Skip to main content
SpringerLink
Log in

Nonconvex Generalized Benders Decomposition

  • Chapter
  • First Online:
Optimization in Science and Engineering

Abstract

This chapter gives an overview of an extension of Benders decomposition (BD) and generalized Benders decomposition (GBD) to deterministic global optimization of nonconvex mixed-integer nonlinear programs (MINLPs) in which the complicating variables are binary. The new decomposition method, called nonconvex generalized Benders decomposition (NGBD), is developed based on convex relaxations of nonconvex functions and continuous relaxations of non-complicating binary variables in the problem. NGBD guarantees finding an ε-optimal solution or indicates the infeasibility of the problem in a finite number of steps. A typical application of NGBD is to solve large-scale stochastic MINLPs that cannot be solved via the decomposition procedures of BD and GBD. Case studies of several industrial problems demonstrate the dramatic computational advantage of NGBD over state-of-the-art commercial solvers.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238–252 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  2. Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Geoffrion, A.M.: Elements of large-scale mathematical programming: Part I: concepts. Manag. Sci. 16(11), 652–675 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  4. Geoffrion, A.M.: Elements of large-scale mathematical programming: Part II: synthesis of algorithms and bibliography. Manag. Sci. 16(11), 652–675 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  5. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I - convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, α-BB, for general twice-differentiable constrained NLPs – I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  8. Gatzke, E.P., Tolsma, J.E., Barton, P.I.: Construction of convex relaxations using automated code generation technique. Optim. Eng. 3, 305–326 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mitsos, A., Chachuat, B., Barton, P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim. 20(2), 573–601 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. IBM. IBM ILOG CPLEX Optimization Studio. http://www-03.ibm.com/software/products/us/en/ibmilogcpleoptistud/ Accessed April 13, 2014

  12. ARKI Consulting and Development. http://www.gams.com/docs/conopt3.pdf Accessed April 13, 2014

  13. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47, 99–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lemaréchal, C., Sagastizábal, C.: Variable metric bundle methods: from conceptual to implementable forms. Math. Program. 76, 393–410 (1997)

    Article  MATH  Google Scholar 

  15. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Cambridge (1999)

    MATH  Google Scholar 

  16. Geoffrion, A.M.: Duality in nonlinear programming: a simplified applications-oriented development. SIAM Rev. 13(1), 1–37 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 2nd edn. Wiley, New York (1993)

    MATH  Google Scholar 

  18. Balas, E., Jeroslow, R.: Canonical cuts on the unit hypercube. SIAM J. Appl. Math. 23(1), 61–69 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  19. Grossmann, I.E., Raman, R., Kalvelagen, E.: DICOPT User’s Manual. http://www.gams.com/dd/docs/solvers/dicopt.pdf Accessed April 13, 2014

  20. Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Floudas, C.A.: Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  22. Bieglerm, L.T., Grossmann, I.E.: Retrospective on optimization. Comput. Chem. Eng. 28, 1169–1192 (2004)

    Article  Google Scholar 

  23. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)

    MATH  Google Scholar 

  24. Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  25. Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: Global optimization of mixed-integer nonlinear problems. AIChE J. 46(9), 1769–1797 (2000)

    Article  Google Scholar 

  26. Kesavan, P., Allgor, R.J., Gatzke, E.P., Barton, P.I.: Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Math. Program. Ser. A 100, 517–535 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Westerlund, T., Pettersson, F., Grossmann, I.E.: Optimization of pump configurations as a MINLP problem. Comput. Chem. Eng. 18(9), 845–858 (1994)

    Article  Google Scholar 

  28. Li, X., Tomasgard, A., Barton, P.I.: Nonconvex generalized Benders decomposition for stochastic separable mixed-integer nonlinear programs. J. Optim. Theory Appl. 151, 425–454 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Selot, A., Kuok, L.K., Robinson, M., Mason, T.L., Barton, P.I.: A short-term operational planning model for natural gas production systems. AIChE J. 54(2), 495–515 (2008)

    Article  Google Scholar 

  30. Selot, A.: Short-Term Supply Chain Management in Upstream Natural Gas Systems. Ph.D. thesis, Massachusetts Institute of Technology (2009)

    Google Scholar 

  31. Sundaramoorthy, A., Evans, J.M.B., Barton, P.I.: Capacity planning under clinical trials uncertainty in continuous pharmaceutical manufacturing, 1: mathematical framework. Ind. Eng. Chem. Res. 51, 13692–13702 (2012)

    Article  Google Scholar 

  32. Sundaramoorthy, A., Li, X., Evans, J.M.B., Barton, P.I.: Capacity planning under clinical trials uncertainty in continuous pharmaceutical manufacturing, 2: solution method. Ind. Eng. Chem. Res. 51, 13703–13711 (2012)

    Article  Google Scholar 

  33. GAMS. General Algebraic and Modeling System. http://www.gams.com/ Accessed April 13, 2014

  34. Li, X., Chen, Y., Barton, P.I.: Nonconvex generalized Benders decomposition with piecewise convex relaxations for global optimization of integrated process design and operation problems. Ind. Eng. Chem. Res. 51, 7287–7299 (2012)

    Article  Google Scholar 

  35. Li, X.: Parallel nonconvex generalized Benders decomposition for natural gas production network planning under uncertainty. Comput. Chem. Eng. 55, 97–108 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada, K7L 3N6

    Xiang Li

  2. Praxair, Inc., Business and Supply Chain Optimization, Tonawanda, NY, 14150, USA

    Arul Sundaramoorthy

  3. Process Systems Engineering Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Paul I. Barton

Authors
  1. Xiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arul Sundaramoorthy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul I. Barton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul I. Barton .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Princeton University Dept. Chemical Engineering, Princeton, New Jersey, USA

    Christodoulos A. Floudas

  3. Dept. Industrial & Systems Engineering, Texas A&M University, College Station, Texas, USA

    Sergiy Butenko

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Li, X., Sundaramoorthy, A., Barton, P.I. (2014). Nonconvex Generalized Benders Decomposition. In: Rassias, T., Floudas, C., Butenko, S. (eds) Optimization in Science and Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0808-0_16

Download citation

Publish with us

Policies and ethics

Search

Navigation