Transforming Mathematical Models Using Declarative Reformulation Rules

  • Antonio Frangioni
  • Luis Perez Sanchez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6683)


Reformulation is one of the most useful and widespread activities in mathematical modeling, in that finding a “good” formulation is a fundamental step in being able so solve a given problem. Currently, this is almost exclusively a human activity, with next to no support from modeling and solution tools. In this paper we show how the reformulation system defined in [13] allows to automatize the task of exploring the formulation space of a problem, using a specific example (the Hyperplane Clustering Problem). This nonlinear problem admits a large number of both linear and nonlinear formulations, which can all be generated by defining a relatively small set of general Atomic Reformulation Rules (ARR). These rules are not problem-specific, and could be used to reformulate many other problems, thus showing that a general-purpose reformulation system based on the ideas developed in [13] could be feasible.


Column Generation Nonlinear Formulation Reformulation System Track Structure Discrete Apply Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  2. 2.
    Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0-1 programming problems. Journal of Optimization Theory and Applications 93(2), 273–300 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben Amor, H., Desrosiers, J., Frangioni, A.: On the Choice of Explicit Stabilizing Terms in Column Generation. Discrete Applied Mathematics 157(6), 1167–1184 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bjorkqvist, J., Westerlund, T.: Automated reformulation of disjunctive constraints in minlp optimization. Computers and Chemical Engineering 23, S11–S14 (1999)CrossRefGoogle Scholar
  5. 5.
    Desaulniers, G., Desrosiers, J., Solomon, M.M. (eds.): Column generation. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  6. 6.
    Frangioni, A., Gendron, B.: 0-1 Reformulations of the Multicommodity Capacitated Network Design Problem. Discrete Applied Mathematics 157(6), 1229–1241 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Frangioni, A., Gentile, C.: SDP Diagonalizations and Perspective Cuts for a Class of Nonseparable MIQP. Operations Research Letters 35(2), 181–185 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frangioni, A., Scutellà, M.G., Necciari, E.: A Multi-exchange Neighborhood for Minimum Makespan Machine Scheduling Problems. Journal of Combinatorial Optimization 8, 195–220 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Judice, J., Mitra, G.: Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods. Journal of Optimization Theory and Applications 57(1), 123–149 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liberti, L.: Reformulation techniques in mathematical programming, in preparation. Thèse d’Habilitation à Diriger des Recherches, Université Paris IXGoogle Scholar
  11. 11.
    Liberti, L.: Reformulations in mathematical programming: Definitions and systematics. RAIRO-RO 43(1), 55–86 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liberti, L., Cafieri, S., Tarissan, F.: Reformulations in mathematical programming: a computational approach. In: Abraham, A., Hassanien, A.-E., Siarry, P., Engelbrecht, A. (eds.) Foundations of Computational Intelligence. SCI, vol. 3, pp. 153–234. Springer, Berlin (2009)Google Scholar
  13. 13.
    Sanchez, L.P.: Artificial Intelligence Techniques for Automatic Reformulation and Solution of Structured Mathematical Models. PhD thesis, University of Pisa (2010)Google Scholar
  14. 14.
    Sherali, D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Dodrecht (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sherali, H.: Personal communication (2007)Google Scholar
  16. 16.
    van Roy, T.J., Wolsey, L.A.: Solving mixed integer programming problems using automatic reformulation. Operations Research 35(1), 45–57 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yang, G., Kifer, M., Wan, H., Zhao, C.: Flora-2: User’s ManualGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Antonio Frangioni
    • 1
  • Luis Perez Sanchez
    • 2
  1. 1.Dipartimento di InformaticaUniversità di Pisa, Polo Universitario della SpeziaLa SpeziaItaly
  2. 2.Dipartimento di InformaticaUniversità di PisaPisaItaly

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