A Penalty Approach for Solving Nonsmooth and Nonconvex MINLP Problems
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.
KeywordsMINLP 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.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.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.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
- 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.G. Nannicini, P. Belotti, Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4(1), 1–31 (2012)Google Scholar
- 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.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.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.S. Lucidi, F. Rinaldi, Exact penalty functions for nonlinear integer programming problems. J. Optim. Theory Appl. 145(3), 479–488 (2010)Google Scholar
- 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
- 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
- 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
- 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.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.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.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.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
- 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.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)CrossRefMATHGoogle Scholar