A Penalty Approach for Solving Nonsmooth and Nonconvex MINLP Problems

  • M. Fernanda P. Costa
  • Ana Maria A. C. Rocha
  • Edite M. G. P. Fernandes
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 223)


This paper presents a penalty approach for globally solving nonsmooth and nonconvex mixed-integer nonlinear programming (MINLP) problems. Both integrality constraints and general nonlinear constraints are handled separately by hyperbolic tangent penalty functions. Proximity from an iterate to a feasible promising solution is enforced by an oracle penalty term. The numerical experiments show that the proposed oracle-based penalty approach is effective in reaching the solutions of the MINLP problems and is competitive when compared with other strategies.


MINLP Penalty function DIRECT Oracle 



The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper.

This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundação para a Ciência e Tecnologia, within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.


  1. 1.
    O. Exler, T. Lehmann, K. Schittkowski, A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization. Math. Program. Comput. 4(4), 383–412 (2012)Google Scholar
  2. 2.
    S. Burer, A.N. Letchford, Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)Google Scholar
  3. 3.
    S. Lee, I.E. Grossmann, A global optimization algorithm for nonconvex generalized disjunctive programming and applications to process systems. Comput. Chem. Eng. 25(11), 1675–1697 (2001)Google Scholar
  4. 4.
    H.S. Ryoo, N.V. Sahinidis, Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19(5), 551–566 (1995)CrossRefGoogle Scholar
  5. 5.
    C.S. Adjiman, I.P. Androulakis, C.A. Floudas, Global optimization of mixed-integer nonlinear problems. AIChE J. 46(9), 1769–1797 (2000)CrossRefGoogle Scholar
  6. 6.
    V.K. Srivastava, A. Fahim, An optimization method for solving mixed discrete-continuous programming problems. Comput. Math. Appl. 53(10), 1481–1491 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    L. Liberti, G. Nannicini, N. Mladenović, A good recipe for solving MINLPs, in Matheuristics: Hybridizing Metaheuristics and Mathematical Programming, vol. 10, ed. by V. Maniezzo, T. Stützle, S. Voß (Springer, US, 2010), pp. 231–244Google Scholar
  8. 8.
    G. Nannicini, P. Belotti, Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4(1), 1–31 (2012)Google Scholar
  9. 9.
    C. D’Ambrosio, A. Frangioni, L. Liberti, A. Lodi, A storm of feasibility pumps for nonconvex MINLP. Math. Program. 136(2), 375–402 (2012)Google Scholar
  10. 10.
    G. Liuzzi, S. Lucidi, F. Rinaldi, Derivative-free methods for bound constrained mixed-integer optimization. Comput. Optim. Appl. 53(2), 505–526 (2012)Google Scholar
  11. 11.
    M.F.P. Costa, A.M.A.C. Rocha, R.B. Francisco, E.M.G.P. Fernandes, Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming. Optimization 65(5), 1085–1104 (2016)Google Scholar
  12. 12.
    S. Lucidi, F. Rinaldi, Exact penalty functions for nonlinear integer programming problems. J. Optim. Theory Appl. 145(3), 479–488 (2010)Google Scholar
  13. 13.
    S. Lucidi, F. Rinaldi, An exact penalty global optimization approach for mixed-integer programming problems. Optim. Lett. 7(2), 297–307 (2013)Google Scholar
  14. 14.
    Y.-C. Lin, K.-S. Hwang, F.-S. Wang, A mixed-coding scheme of evolutionary algorithms to solve mixed-integer nonlinear programming problems. Comput. Math. Appl. 47(8–9), 1295–1307 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Yiqing, Y. Xigang, L. Yongjian, An improved PSO algorithm for solving non-convex NLP/MINLP problems with equality constraints. Comput. Chem. Eng. 31(3), 153–162 (2007)CrossRefGoogle Scholar
  16. 16.
    K. Deep, K.P. Singh, M. L. Kansal, C. Mohan, A real coded genetic algorithm for solving integer and mixed integer optimization problems. Appl. Math. Comput. 212(2), 505–518 (2009)Google Scholar
  17. 17.
    A. Hedar, A. Fahim, Filter-based genetic algorithm for mixed variable programming. Numer. Algebr. Control Optim. 1(1), 99–116 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    F.P. Fernandes, M.F.P. Costa, E.M.G.P. Fernandes, Branch and bound based coordinate search filter algorithm for nonsmooth nonconvex mixed-integer nonlinear programming problems, in Computational Science and Its Applications – ICCSA 2014, Part II, LNCS, vol. 8580, ed. by B. Murgante, S. Misra, A.M.A.C. Rocha, C. Torre, J.G. Rocha, M.I. Falcão, D. Taniar, B.O. Apduhan, O. Gervasi (Springer, Berlin, 2014), pp. 140–153Google Scholar
  19. 19.
    M. Schlüter, J.A. Egea, J.R. Banga, Extended ant colony optimization for non-convex mixed integer nonlinear programming. Comput. Oper. Res. 36(7), 2217–2229 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M.F.P. Costa, A.M.A.C. Rocha, R.B. Francisco, E.M.G.P. Fernandes, Extension of the firefly algorithm and preference rules for solving MINLP problems, in International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2016), AIP. Conf. Proc. 1863, 270003-1–270003-4 (2017)Google Scholar
  21. 21.
    P. Belotti, C. Kirches, S. Leyffer, J. Linderoth, J. Luedtke, A. Mahajan, Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)Google Scholar
  22. 22.
    F. Boukouvala, R. Misener, C.A. Floudas, Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO. Eur. J. Oper. Res. 252(3), 701–727 (2016)Google Scholar
  23. 23.
    M.R. Bussieck, S. Vigerske, MINLP solver software, in Wiley Encyclopedia of Operations Research and Management Science, ed. by J.J. Cochran, L.A. Cox, P. Keskinocak, J.P. Kharoufeh, J.C. Smith (Wiley, New York, 2011)Google Scholar
  24. 24.
    A.M.A.C. Rocha, M.F.P. Costa, E.M.G.P. Fernandes, Solving MINLP problems by a penalty framework, in Proceedings of XIII Global Optimization Workshop, ed. by A.M. Rocha, M.F. Costa, E. Fernandes, (2016), pp. 97–100Google Scholar
  25. 25.
    D.R. Jones, C.D. Perttunen, B.E. Stuckman, Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    M. Schlüter, M. Gerdts, The oracle penalty method. J. Glob. Optim. 47(2), 293–325 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    D.E. Finkel, DIRECT Optimization Algorithm User Guide, Center for Research in Scientific Computation. (CRSC-TR03-11, North Carolina State University, Raleigh, NC 27695-8205, March 2003)Google Scholar
  28. 28.
    C.A. Floudas, P.M. Pardalos, C. Adjiman, W.R. Esposito, Z.H. Gümüs, S.T. Harding, J.L. Klepeis, C.A. Meyer, C.A. Schweiger, Handbook of Test Problems in Local and Global Optimization, Nonconvex Optimization and its Applications (Springer Science & Business Media, Dordrecht, 1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • M. Fernanda P. Costa
    • 1
  • Ana Maria A. C. Rocha
    • 2
  • Edite M. G. P. Fernandes
    • 2
  1. 1.Centre of MathematicsUniversity of Minho, Campus de GualtarBragaPortugal
  2. 2.Algoritmi Research CentreUniversity of Minho, Campus de GualtarBragaPortugal

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