Journal of Global Optimization

, Volume 43, Issue 2–3, pp 277–297 | Cite as

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

  • Steffen RebennackEmail author
  • Josef Kallrath
  • Panos M. Pardalos


We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.


MINLP Column enumeration Decomposition Packing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adjiman C.S., Androulakis I.P. and Floudas C.A. (1997). Global optimization of MINLP problems in process synthesis and design. Comput. Chem. Eng. 21(Suppl. S): S445–S450 Google Scholar
  2. Adjiman C.S., Androulakis I.P. and Floudas C.A. (2000). Global optimization of mixed-integer nonlinear problems. AICHE J. 46(9): 1769–1797 CrossRefGoogle Scholar
  3. Androulakis I.P., Maranas C.D. and Floudas C.A. (1995). αBB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4): 337–363 CrossRefGoogle Scholar
  4. Dowsland K.A. and Dowsland W.B. (1992). Packing Problems. Euro. J. Oper. Res. 56: 2–14 CrossRefGoogle Scholar
  5. Dyckhoff H. (1990). A Typology of Cutting and Packing Problems. Euro. J. Oper. Res. 44: 145–159 CrossRefGoogle Scholar
  6. Fraser H.J. and George J.A. (1994). Integrated Container Loading Software for Pulp and Paper Industry. Euro. J. Oper. Res. 77: 466–474 CrossRefGoogle Scholar
  7. Floudas, C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications, vol. 37 of Nonconvex Optimization and Its Applications, pp. 309–554. Kluwer Academic Publishers (2000)Google Scholar
  8. George J.A., George J.M. and Lamar B.W. (1995). Packing Different-sized Circles into a Rectangular Container. Euro. J. Oper. Res. 84: 693–712 CrossRefGoogle Scholar
  9. Huang W.Q., Li Y., Akeb H. and Li C.M. (2005). Greedy Algorithms for Packing Unequal Circles into a Rectangular Container. J. Oper. Res. Soc. 56(5): 539–548 CrossRefGoogle Scholar
  10. Kallrath, J.: Online Storage Systems and Transportation Problems with Applications: Optimization Models and Mathematical Solutions, vol. 91 of Applied Optimization, pp. 92–104. Kluwer Academic Publishers, Norwell, MA (2004)Google Scholar
  11. Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Glob. Optim (2008). doi:  10.1007/s10898-007-9274-6 CrossRefGoogle Scholar
  12. Lindo Systems: Lindo API: User’s Manual. Lindo Systems, Inc., Chicago (2004)Google Scholar
  13. Lenstra J.K. and Rinnooy Kan A.H.G. (1979). Complexity of Packing, Covering and Partitioning Problems. In: Schrijver, A. (eds) Packing and Covering in Combinatorics, pp 275–291. Mathematisch Centrum, Amsterdam Google Scholar
  14. Liberti, L., Maculan, N. (eds.): Global Optimization: From Theory to Implementation, vol. 84 of Nonconvex Optimization and Its Applications, pp. 223–232. Springer (2006)Google Scholar
  15. Pintér, J.D.: Continuous global optimization software: A brief review. Optima, 52, 1–8 (1996a). See also
  16. Pintér J.D. Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications, vol. 6 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers (1996b)Google Scholar
  17. Pardalos P.M., Resende M.G.C. (eds.): Handbook of Applied Optimization pp. 337–351. Oxford University Press (2002)Google Scholar
  18. Ryoo H.S. and Sahinidis N.V. (1995). Global optimization of non-convex NLPs and MINLPs with application in process design. Comput. Chem. Eng. 19(5): 551–566 CrossRefGoogle Scholar
  19. Ryoo H.S. and Sahinidis N.V. (1996). A branch-and-reduce approach to global optimization. J. Glo. Optim. 8(2): 107–138 CrossRefGoogle Scholar
  20. Sahinidis N.V. (1996). BARON: A general purpose global optimization software package. J. Glob. Optim. 8(2): 201–205 CrossRefGoogle Scholar
  21. Schrage L. (2006). Optimization Modeling with LINGO. LINDO Systems, Inc., Chicago, IL Google Scholar
  22. Stoyan Y.G. and Yaskov G.N. (1998). Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints. Int. Trans. Oper. Res. 5(1): 45–57 Google Scholar
  23. Stoyan Y.G. and Yaskov G.N. (2004). A mathematical model and a solution method for the problem of placing various-sized circles into a strip. Euro. J. Oper. Res. 156: 590–600 CrossRefGoogle Scholar
  24. Wilhelm W. E. (2001). A technical review of column generation in integer programming. Optim. Eng. 2: 159–200 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  • Steffen Rebennack
    • 1
    Email author
  • Josef Kallrath
    • 2
  • Panos M. Pardalos
    • 1
  1. 1.Center of Applied OptimizationUniversity of FloridaGainesvilleUSA
  2. 2.Department of AstrononyUniversity of FloridaGainesvilleUSA

Personalised recommendations