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Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm

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Abstract

In this article we look at a new algorithm for solving convex mixed integer nonlinear programming problems. The algorithm uses an integrated approach, where a branch and bound strategy is mixed with solving nonlinear programming problems at each node of the tree. The nonlinear programming problems, at each node, are not solved to optimality, rather one iteration step is taken at each node and then branching is applied. A Sequential Cutting Plane (SCP) algorithm is used for solving the nonlinear programming problems by solving a sequence of linear programming problems. The proposed algorithm generates explicit lower bounds for the nodes in the branch and bound tree, which is a significant improvement over previous algorithms based on QP techniques. Initial numerical results indicate that the described algorithm is a competitive alternative to other existing algorithms for these types of problems.

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References

  1. A.S. Manne, “GAMS/MINOS: Three examples,” Technical Report, Department of Operations Research, Stanford University, Stanford: California, 1986.,

  2. E.M.L. Beale and J.A. Tomlin, “Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables,” in: OR69 Proceedings of the Fifth International Conference on Operational Research, J. Lawrence (Ed.), Tavistock Publications, London, 1970, pp. 447–454.,

  3. M. Benichou, J.M. Gauthier, P. Girodet, G. Hentges, G. Ribiere, and O. Vincent, “Experiments in mixed-integer linear programming,'' Mathematical Programming, vol. 1, pp. 76–94, 1971.,

  4. R.E. Bixby, M. Fenelon, Z. Gu, E. Rothberg, and R. Wunderling, “MIP: Theory and practice—closing the gap,” in: System Modelling and Optimization: Methods, Theory and Applications, M.J.D. Powell, S. Scholtes (Eds.), Kluwer Academic Publishers: Dordrecht, 2000, pp. 19–49.,

  5. K.-M. Björk, P.O. Lindberg, and T. Westerlund, “Some convexifications in global optimization of problems containing signomial terms,” Computers and Chemical Engineering, vol. 27, pp. 669–679, 2003.,

  6. B. Borchers and J.E. Mitchell, “An improved branch and bound algorithm for mixed integer nonlinear programs,” Computers and Operations Research, vol. 21, pp. 359–367, 1994.,

  7. M.R. Bussieck, A.S. Drud, and A. Meeraus, “MINLPLib—A collection of test models for mixed-integer nonlinear programming,” GAMS Development Corporation, Washington DC, 2001. Web page at < http://www.gamsworld.org/minlp/minlplib.htm >.,

  8. J. Czyzyk, M.P. Mesnier, and J.J. Moré, “The NEOS Server,” IEEE Journal on Computational Science and Engineering, vol. 5, pp. 68–75, 1998.,

  9. H. Dahl, A. Meeraus, and S.A. Zenios, “Some financial optimization models: I risk management,” in: Financial Optimization, S.A. Zenios (Ed.), Cambridge University Press, Cambridge, 1993, pp. 3–36.,

  10. R.J. Dakin, “A tree-search algorithm for mixed integer programming problems,” Computer Journal, vol. 8, pp. 250–255, 1965.,

  11. E.D. Dolan, “The NEOS Server 4.0 Administrative guide, technical memorandum ANL/MCS-TM-250,” Mathematics and Computer Science Division, Argonne National Laboratory, Argonne: IL, 2001.,

  12. E.D. Dolan and J.J. Moré, “Benchmarking optimization software with performance profiles,'' Mathematical Programming, vol. 91, pp. 201–213, 2002.,

  13. M.A. Duran and I.E. Grossmann, “An outer-approximation algorithm for a class of mixed-integer nonlinear programs,” Mathematical Programming, vol. 36, pp. 307–339, 1986.,

  14. R. Fletcher and S. Leyffer, “User manual for filterSQP,” Numerical Analysis Report NA/181, Department of Mathematics, University of Dundee, Dundee, 1998.,

  15. R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function,” Mathematical Programming, vol. 91, pp. 239–269, 2002.,

  16. C.A. Floudas, “Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications, Topics in Chemical Engineering,” Oxford University Press: New York, 1995.,

  17. C.A. Floudas and P.M. Pardalos, Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht, 2001.,

  18. B. Gavish, D. Horsky, and K. Srikanth, “An approach to the optimal positioning of a new product,” Management Science, vol. 29, pp. 1277–1297, 1983.,

  19. P.E. Gill, W. Murray, and M.H. Wright, Practical Optimization, Academic Press, London, 1981.,

  20. O.K. Gupta and A. Ravindran, “Branch and bound experiments in convex nonlinear integer programming,” Management Science, vol. 31, pp. 1533–1546, 1985.,

  21. I. Harjunkoski, T. Westerlund, R. Pörn, and H. Skrifvars, “Different transformations for solving non-convex trim-loss problems by MINLP,” European Journal of Operational Research, vol. 105, pp. 594–603, 1998.,

  22. I. Harjunkoski, T. Westerlund, and R. Pörn, “Numerical and environmental considerations on a complex industrial mixed integer non-linear programming (MINLP) problem,” Computers and Chemical Engineering, vol. 23, pp. 1545–1561, 1999.,

  23. G.R. Kocis and I.E. Grossmann, “Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problems in process synthesis,” Industrial and Engineering Chemical Research, vol. 27, pp. 1407–1421, 1988.,

  24. A.H. Land and A.G. Doig, “An automatic method of solving discrete programming problems,'' Econometrica, vol. 28, pp. 497–520, 1960.,

  25. S. Leyffer, “MacMINLP : AMPL collection of mixed integer nonlinear programs,” University of Dundee, Dundee, 2000. Available from < http://www.maths.dundee.ac.uk/∼sleyffer/MacMINLP/ > .,

  26. S. Leyffer, “Integrating SQP and branch-and-bound for mixed integer nonlinear programming,” Computational Optimization and Applications, vol. 18, pp. 295–309, 2001.,

  27. J.C.T. Mao and B.A. Wallingford, “An extension of Lawler and Bell's method of discrete optimization with examples from capital budgeting,” Management Science, vol. 15, pp. 51–60, 1968.,

  28. R. Pörn and T. Westerlund, “A cutting plane method for minimizing pseudo-convex functions in the mixed integer case,” Computers and Chemical Engineering, vol. 24, pp. 2655–2665, 2000.,

  29. I. Quesada and I.E. Grossmann, “An LP/NLP based branch and bound algorithm for convex MINLP optimization problems,” Computers and Chemical Engineering, vol. 16, pp. 937–947, 1992.,

  30. N.V. Sahinidis and I.E. Grossmann, “Convergence properties of generalized Benders decomposition,” Computers and Chemical Engineering, vol. 15, pp. 481–491, 1991.,

  31. N.V. Sahinidis, “BARON: A general purpose global optimization software package,” Journal of Global Optimization, vol. 8, pp. 201–205, 1996.,

  32. M.W.P. Savelsbergh, “Preprocessing and probing techniques for mixed integer programming problems,” ORSA Journal on Computing, vol. 6, pp. 445–454, 1994.,

  33. C. Still and T. Westerlund, “Extended cutting plane algorithm,” in: Encyclopedia of Optimization, C.A. Floudas, P.M. Pardalos (Eds.), Kluwer Academic Publishers, Dordrecht, 2001, Vol. 2, pp. 53–61.,

  34. C. Still and T. Westerlund, “A sequential cutting plane algorithm for solving convex NLP problems,” European Journal of Operational Research, 2005, (Accepted).,

  35. T. Westerlund and F. Pettersson, “An extended cutting plane method for solving convex MINLP problems,” Computers and Chemical Engineering, vol. 19(Suppl.), pp. S131–S136, 1995.,

  36. T. Westerlund, H. Skrifvars, I. Harjunkoski, and R. Pörn, “An extended cutting plane method for a class of non-convex MINLP problems,” Computers and Chemical Engineering, vol. 22, pp. 357–365, 1998.,

  37. T. Westerlund and K. Lundqvist, “Alpha-ECP Version 5.01. An interactive MINLP-Solver based on the extended cutting plane method,” Report 01-178-A, Process Design Laboratory, Åbo Akademi University, Åbo, 2001.,

  38. T. Westerlund and R. Pörn, “Solving pseudo-convex mixed integer optimization problems by cutting plane techniques,” Optimization and Engineering, vol. 3, pp. 253–280, 2002.,

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Authors and Affiliations

  1. Department of Mathematics, Åbo Akademi University, Fänriksgatan 3B, FIN-20500, Åbo, Finland

    Claus Still

  2. Process Design Laboratory, Åbo Akademi University, Biskopsgatan 8, FIN-20500, Åbo, Finland

    Tapio Westerlund

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  1. Claus Still
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  2. Tapio Westerlund
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Still, C., Westerlund, T. Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm. Comput Optim Applic 34, 63–83 (2006). https://doi.org/10.1007/s10589-005-3076-x

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  • DOI: https://doi.org/10.1007/s10589-005-3076-x

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