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Using regularization and second order information in outer approximation for convex MINLP

Mathematical Programming volume 180, pages 285–310 (2020)Cite this article

Abstract

In this paper, we present two new methods for solving convex mixed-integer nonlinear programming problems based on the outer approximation method. The first method is inspired by the level method and uses a regularization technique to reduce the step size when choosing new integer combinations. The second method combines ideas from both the level method and the sequential quadratic programming technique and uses a second order approximation of the Lagrangean when choosing the new integer combinations. The main idea behind the methods is to choose the integer combination more carefully at each iteration, in order to obtain the optimal solution in fewer iterations compared to the original outer approximation method. We prove rigorously that both methods will find and verify the optimal solution in a finite number of iterations. Furthermore, we present a numerical comparison of the methods based on 109 test problems to illustrate their advantages.

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Notes

  1. http://www.gamsworld.org/minlp/minlplib2/html/index.html.

  2. http://www.gamsworld.org/minlp/minlplib2/html/cvxnonsep_nsig40.html.

  3. http://www.gamsworld.org/minlp/minlplib2/html/ibs2.html.

References

  1. Bagirov, A., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Berlin (2014)

    Book  Google Scholar 

  2. Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013). https://doi.org/10.1017/S0962492913000032

    MathSciNet  Article  MATH  Google Scholar 

  3. Biegler, L.T., Grossmann, I.E.: Retrospective on optimization. Comput. Chem. Eng. 28(8), 1169–1192 (2004)

    Article  Google Scholar 

  4. Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discr. Optim. 5(2), 186–204 (2008). https://doi.org/10.1016/j.disopt.2006.10.011

    MathSciNet  Article  MATH  Google Scholar 

  5. Bonami, P., Cornuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119(2), 331–352 (2009)

    MathSciNet  Article  Google Scholar 

  6. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  7. Currie, J., Wilson, D.I.: OPTI: lowering the barrier between open source optimizers and the industrial MATLAB user. In: Sahinidis, N., Pinto, J. (eds.) Foundations of Computer-Aided Process Operations. Savannah, Georgia (2012)

    Google Scholar 

  8. Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8(3), 250–255 (1965)

    MathSciNet  Article  Google Scholar 

  9. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. B 91(2), 201–213 (2002). https://doi.org/10.1007/s101070100263

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  11. den Hertog, D., Kaliski, J., Roos, C., Terlaky, T.: A logarithmic barrier cutting plane method for convex programming. Ann. Oper. Res. 58(2), 67–98 (1995)

    MathSciNet  Article  Google Scholar 

  12. de Oliveira, W.: Regularized optimization methods for convex MINLP problems. TOP 24(3), 665–692 (2016)

    MathSciNet  Article  Google Scholar 

  13. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66(1), 327–349 (1994)

    MathSciNet  Article  Google Scholar 

  14. Floudas, C.A.: Deterministic global optimization. Theory, methods and applications In: Nonconvex Optimization and its Applications, vol. 37. Springer, US (2000)

    Book  Google Scholar 

  15. GAMSWorld: mixed-integer nonlinear programming library. http://www.gamsworld.org/minlp/minlplib2/html/ (2016). Accessed 24 Nov 2016

  16. Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)

    MathSciNet  Article  Google Scholar 

  17. Gershgorin, S.A.: Uber die Abgrenzung der Eigenwerte einer Matrix. Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na 6, 749–754 (1931)

    MATH  Google Scholar 

  18. Grossmann, I.E., Viswanathan, J., Vecchietti, A., Raman, R., Kalvelagen, E.: GAMS/DICOPT: A Discrete Continuous Optimization Package (2002)

  19. Gurobi Optimization, I.: Gurobi optimizer reference manual. http://www.gurobi.com (2016)

  20. IBM Corp., IBM: V12.6: User’s Manual for CPLEX. Int. Bus. Mach. Corp. 12(1), 481 (2009)

    Google Scholar 

  21. Kelley Jr., J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960)

    MathSciNet  Article  Google Scholar 

  22. Kiwiel, K.C.: Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities. Math. Program. 69(1–3), 89–109 (1995)

    MathSciNet  MATH  Google Scholar 

  23. 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)

    MathSciNet  Article  Google Scholar 

  24. Kronqvist, J., Lundell, A., Westerlund, T.: Reformulations for utilizing separability when solving convex MINLP problems. J. Glob. Optim. 71(3), 571–592 (2018). https://doi.org/10.1007/s10898-018-0616-3

    MathSciNet  Article  MATH  Google Scholar 

  25. Lee, J., Leyffer, S. (eds.): Mixed Integer Nonlinear Programming, vol. 154. Springer, Berlin (2011)

    Google Scholar 

  26. Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69(1–3), 111–147 (1995)

    MathSciNet  Article  Google Scholar 

  27. Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer, Berlin (2004)

    Book  Google Scholar 

  28. Quesada, I., Grossmann, I.E.: An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16(10–11), 937–947 (1992)

    Article  Google Scholar 

  29. Slater, M., et al.: Lagrange multipliers revisited. Technical report, Cowles Foundation for Research in Economics, Yale University (1959)

  30. Trespalacios, F., Grossmann, I.E.: Review of mixed-integer nonlinear and generalized disjunctive programming methods. Chem. Ing. Tech. 86(7), 991–1012 (2014). https://doi.org/10.1002/cite.201400037

    Article  Google Scholar 

  31. Viswanathan, J., Grossmann, I.E.: A combined penalty function and outer-approximation method for MINLP optimization. Comput. Chem. Eng. 14(7), 769–782 (1990)

    Article  Google Scholar 

  32. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    MathSciNet  Article  Google Scholar 

  33. Wei, Z., Ali, M.M.: Outer approximation algorithm for one class of convex mixed-integer nonlinear programming problems with partial differentiability. J. Optim. Theory Appl. 167(2), 644–652 (2015). https://doi.org/10.1007/s10957-015-0715-y

    MathSciNet  Article  MATH  Google Scholar 

  34. Westerlund, T., Petterson, F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, S131–S136 (1995)

    Article  Google Scholar 

  35. Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3(3), 253–280 (2002)

    MathSciNet  Article  Google Scholar 

  36. Wolfe, P.: A duality theorem for non-linear programming. Q. Appl. Math. 19(3), 239–244 (1961)

    MathSciNet  Article  Google Scholar 

  37. Zaourar, S., Malick, J.: Quadratic stabilization of Benders decomposition. https://hal.archives-ouvertes.fr/hal-01181273 (2014). Working paper or preprint

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Acknowledgements

Jan Kronqvist is grateful for the grants given by Walter Ahlström foundation, Svenska tekniska vetenskapsakademien i Finland, Tekniikan edistämissäätiö, TFIF and Waldemar von Frenckells stiftelse, which made the research visit at Carnegie Mellon University possible. David E. Bernal and Ignacio E. Grossmann would like to thank the Center Advanced Process Decision Making (CAPD) for its financial support. The authors would like to acknowledge the Dagstuhl Seminar 18081 on Designing and Implementing Algorithms for Mixed-Integer Nonlinear Optimization, where an early version of the results shown in this manuscript were presented and discussed.

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Authors and Affiliations

  1. Process Design and Systems Engineering, Åbo Akademi University, Turku, Finland

    Jan Kronqvist

  2. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

    David E. Bernal & Ignacio E. Grossmann

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  1. Jan Kronqvist
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  2. David E. Bernal
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  3. Ignacio E. Grossmann
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Correspondence to Ignacio E. Grossmann.

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Kronqvist, J., Bernal, D.E. & Grossmann, I.E. Using regularization and second order information in outer approximation for convex MINLP. Math. Program. 180, 285–310 (2020). https://doi.org/10.1007/s10107-018-1356-3

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Mathematics Subject Classification

  • 65K05
  • 90C11
  • 90C30
  • 90C55