Reformulations for utilizing separability when solving convex MINLP problems



Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.


Convex MINLP Lifted polyhedral approximation Separable MINLP Extended cutting plane algorithm Extended supporting hyperplane algorithm Outer approximation 



Financial support from the Finnish Graduate School in Chemical Engineering is gratefully acknowledged, as is support from GAMS Development Corporation.


  1. 1.
    Balas, E.: Projection, lifting and extended formulation in integer and combinatorial optimization. Ann. Oper. Res. 140(1), 125–161 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berenguel, J.L., Casado, L., García, I., Hendrix, E.M., Messine, F.: On interval branch-and-bound for additively separable functions with common variables. J. Glob. Optim. 56(3), 1101–1121 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., et al.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5(2), 186–204 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boyd, S., Kim, S.J., Vandenberghe, L., Hassibi, A.: A tutorial on geometric programming. Optim. Eng. 8(1), 67 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Bussieck, M.R., Vigerske, S.: MINLP solver software. Wiley Encyclopedia of Operations Research and Management Science (2010)Google Scholar
  7. 7.
    Candes, E., Romberg, J.: l1-Magic: Recovery of Sparse Signals Via Convex Programming. (2005). Accessed 1 Dec 2016
  8. 8.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8(1), 1–48 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8(3), 250–255 (1965)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36(3), 307–339 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66(1–3), 327–349 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Floudas, C.A.: Deterministic global optimization: theory, methods and applications, vol. 37. Springer Science & Business Media (2013).
  13. 13.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3–38 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    GAMSWorld: Mixed-Integer Nonlinear Programming Library (2016). Accessed 24 Nov 2016
  15. 15.
    Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Griewank, A., Toint, P.L.: On the unconstrained optimization of partially separable functions. Nonlinear Optim. 1982, 247–265 (1981)MATHGoogle Scholar
  17. 17.
    Grossmann, I., Viswanathan, J., Vecchietti, A., Raman, R., Kalvelagen, E.: DICOPT. Engineering Research Design Center. GAMS Development Corporation, Pittsburgh (2009)Google Scholar
  18. 18.
    Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3(3), 227–252 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Grossmann, I.E., Kravanja, Z.: Mixed-integer nonlinear programming: a survey of algorithms and applications. In: Biegler, L.T., Coleman, T.F., Conn, A.R., Santosa, F.N. (eds.) Large-Scale Optimization with Applications, pp. 73–100. Springer (1997).
  20. 20.
    Gurobi: Gurobi 6.5 Performance Benchmarks (2015).
  21. 21.
    Hijazi, H., Bonami, P., Ouorou, A.: An outer-inner approximation for separable MINLPs. LIF, Faculté des Sciences de Luminy, Université de Marseille, Technical Report (2010)Google Scholar
  22. 22.
    Kronqvist, J., Lundell, A., Westerlund, T.: The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64(2), 249–272 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kronqvist, J., Lundell, A., Westerlund, T.: Lifted polyhedral approximations in convex mixed integer nonlinear programming. In: XIII Global Optimization Workshop GOW16 4–8 Sept 2016, vol. 16, pp. 117–120 (2016)Google Scholar
  24. 24.
    Lastusilta, T., Bussieck, M.R., Westerlund, T.: An experimental study of the GAMS/AlphaECP MINLP solver. Ind. Eng. Chem. Res. 48(15), 7337–7345 (2009)CrossRefGoogle Scholar
  25. 25.
    Lee, J., Leyffer, S. (eds.): Mixed Integer Nonlinear Programming, vol. 154. Springer, Berlin (2011)Google Scholar
  26. 26.
    Li, H.L., Tsai, J.F., Floudas, C.A.: Convex underestimation for posynomial functions of positive variables. Optim. Lett. 2(3), 333–340 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lubin, M., Yamangil, E., Bent, R., Vielma, J.P.: Extended formulations in mixed-integer convex programming. In: International Conference on Integer Programming and Combinatorial Optimization, pp. 102–113. Springer (2016)Google Scholar
  28. 28.
    Lundell, A.: Transformation Techniques for Signomial Functions in Global Optimization. Ph.D. thesis, Åbo Akademi University (2009)Google Scholar
  29. 29.
    Lundell, A., Skjäl, A., Westerlund, T.: A reformulation framework for global optimization. J. Glob. Optim. 57(1), 115–141 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Nowak, I.: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming, vol. 152. Springer, Berlin (2006)Google Scholar
  31. 31.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  32. 32.
    Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8(2), 201–205 (1996)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, vol. 65. Springer, Berlin (2002)MATHGoogle Scholar
  34. 34.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Vielma, J.P., Dunning, I., Huchette, J., Lubin, M.: Extended formulations in mixed integer conic quadratic programming. arXiv preprint arXiv:1505.07857 (2015)
  36. 36.
    Vinel, A., Krokhmal, P.A.: Polyhedral approximations in \(p\)-order cone programming. Optim. Methods Softw. 29(6), 1210–1237 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Westerlund, T., Petterson, F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, S131–S136 (1995)CrossRefGoogle Scholar
  38. 38.
    Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3(3), 253–280 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Jan Kronqvist
    • 1
  • Andreas Lundell
    • 2
  • Tapio Westerlund
    • 1
  1. 1.Process Design and Systems EngineeringÅbo Akademi UniversityTurkuFinland
  2. 2.Mathematics and StatisticsÅbo Akademi UniversityTurkuFinland

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