Journal of Global Optimization

, Volume 33, Issue 2, pp 157–196 | Cite as

Linearity Embedded in Nonconvex Programs

  • Leo Liberti


Nonconvex programs involving bilinear terms and linear equality constraints often appear more nonlinear than they really are. By using an automatic symbolic reformulation we can substitute some of the bilinear terms with linear constraints. This has a dramatically improving effect on the tightness of any convex relaxation of the problem, which makes deterministic global optimization algorithms like spatial Branch-and-Bound much more eff- cient when applied to the problem.


Bilinear Convex relaxation Global optimization MINLP Reduction constraint Reformulation RLT 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adhya, N., Tawarmalani, M., Sahinidis, N. 1999A Lagrangian approach to the pooling problemIndustrial and Engineering Chemistry Research3819561972CrossRefGoogle Scholar
  2. 2.
    Adjiman, C. (1998), Global Optimization Techniques for Process Systems Engineering. Ph.D. thesis, Princeton University.Google Scholar
  3. 3.
    Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A. 1998A global optimization method, α BB, for general twice-differentiable constrained NLPs: I. Theoretical advancesComputers and Chemical Engineering2211371158CrossRefGoogle Scholar
  4. 4.
    Aho, A., Hopcroft, J., Ullman, J. 1983Data Structures and AlgorithmsAddison-WesleyReading, MAGoogle Scholar
  5. 5.
    Al-Khayyal, F., Falk, J. 1983Jointly constrained biconvex programmingMathematics of Operations Research8273286Google Scholar
  6. 6.
    Epperly, T. (1995), Global optimization of nonconvex nonlinear programs using parallel branch and bound. Ph.D. thesis, University of Winsconsin, Madison.Google Scholar
  7. 7.
    Epperly, T., Pistikopoulos, E. 1997A reduced space branch and bound algorithm for global optimizationJournal of Global Optimization11287311CrossRefGoogle Scholar
  8. 8.
    Gill, P. (1999), User's Guide for SNOPT 5.3. Systems Optimization Laboratory, Department of EESOR, Stanford University, California.Google Scholar
  9. 9.
    Kesavan, P., Barton, P. 2000Decomposition algorithms for nonconvex mixed-integer nonlinear programsAIChE Symposium Series96458461Google Scholar
  10. 10.
    Korte, B., Vygen, J. 2000Combinatorial Optimization, Theory and AlgorithmsSpringer-VerlagBerlinGoogle Scholar
  11. 11.
    Liberti, L. 2004Reduction constraints for the global optimization of NLPsInternational Transactions in Operations Research113341CrossRefGoogle Scholar
  12. 12.
    Liberti, L., Tsiakis, P., Keeping, B. and Pantelides, C. (2001),ooOPS , 1.24 ed., Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK.Google Scholar
  13. 13.
    McCormick, G. 1976Computability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problemsMathematical Programming10146175CrossRefGoogle Scholar
  14. 14.
    Ryoo, H.S., Sahinidis, N.V. 1995Global optimization of nonconvex NLPs and MINLPs with applications in process designComputers and Chemical Engineering19551566CrossRefGoogle Scholar
  15. 15.
    Sherali, H., Alameddine, A. 1992A new Reformulation–Linearization Technique for bilinear programming problemsJournal of Global Optimization2379410CrossRefGoogle Scholar
  16. 16.
    Sherali, H., Smith, J., Adams, W. 2000Reduced first-level representations via the reformulation–linearization technique: Results, counterexamplesand computations. Discrete Applied Mathematics101247267CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Smith, E., Pantelides, C. 1997Global optimisation of nonconvex MINLPsComputers and Chemical Engineering21S791S796Google Scholar
  19. 19.
    Smith, E., Pantelides, C. 1999A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPsComputers and Chemical Engineering23457478CrossRefGoogle Scholar
  20. 20.
    Vaidyanathan, R., El-Halwagi, M. 1996Global optimization of nonconvex MINLPs by interval analysisGrossmann, I. eds. Global Optimization in Engineering DesignKluwer Academic PublishersDordrecht175193Google Scholar
  21. 21.
    Wolsey, L. 1998Integer ProgrammingWileyNew YorkGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Leo Liberti
    • 1
  1. 1.Politecnico di MilanoDEIMilanoItaly

Personalised recommendations