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Computing feasible points for binary MINLPs with MPECs

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Abstract

Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use an iterative solution procedure for solving series of regularized problems. In the case of success, these procedures result in a feasible solution of the original mixed-binary nonlinear problem. Since we rely on local nonlinear programming solvers the resulting method is fast and we further improve its reliability by additional algorithmic techniques. We show the strength of our method by an extensive computational study on 662 MINLPLib2instances, where our methods are able to produce feasible solutions for \({60}{\%}\) of all instances in at most \({10}\,{\hbox {s}}\).

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Notes

  1. For CONOPT4, the failed instances are crudeoil_lee1_08, crudeoil_lee2_05, crudeoil_lee2_06, crudeoil_lee3_05, crudeoil_lee3_06, crudeoil_lee4_10, nuclear25a, telecomsp_njlata, telecomsp_pacbell for the reformulation scheme of Scholtes (9); crudeoil_lee1_08, crudeoil_lee3_06, crudeoil_lee3_10, gasprod_sarawak81, nuclear25a, sepasequ_convent, telecomsp_njlata, telecomsp_pacbell for the Fischer–Burmeister reformulation scheme (8) and crudeoil_lee1_06, crudeoil_lee1_09, crudeoil_lee2_06, crudeoil_lee2_07, crudeoil_lee2_08, crudeoil_lee2_09, crudeoil_lee2_10, crudeoil_lee3_06, crudeoil_lee3_08, crudeoil_lee4_05, crudeoil_lee4_06, crudeoil_lee4_07, crudeoil_lee4_08, nuclear49a, nuclear49b, squfl025-040persp, telecomsp_njlata, telecomsp_pacbell for the penalty-based reformulation (18). For Ipopt, the failed instances are faclay60, faclay70, faclay80 for the reformulation scheme of Scholtes (3), whereas it fails on faclay75 for the Fischer–Burmeister reformulation and on faclay33 and for the penalty-based reformulation.

References

  1. Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009). https://doi.org/10.1007/s12532-008-0001-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Baumrucker, B.T., Renfro, J.G., Biegler, L.T.: MPEC problem formulations and solution strategies with chemical engineering applications. Comput. Chem. Eng. 32(12), 2903–2913 (2008). https://doi.org/10.1016/j.compchemeng.2008.02.010

    Article  Google Scholar 

  3. Benoist, T., Estellon, B., Gardi, F., Megel, R., Nouioua, K.: Localsolver 1.x: a black-box local-search solver for 0–1 programming. 4OR 9(3), 299 (2011). https://doi.org/10.1007/s10288-011-0165-9

    Article  MathSciNet  MATH  Google Scholar 

  4. Berthold, T.: Heuristic algorithms in global MINLP solvers. Ph.D. thesis, Technische Universität Berlin (2014)

  5. Berthold, T., Gleixner, A.M.: Undercover: a primal MINLP heuristic exploring a largest sub-MIP. Math. Program. 144(1), 315–346 (2014). https://doi.org/10.1007/s10107-013-0635-2

    Article  MathSciNet  MATH  Google Scholar 

  6. Berthold, T., Heinz, S., Pfetsch, M.E., Vigerske, S.: Large neighborhood search beyond MIP. In: Proceedings of the 9th Metaheuristics International Conference (MIC 2011), pp. 51–60 (2011)

  7. Berthold, T., Heinz, S., Vigerske, S.: Extending a CIP framework to solve MIQCPs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 427–444. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1927-3_15

    Chapter  Google Scholar 

  8. Bonami, P., Cornuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119(2), 331–352 (2009). https://doi.org/10.1007/s10107-008-0212-2

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonami, P., Gonçalves, J.P.M.: Heuristics for convex mixed integer nonlinear programs. Comput. Optim. Appl. 51(2), 729–747 (2012). https://doi.org/10.1007/s10589-010-9350-6

    Article  MathSciNet  MATH  Google Scholar 

  10. D’Ambrosio, C., Frangioni, A., Liberti, L., Lodi, A.: Experiments with a feasibility pump approach for nonconvex MINLPs. In: Festa, P. (ed.) Experimental Algorithms. Lecture Notes in Computer Science, vol. 6049, pp. 350–360. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-13193-6_30

    Chapter  Google Scholar 

  11. D’Ambrosio, C., Frangioni, A., Liberti, L., Lodi, A.: A storm of feasibility pumps for nonconvex MINLP. Math. Program. 136(2), 375–402 (2012). https://doi.org/10.1007/s10107-012-0608-x

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Drud, A.S.: CONOPT—a large-scale GRG code. INFORMS J. Comput. 6(2), 207–216 (1994). https://doi.org/10.1287/ijoc.6.2.207

    Article  MATH  Google Scholar 

  14. Drud, A.S.: CONOPT: a system for large scale nonlinear optimization, tutorial for CONOPT subroutine library. Technical Report, ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1995)

  15. Drud, A.S.: CONOPT: a system for large scale nonlinear optimization, reference manual for CONOPT subroutine library. Technical Report, ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1996)

  16. Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992). https://doi.org/10.1080/02331939208843795

    Article  MathSciNet  MATH  Google Scholar 

  17. Fischetti, M., Lodi, A.: Local branching. Math. Program. 98(1–3), 23–47 (2003). https://doi.org/10.1007/s10107-003-0395-5

    Article  MathSciNet  MATH  Google Scholar 

  18. GAMS Development Corporation: General Algebraic Modeling System (GAMS) Release 24.5.4. Washington, DC, USA (2015). http://www.gams.com. Accessed 10 Aug 2018

  19. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005). https://doi.org/10.1137/S0036144504446096

    Article  MathSciNet  MATH  Google Scholar 

  20. Gleixner, A., Eifler, L., Gally, T., Gamrath, G., Gemander, P., Gottwald, R.L., Hendel, G., Hojny, C., Koch, T., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schlösser, F., Serrano, F., Shinano, Y., Viernickel, J.M., Vigerske, S., Weninger, D., Witt, J.T., Witzig, J.: The SCIP Optimization Suite 5.0. Technical Report 17-61, ZIB, Takustr.7, 14195 Berlin (2017)

  21. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1), 257–288 (2013). https://doi.org/10.1007/s10107-011-0488-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu, X.M., Ralph, D.: Convergence of a penalty method for mathematical programming with complementarity constraints. J. Optim. Theory Appl. 123(2), 365–390 (2004). https://doi.org/10.1007/s10957-004-5154-0

    Article  MathSciNet  Google Scholar 

  23. Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math. Program. Comput. 3(2), 103–163 (2011). https://doi.org/10.1007/s12532-011-0025-9

    Article  MathSciNet  Google Scholar 

  24. Kraemer, K., Kossack, S., Marquardt, W.: An efficient solution method for the MINLP optimization of chemical processes. Comput. Aided Chem. Eng. 24, 105 (2007). https://doi.org/10.1016/S1570-7946(07)80041-1

    Article  Google Scholar 

  25. Kraemer, K., Marquardt, W.: Continuous reformulation of MINLP problems. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and Its Applications in Engineering: The 14th Belgian-French-German Conference on Optimization, pp. 83–92. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-12598-0_8

  26. Liberti, L., Mladenović, N., Nannicini, G.: A recipe for finding good solutions to MINLPs. Math. Program. Comput. 3(4), 349–390 (2011). https://doi.org/10.1007/s12532-011-0031-y

    Article  MathSciNet  MATH  Google Scholar 

  27. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  28. Maher, S.J., Fischer, T., Gally, T., Gamrath, G., Gleixner, A., Gottwald, R.L., Hendel, G., Koch, T., Lübbecke, M.E., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schenker, S., Schwarz, R., Serrano, F., Shinano, Y., Weninger, D., Witt, J.T., Witzig, J.: The SCIP optimization suite 4.0. Technical Report 17-12, ZIB, Takustr.7, 14195 Berlin (2017)

  29. Nannicini, G., Belotti, P.: Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4(1), 1–31 (2012). https://doi.org/10.1007/s12532-011-0032-x

    Article  MathSciNet  MATH  Google Scholar 

  30. Nannicini, G., Belotti, P., Liberti, L.: A local branching heuristic for MINLPs (2008). arXiv preprint arXiv:0812.2188

  31. Rose, D., Schmidt, M., Steinbach, M.C., Willert, B.M.: Computational optimization of gas compressor stations: MINLP models versus continuous reformulations. Math. Methods Oper. Res. 83(3), 409–444 (2016). https://doi.org/10.1007/s00186-016-0533-5

    Article  MathSciNet  MATH  Google Scholar 

  32. Sahinidis, N.V.: BARON 14.3.1: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2014)

  33. Schmidt, M., Steinbach, M.C., Willert, B.M.: A primal heuristic for nonsmooth mixed integer nonlinear optimization. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 295–320. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-38189-8_13

    Chapter  MATH  Google Scholar 

  34. Schmidt, M., Steinbach, M.C., Willert, B.M.: An MPEC based heuristic. In: Koch, T., Hiller, B., Pfetsch, M.E., Schewe, L. (eds.) Evaluating Gas Network Capacities, SIAM-MOS series on Optimization, Chapter 9, pp. 163–180. SIAM (2015). https://doi.org/10.1137/1.9781611973693.ch9

  35. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001). https://doi.org/10.1137/S1052623499361233

    Article  MathSciNet  MATH  Google Scholar 

  36. Stein, O., Oldenburg, J., Marquardt, W.: Continuous reformulations of discrete-continuous optimization problems. Comput. Chem. Eng. 28(10), 1951–1966 (2004). https://doi.org/10.1016/j.compchemeng.2004.03.011. (Special Issue for Professor Arthur W. Westerberg)

    Article  Google Scholar 

  37. Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl. 13(1), 201–220 (1999). https://doi.org/10.1023/A:1008669226453

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Book  MATH  Google Scholar 

  39. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004). https://doi.org/10.1007/s10107-003-0467-6

    Article  MathSciNet  MATH  Google Scholar 

  40. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005). https://doi.org/10.1007/s10107-005-0581-8

    Article  MathSciNet  MATH  Google Scholar 

  41. Ugray, Z., Lasdon, L., Plummer, J., Glover, F., Kelly, J., Martí, R.: Scatter search and local NLP solvers: a multistart framework for global optimization. INFORMS J. Comput. 19(3), 328–340 (2007). https://doi.org/10.1287/ijoc.1060.0175

    Article  MathSciNet  MATH  Google Scholar 

  42. 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). https://doi.org/10.1007/s10107-004-0559-y

    Article  MathSciNet  MATH  Google Scholar 

  43. Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005). https://doi.org/10.1137/S1052623403426556

    Article  MathSciNet  MATH  Google Scholar 

  44. Wu, B., Ghanem, B.: \(l_p\)-box ADMM: a versatile framework for integer programming. Technical Report (2016). http://arxiv.org/abs/1604.07666. Accessed 10 Aug 2018

  45. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995). https://doi.org/10.1080/02331939508844060

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research has been performed as part of the Energie Campus Nürnberg and supported by funding through the “Aufbruch Bayern (Bavaria on the move)” initiative of the state of Bavaria. Both authors acknowledge funding through the DFG Transregio TRR 154, subprojects A05, B07, and B08. Last but not least, we want to express our sincere gratefulness to Stefan Vigerske from GAMS. Without his patient help, the implementations underlying this paper would not have been possible. Thanks a lot, Stefan.

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  1. Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Discrete Optimization, Cauerstr. 11, 91058, Erlangen, Germany

    Lars Schewe & Martin Schmidt

  2. Energie Campus Nürnberg, Fürther Str. 250, 90429, Nuremberg, Germany

    Lars Schewe & Martin Schmidt

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Schewe, L., Schmidt, M. Computing feasible points for binary MINLPs with MPECs. Math. Prog. Comp. 11, 95–118 (2019). https://doi.org/10.1007/s12532-018-0141-x

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