Abstract
We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.
References
Audet C., Dennis J.E. (2001): Pattern search algorithm for mixed variable programming. SIAM J. Optim. 11, 573–594
Borchers B., Mitchell J.E. (1994): An improved branch and bound algorithm for mixed integer nonlnear programming. Comput. Oper. Res. 21(4): 359–367
Bünner M.J., Schittkowski K., van de Braak G. (2004): Optimal design of electronic components by mixed-integer nonlinear programming. Optim. Eng. 5, 271–294
Burke J.V. (1992): A robust trust region method for constrained nonlinear programming problems. SIAM J. Optim. 2, 325–347
Byrd R., Schnabel R.B., Schultz G.A. (1987): A trust region algorithm for nonlinearly constrained optimization. SIAM J. Numer. Anal. 24, 1152–1170
Celis, M.R.: A trust region strategy for nonlinear equality constrained optimization. Ph.D. Thesis, Department of Mathematics. Rice University (1983)
Conn A.R., Gould I.M., Toint P.L. (2000): Trust-Region Methods. SIAM, Philadelphia
Duran M., Grossmann I.E. (1986): An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathe. Program. 36, 307–339
Exler, O., Schittkowksi, K.: MISQP: a Fortran implementation of a trust region SQP algorithm for mixed-integer nonlinear programming—user’s guide, version 2.1, Report, Department of Computer Science, University of Bayreuth (2006). http://www.uni-bayreuth.de/departments/math/~kschittkowski/MISQP.pdf
Fletcher R. (1981): Practical Methods of Optimization, vol. 2, Constrained Optimization. Wiley, Chichester
Fletcher R. (1982): Second order correction for nondifferentiable optimization. In: Watson G.A. (eds) Numerical Analysis. Springer, Berlin Heidelberg New York, pp. 85–114
Fletcher R., Leyffer S. (1994): Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349
Floudas C.A. (1995): Nonlinear and Mixed-Integer Optimization. Oxford University Press, New York, Oxford
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer (1999)
Gill P.E., Murray W., Wright M. (1981): Practical Optimization. Academic, New York
Grossmann I.E., Kravanja Z. (1997): Mixed-integer nonlinear programming: A survey of algorithms and applications. In: Conn A.R., Biegler L.T., Coleman T.F., Santosa F.N. (eds) Large-Scale Optimization with Applications, Part II: Optimal Design and Control. Springer, Berlin Heidelberg New York
Gupta O.K., Ravindran V. (1985): Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31, 1533–1546
Leyffer S. (2001): Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309
Li H.-L., Chou C.-T. (1994): A global approach for nonlinear mixed discrete programming in design optimization. Eng. Optim. 22, 109–122
Moré J.J. (1983): Recent developments in algorithms and software for trust region methods. In: Bachem A., Grötschel M., Korte B. (eds) Mathematical Programming: The State of the Art. Springer, Berlin Hedelberg New York, pp. 258–287
Powell M.J.D. (1984): On the global convergence of trust region algorithms for unconstrained minimization. Math. Program. 29, 297–303
Powell M.J.D., Yuan Y. (1991): A trust region algorithm for equality constrained optimization. Math. Program. 49, 189–211
Schittkowski, K.: QL: A Fortran code for convex quadratic programming—user’s guide, version 2.11. Report, Department of Mathematics, University of Bayreuth (2003) http://www.uni-bayreuth.de/departments/math/~kschittkowski/QL.pdf
Spickenreuther, T.: Entwicklung eines allgemeinen Branch & Bound Ansatzes zur gemischt-ganzzahligen Optimierung. Diploma Thesis, Department of Mathematics, University of Bayreuth (2005)
Stoer, J.: Foundations of recursive quadratic programming methods for solving nonlinear programs. In: Schittkowski, K. (ed.) Computational Mathematical Programming, vol. 15. NATO ASI Series, Series F: Computer and Systems Sciences. Springer, Berlin Heidelberg New York (1985)
Toint P.L. (1997): A nonmonotone trust-region algorithm for nonlinear optimization subject to convex constraints. Math. Program. 77, 69–94
Westerlund P. (2002): Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280
Yuan Y.-X. (1984): An example of only linearly convergence of trust region algorithms for nonsmooth optimization. IMA J. Numer. Anal. 4, 327–335
Yuan Y.-X. (1985): On the superlinear convergence of a trust region algorithm for nonsmooth optimization. Math. Program. 31, 269–285
Yuan Y.-X. (1995): On the convergence of a new trust region algorithm. Numer. Math. 70, 515–539
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Exler, O., Schittkowski, K. A trust region SQP algorithm for mixed-integer nonlinear programming. Optimization Letters 1, 269–280 (2007). https://doi.org/10.1007/s11590-006-0026-1
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DOI: https://doi.org/10.1007/s11590-006-0026-1