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Journal of Global Optimization

, Volume 72, Issue 2, pp 305–321 | Cite as

Decomposition-based Inner- and Outer-Refinement Algorithms for Global Optimization

  • Ivo Nowak
  • Norman Breitfeld
  • Eligius M. T. Hendrix
  • Grégoire Njacheun-Njanzoua
Article

Abstract

Traditional deterministic global optimization methods are often based on a Branch-and-Bound (BB) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present a new deterministic decomposition-based successive approximation method for general modular and/or sparse MINLPs. The new method, called Decomposition-based Inner- and Outer-Refinement, is based on a block-separable reformulation of the model into sub-models. It generates inner- and outer-approximations using column generation, which are successively refined by solving many easier MINLP and MIP subproblems in parallel (using BB), instead of searching over one (global) BB search tree. We present preliminary numerical results with Decogo (Decomposition-based Global Optimizer), a new parallel decomposition MINLP solver implemented in Python and Pyomo.

Keywords

Global optimization Decomposition method MINLP Successive approximation Column generation 

References

  1. 1.
    Adjiman, C.S., Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)-BB. http://titan.princeton.edu (2002)
  2. 2.
    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-Tal, A., Eiger, G., Gershovitz, V.: Global minimization by reducing the duality gap. Math. Program. 63, 193–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borndörfer, R., Löbel, A., Reuther, M., Schlechte, T., Weider, S.: Rapid branching. Public Transp. 5, 3–23 (2013)CrossRefGoogle Scholar
  6. 6.
    Burer, S., Letchford, A.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)MathSciNetGoogle Scholar
  7. 7.
    Bussieck, M. R., Vigerske, S.: MINLP solver software. http://www.math.hu-berlin.de/~stefan/minlpsoft.pdf (2014)
  8. 8.
    Desrosiers, J., Lübbecke, M.: Branch-price-and-cut algorithms. In: Cochran, J., Cox, L., Keskinocak, P., Kharoufeh, J., Smith, J. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)Google Scholar
  9. 9.
    Desrosiers, J., Lübbecke, M.E.: Selected topics in column generation. Oper. Res. 53, 1007–1023 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Domschke, P., Geißler, B., Kolb, O., Lang, J., Martin, A., Morsi, A.: Combination of nonlinear and linear optimization of transient gas networks. INFORMS J. Comput. 23, 605–617 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duran, M., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Engineer, F., Nemhauser, G., Savelsbergh, M.: Shortest path based column generation on large networks with many resource constraints. Technical report, Georgia Tech (2008)Google Scholar
  13. 13.
    Feltenmark, S., Kiwiel, Krzysztof C.: Dual applications of proximal bundle methods including lagrangian relaxation of nonconvex problems. SIAM J. Optim. 10(3), 697–721 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66(3(A)), 327–349 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Flippo, O.E., Rinnoy Kan, A.H.G.: Decomposition in general mathematical programming. Math. Program. 60, 361–382 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Geoffrion, A.M.: General Benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Geoffrion, A.M.: Lagrangian relaxation for integer programming. Math. Program. Stud. 2, 82–114 (1974)CrossRefGoogle Scholar
  18. 18.
    Goderbauer, S., Bahl, B., Voll, P., Lübbecke, M., Bardow, A., Koster, A.: An adaptive discretization MINLP algorithm for optimal synthesis of decentralized energy supply systems. Comput. Chem. Eng. 95, 38–48 (2016)CrossRefGoogle Scholar
  19. 19.
    Hart, W., Laird, C., Watson, J.P., Woodruff, D.: Pyomo—Optimization Modeling in Python, vol. 67. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  20. 20.
    Houska, B., Frasch, J., Diehl, M.: An augmented Lagrangian based algorithm for distributed non-convex optimization. http://www.optimization-online.org/DB_HTML/2014/07/4427.html (2014)
  21. 21.
    Koch, T., Ralphs, T., Shinano, Y.: What could a million cores do to solve integer programs? Math. Methods Oper. Res. 76, 67–93 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kojima, M., Matsumoto, T., Shid, M.: Moderate nonconvexity = convexity + quadratic concavity. Technical report. http://www.is.titech.ac.jp/~kojima/sdp.html (1999)
  23. 23.
    Lemaréchal, C., Renaud, A.: A geometric study of duality gaps, with applications. Math. Program. 90, 399–427 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Leyffer, S., Sartenaer, A., Wanufelle, E.: Branch-and-refine for mixed integer nonconvex global optimization. Technical report, Preprint ANL/MCS-P1547-0908, Mathematics and Computer Science Division, Argonne National Laboratory (2008)Google Scholar
  25. 25.
    Lin, Y., Schrage, L.: The global solver in the LINDO API. Optim. Methods Softw. 24, 657–668 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Misener, R., Floudas, C.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59, 503–526 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nowak, I.: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  28. 28.
    Nowak, I.: A dynamic reduce and generate approach for airline crew scheduling. GERAD International Workshop on Column Generation, Aussois. http://www.gerad.ca/colloques/ColumnGeneration2008/slides/IvoNowak.pdf (2008)
  29. 29.
    Nowak, I.: Parallel decomposition methods for nonconvex optimization—recent advances and new directions. In: Proceedings of MAGO (2014)Google Scholar
  30. 30.
    Nowak, I.: Column generation based alternating direction methods for solving MINLPs. http://www.optimization-online.org/DB_HTML/2015/12/5233.html (2015)
  31. 31.
    Ralphs, T., Galati, M.: Decomposition and dynamic cut generation in integer linear programming. Math. Program. 106(2), 261–285 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tawarmalani, M., Sahinidis, N.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    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 (2002)CrossRefzbMATHGoogle Scholar
  34. 34.
    Vigerske, S.: Decomposition in multistage stochastic programming and a constraint integer programming approach to mixed-integer nonlinear programming. Ph.D. thesis, Humboldt-Universität zu Berlin (2012)Google Scholar
  35. 35.
    Vigerske, S.: MINLP Library 2. http://www.gamsworld.org/minlp/minlplib2/html (2017)
  36. 36.
    Wächter, A.: An interior point algorithm for large-scale nonlinear optimization with applications in process engineering. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, USA. http://researcher.watson.ibm.com/researcher/files/us-andreasw/thesis.pdf (2002)
  37. 37.
    Westerlund, T., Petterson, F.: An extended cutting plane method for solving convex MINLP problems. Compu. Chem. Eng. 21, 131–136 (1995)CrossRefGoogle Scholar
  38. 38.
    Yuan, X., Zhang, S., Piboleau, L., Domenech, S.: Une methode d’optimisation nonlineare en variables mixtes pour la conception de procedes. RAIRO 22, 331 (1988)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hamburg University of Applied SciencesHamburgGermany
  2. 2.Technische Universität BerlinBerlinGermany
  3. 3.Universidad de MálagaMálagaSpain
  4. 4.Wageningen UniversityWageningenThe Netherlands

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