On solving generalized convex MINLP problems using supporting hyperplane techniques

  • Tapio Westerlund
  • Ville-Pekka Eronen
  • Marko M. Mäkelä
Article
  • 32 Downloads

Abstract

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke’s subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable \(f^{\circ }\)-quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an \(f^{\circ }\)-pseudoconvex function, subject to \(f^{\circ }\)-pseudoconvex constraints. With some additional conditions, the proof is also valid for \(f^{\circ }\)-quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443–459, 2017) to also solve problems with \(f^{\circ }\)-pseudoconvex objective function.

Keywords

Nonsmooth optimization Mixed-integer nonlinear programming Generalized convexities Supporting hyperplanes Cutting planes 

Mathematics Subject Classification

90C11 90C25 

Notes

Acknowledgements

This research was supported by the Grant No. 294002 of the Academy of Finland. The authors also acknowledge GAMS Development Corporation for providing us license to use different GAMS solvers.

References

  1. 1.
    Androulakis, I., Maranas, C., Floudas, C.A.: \(\alpha \)BB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bagirov, A., Mäkelä, M.M., Karmitsa, N.: Introduction to Nonsmooth Optimization: Theory Practice and Software. Springer International Publishing, Cham, Heidelberg (2014)CrossRefMATHGoogle Scholar
  3. 3.
    Bonami, P., Kilinc, M., Linderoth, J.: Algorithms and software for convex mixed-integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Programming, The IMA Volumes in Mathematics and Its Applications, pp. 1–39. Springer, New York (2012)Google Scholar
  4. 4.
    Bussieck, M.R., Vigerske, S.: MINLP solver software. In: Wiley Encyclopedia of Operations Research and Management Science. Wiley (2011).  https://doi.org/10.1002/9780470400531.eorms0527
  5. 5.
    Cambini, A., Martein, L.: Generalized convexity and optimization—theory and applications. In: Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2009)Google Scholar
  6. 6.
    Castillo, I., Westerlund, J., Emet, S., Westerlund, T.: Optimization of block layout design problems with unequal areas: a comparison of MILP and MINLP optimization methods. Comput. Chem. Eng. 30, 54–69 (2005)CrossRefGoogle Scholar
  7. 7.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  8. 8.
    de Oliveira, W.: Regularized optimization methods for convex MINLP problems. TOP 24, 665–692 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Eronen, V.-P., Mäkelä, M.M., Westerlund, T.: On the generalization of ECP and OA methods to nonsmooth MINLP problems. Optimization 63(7), 1057–1073 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eronen, V.-P., Mäkelä, M.M., Westerlund, T.: Extended cutting plane method for a class of nonsmooth nonconvex MINLP problems. Optimization 64(3), 641–661 (2015)MathSciNetMATHGoogle Scholar
  12. 12.
    Eronen, V.-P., Kronqvist, J., Westerlund, T., Mäkelä, M.M., Karmitsa, N.: Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems. J. Glob. Optim. 69(2), 443–459 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fletcher, R., Leyffer, S.: Numerical experience with lower bounds for MIQP branch-and-bound. SIAM J. Optim. 8, 604–616 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1973)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jain, V., Grossmann, I.: Cyclic scheduling of continuous parallel-process units with decaying performance. AIChE J. 44, 1623–1636 (1999)Google Scholar
  18. 18.
    Kelley, J.E.: The cutting plane method for solving convex programs. J. SIAM 8, 703–712 (1960)MathSciNetMATHGoogle Scholar
  19. 19.
    Kronqvist, J., Lundell, A., Westerlund, T.: The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64, 249–272 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lee, J., Leyffer, S.: Mixed Integer Nonlinear Programming. Springer, New York (2012)CrossRefMATHGoogle Scholar
  21. 21.
    Leyffer, S.: Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lundell, A., Skjäl, A., Westerlund, T.: A reformulation framework for global optimization. J. Glob. Optim. 57, 115–141 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Meyer, C.A., Floudas, C.A.: Convex underestimation of twice continuously differential functions by piecewise quadratic perturbations: spline \(\alpha \)BB underestimators. J. Glob. Optim. 32, 221–258 (2005)CrossRefMATHGoogle Scholar
  24. 24.
    Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific Publishing Co., Singapore (1992)CrossRefMATHGoogle Scholar
  25. 25.
    Nestorov, Y., Nemirowskii, A.: Interior-point polynomial algorithms in convex programming. In: SIAM Studies in Applied Mathematics, vol. 13. Philadelphia (1994)Google Scholar
  26. 26.
    Pörn, R.: Mixed-Integer Non-Linear Programming: Convexification Techniques and Algorithm Development. Ph.D. Thesis, Åbo Akademi University (2000)Google Scholar
  27. 27.
    Quesada, I., Grossmann, I.E.: An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1999)CrossRefGoogle Scholar
  28. 28.
    Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York, London (1973)MATHGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (1997)Google Scholar
  30. 30.
    Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–138 (1996)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Veinott Jr., A.F.: The supporting hyperplane method for unimodal programming. Oper. Res. 15(1), 147–152 (1967)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Westerlund, T., Skrifvars, H., Harjunkoski, I., Pörn, R.: An extended cutting plane method for solving a class of non-convex MINLP problems. Comput. Chem. Eng. 22, 357–365 (1998)CrossRefMATHGoogle Scholar
  33. 33.
    Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Westerlund, T., Pettersson, F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, 131–136 (1995)CrossRefGoogle Scholar
  35. 35.
    Westerlund, T.: User’s guide for GAECP, version 5.537. An Interactive Solver for Generalized Convex MINLP-Problems Using Cutting Plane and Supporting Hyperplane Techniques. Åbo Akademi University. www.abo.fi/~twesterl/GAECPDocumentation.pdf (2017)

Copyright information

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

Authors and Affiliations

  1. 1.Faculty of Science and EngineeringÅbo Akademi UniversityTurkuFinland
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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