonconvex optimization problems using the reformulation-linearization/convexification technique (RLT). In (Pardalos and Romeijn, 2002), pp. 1–63.Google Scholar
  19. Sherali H., Adams W. (1986), A tight linearization and an algorithm for 0–1 quadratic programming problems. Management Science 32(10): 1274–1290CrossRefGoogle Scholar
  20. Sherali H., Adams W. (1999). A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, DordrechtGoogle Scholar
  21. Sherali H., Alameddine A. (1992), A new reformulation-linearization technique for bilinear programming problems. Journal of Global Optimization 2, 379–410CrossRefGoogle Scholar
  22. Sherali H., Tuncbilek C. (1997), New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems. Operations Research Letters 21, 1–9CrossRefGoogle Scholar
  23. Smith, E. (1996), On the optimal design of continuous processes. Ph.D. thesis, Imperial College of Science, Technology and Medicine, University of London.Google Scholar
  24. Smith E., Pantelides C. (1999), A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Computers & Chemical Engineering 23, 457–478CrossRefGoogle Scholar
  25. Tawarmalani M., Sahinidis N. (2002a), Convexification and Global Optimization in continuous and mixed-integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, DordrechtGoogle Scholar
  26. Tawarmalani M., Sahinidis, N. (2002b), Exact algorithms for global optimization of mixed-integer nonlinear programs. In (Pardalos and Romeijn, 2002), pp. 1–63.Google Scholar
  27. Tawarmalani M., Sahinidis N. (2004), Global optimization of mixed integer nonlinear programs: a theoretical and computational study. Mathematical Programming 99, 563–591CrossRefGoogle Scholar
  28. Tuy H. (1998), Convex Analysis and Global Optimization. Kluwer Academic Publishers, DordrechtGoogle Scholar
  29. Visweswaran V., Floudas C. (1993), New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints. Journal of Global Optimization 3, 439-462CrossRefGoogle Scholar
  30. Visweswaran V., Floudas C. (1996), New formulations and branching strategies for the GOP algorithm. In: Grossmann I.(eds). Global Optimization in Engineering Design. Kluwer Academic Publishers, DordrechtGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.CNRS LIXÉcole PolytechniquePalaiseauFrance
  2. 2.Centre for Process Systems Engineering, Department of Chemical Engineering and Chemical TechnologyImperial College LondonLondonUK

Personalised recommendations