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Mixed integer nonlinear programming tools: an updated practical overview

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Abstract

We present a review of available tools for solving mixed integer nonlinear programming problems. Our aim is to give the reader a flavor of the difficulties one could face and to discuss the tools one could use to try to overcome such difficulties.

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Notes

  1. Note that we do not consider MINLPs that can be exactly reformulated as MILP problems.

  2. If k=1, no fixing is performed.

  3. A pseudo-convex MINLP is an MINLP involving pseudo-convex functions. Informally, a function is pseudo-convex if behaves like a convex function with respect to finding its local minima, but need not actually be convex, see, e.g., Mangasarian (1965).

  4. The development of LAGO is currently ceased but the software is open-source, so we prefer to keep it into the list because future development is possible.

References

  • Abhishek, K. (2008). Topics in mixed integer nonlinear programming. Ph.D. thesis, Lehigh University.

  • Abhishek, K., Leyffer, S., & Linderoth, J. (2010). FilMINT: an outer-approximation-based solver for nonlinear mixed integer programs. INFORMS Journal on Computing, 22, 555–567.

    Article  Google Scholar 

  • Achterberg, T. (2007). Constraint integer programming. Ph.D. thesis, Technische Universität Berlin.

  • Adjiman, C., Androulakis, I., & Floudas, C. (1997). Global optimization of MINLP problems in process synthesis and design. Computers & Chemical Engineering, 21, 445–450.

    Article  Google Scholar 

  • Adjiman, C., Androulakis, I., & Floudas, C. (2000). Global optimization of mixed-integer nonlinear problems. AIChE Journal, 46, 1769–1797.

    Article  Google Scholar 

  • Androulakis, I., Maranas, C., & Floudas, C. (1995). αBB: a global optimization method for general constrained nonconvex problems. Journal of Global Optimization, 7, 337–363.

    Article  Google Scholar 

  • Beale, E., & Tomlin, J. (1970). Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In J. Lawrence (Ed.), Proceedings of the Fifth International Conference on Operational Research: OR 69 (pp. 447–454). London: Tavistock.

    Google Scholar 

  • Belotti, P., Lee, J., Liberti, L., Margot, F., & Wächter, A. (2009). Branching and bounds tightening techniques for non-convex MINLP. Optimization Methods & Software, 24, 597–634.

    Article  Google Scholar 

  • Benders, J. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 267–299.

    Article  Google Scholar 

  • Berthold, T., Heinz, S., & Vigerske, S. (2012). Extending a CIP framework to solve MIQCPs. In J. Lee & S. Leyffer (Eds.), IMA volumes in mathematics and its applications: Vol. 154. Mixed-integer nonlinear optimization: algorithmic advances and applications (pp. 427–444). Berlin: Springer.

    Google Scholar 

  • Bonami, P., & Gonçalves, J. (2008). Primal heuristics for mixed integer nonlinear programs (Tech. Rep.). IBM Research Report RC24639.

  • Bonami, P., Forrest, J., Lee, J., & Wächter, A. (2007). Rapid development of an MINLP solver with COIN-OR. Optima, 75, 1–5.

    Google Scholar 

  • Bonami, P., Biegler, L., Conn, A., Cornuéjols, G., Grossmann, I., Laird, C., Lee, J., Lodi, A., Margot, F., Sawaya, N., & Wächter, A. (2008). An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optimization, 5, 186–204.

    Article  Google Scholar 

  • Bonami, P., Cornuéjols, G., Lodi, A., & Margot, F. (2009). A feasibility pump for mixed integer nonlinear programs. Mathematical Programming, 119, 331–352.

    Article  Google Scholar 

  • Bongartz, I., Conn, A. R., Gould, N., & Toint, P. L. (1995). CUTE: constrained and unconstrained testing environment. ACM Transactions on Mathematical Software, 21, 123–160. doi:10.1145/200979.201043.

    Article  Google Scholar 

  • Brooke, A., Kendrick, D., & Meeraus, A. (1992). GAMS: a user’s guide. URL citeseer.ist.psu.edu/brooke92gams.html.

  • Bussieck, M., & Drud, A. SSB: a new solver for mixed integer nonlinear programming. In Recent advances in nonlinear mixed integer optimization, INFORMS Fall, Invited talk.

  • Bussieck, M., & Vigerske, S. (2011). MINLP solver software. In J. Cochran (Ed.), Wiley encyclopedia of operations research and management science. New York: Wiley.

    Google Scholar 

  • CBC. URL https://projects.coin-or.org/Cbc.

  • Conn, A., Scheinberg, K., & Vicente, L. (2008). MPS/SIAM book series on optimization. Introduction to derivative free optimization. Philadelphia: SIAM.

    Google Scholar 

  • Dakin, R. (1965). A tree-search algorithm for mixed integer programming problems. Computer Journal, 8(3), 250–255. doi:10.1093/comjnl/8.3.250. URL http://comjnl.oxfordjournals.org/content/8/3/250.abstract.

    Article  Google Scholar 

  • D’Ambrosio, C. (2010). Application-oriented mixed integer non-linear programming. 4OR, 8, 319–322.

    Article  Google Scholar 

  • D’Ambrosio, C., Frangioni, A., Liberti, L., & Lodi, A. (2010). A storm of feasibility pumps for nonconvex MINLP (Tech. Rep. OR/10/13). Università di Bologna. To appear in Mathematical Programming.

  • D’Ambrosio, C., & Lodi, A. (2011). Mixed integer non-linear programming tools: a practical overview. 4OR: A. 4OR, 9, 329–349.

    Article  Google Scholar 

  • Duran, M., & Grossmann, I. (1986). An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming, 36, 307–339.

    Article  Google Scholar 

  • Fourer, R., Gay, D., & Kernighan, B. (2003). AMPL: a modeling language for mathematical programming (2nd ed.). Monterey: Duxbury Press/Brooks/Cole Publishing Co.

    Google Scholar 

  • Geoffrion, A. (1972). Generalized Benders decomposition. Journal of Optimization Theory and Applications, 10, 237–260.

    Article  Google Scholar 

  • Grossmann, I. (2002). Review of nonlinear mixed-integer and disjunctive programming techniques. Optimization and Engineering, 3, 227–252.

    Article  Google Scholar 

  • Gupta, O., & Ravindran, V. (1985). Branch and bound experiments in convex nonlinear integer programming. Management Science, 31, 1533–1546.

    Article  Google Scholar 

  • GUROBI. URL http://www.gurobi.com/.

  • IBM-CPLEX. URL http://www-01.ibm.com/software/integration/optimization/cplex/. (v. 12.0).

  • Jeroslow, R. (1973). There cannot be any algorithm for integer programming with quadratic constraints. Operations Research, 21, 221–224.

    Article  Google Scholar 

  • Kelley, J. E. Jr. (1960). The cutting-plane method for solving convex programs. Journal of the Society for Industrial and Applied Mathematics, 8, 703–712.

    Article  Google Scholar 

  • Kesavan, P., & Barto, P. (2000). Generalized branch-and-cut framework for mixed-integer nonlinear optimization problems. Computers & Chemical Engineering, 24, 1361–1366.

    Article  Google Scholar 

  • Kocis, G., & Grossmann, I. (1989). Computational experience with DICOPT solving MINLP problems in process systems engineering. Computers & Chemical Engineering, 13, 307–315.

    Article  Google Scholar 

  • Land, A., & Doig, A. (1960). An automatic method of solving discrete programming problems. Econometrica, 28(3), 497–520. URL http://www.jstor.org/stable/1910129.

    Article  Google Scholar 

  • Lee, J., & Leyffer, S. (Eds.) (2012). IMA volumes in mathematics and its applications: Vol. 154. Mixed integer nonlinear programming. Berlin: Springer.

    Google Scholar 

  • Leyffer, S. (1999). User manual for MINLP_BB (Tech. Rep.). University of Dundee.

  • Leyffer, S. (2001). Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Computational Optimization and Applications, 18, 295–309.

    Article  Google Scholar 

  • Leyffer, S., & Mahajan, A. (2011). Software for nonlinearly constrained optimization. New York: Wiley.

    Google Scholar 

  • Liberti, L. (2004a). Reformulation and convex relaxation techniques for global optimization. Ph.D. thesis, Imperial College, London, UK.

  • Liberti, L. (2004b). Reformulation and convex relaxation techniques for global optimization. 4OR, 2, 255–258.

    Article  Google Scholar 

  • Liberti, L. (2006). Writing global optimization software. In L. Liberti & N. Maculan (Eds.), Global optimization: from theory to implementation (pp. 211–262). Berlin: Springer.

    Chapter  Google Scholar 

  • Liberti, L., Cafieri, S., & Tarissan, F. (2009a). Reformulations in mathematical programming: a computational approach. In A. Abraham, A. Hassanien, & P. Siarry (Eds.), Studies in computational intelligence: Vol. 203. Foundations on computational intelligence, vol. 3 (pp. 153–234). New York: Springer.

    Chapter  Google Scholar 

  • Liberti, L., Nannicini, G., & Mladenovic, N. (2009b). A good recipe for solving MINLPs. In V. Maniezzo, T. Stützle, & S. Voss (Eds.), Annals of information systems: Vol. 10. MATHEURISTICS: hybridizing metaheuristics and mathematical programming (pp. 231–244). Berlin: Springer.

    Google Scholar 

  • Linderoth, J., & Lodi, A. (2011). MILP software. In J. Cochran (Ed.), Wiley encyclopedia of operations research and management science (Vol. 5, pp. 3239–3248). New York: Wiley.

    Google Scholar 

  • Lodi, A. (2009). Mixed integer programming computation. In M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, & L. Wolsey (Eds.), 50 Years of integer programming 1958–2008: from the early years to the state-of-the-art (pp. 619–645). Berlin: Springer.

    Google Scholar 

  • Mangasarian, O. (1965). Pseudo-convex functions. Journal of the Society for Industrial and Applied Mathematics, 3, 281–290.

    Google Scholar 

  • McCormick, G. (1976). Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Mathematical Programming, 10, 147–175.

    Article  Google Scholar 

  • Nannicini, G., & Belotti, P. (2011). Rounding based heuristics for nonconvex MINLPs (Tech. Rep.). Tepper, School of Business, Carnegie Mellon University. March.

  • Nemhauser, G., Savelsbergh, M., & Sigismondi, G. (1994). MINTO, a mixed INTeger optimizer. Operations Research Letters, 15, 47–585.

    Article  Google Scholar 

  • NEOS. URL www-neos.mcs.anl.gov/neos (v. 5.0).

  • Nocedal, J., & Wright, S. (2006). Springer series in operations research. Numerical optimization.

    Google Scholar 

  • Nowak, I. (2005). International series of numerical mathematics. Relaxation and decomposition methods for mixed integer nonlinear programming. Berlin: Birkhäuser.

    Google Scholar 

  • Nowak, I., & Vigerske, S. (2008). LaGO—a (heuristic) branch and cut algorithm for nonconvex MINLPs. Central European Journal of Operations Research, 16, 127–138.

    Article  Google Scholar 

  • Quesada, I., & Grossmann, I. (1992). An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Computers & Chemical Engineering, 16, 937–947.

    Article  Google Scholar 

  • Ryoo, H., & Sahinidis, N. (1996). A branch-and-reduce approach to global optimization. Journal of Global Optimization, 8, 107–138.

    Article  Google Scholar 

  • Sahinidis, N. (1996). BARON: a general purpose global optimization software package. Journal of Global Optimization, 8, 201–205.

    Article  Google Scholar 

  • Schweiger, C., & Floudas, C. (1998a). MINOPT: a modeling language and algorithmic framework for linear, mixed-integer, nonlinear, dynamic, and mixed-integer nonlinear optimization. Princeton: Princeton University Press.

    Google Scholar 

  • Schweiger, C., & Floudas, C. (1998b). MINOPT: a software package for mixed-integer nonlinear optimization (3rd ed.).

  • SCIP. URL http://scip.zib.de/scip.shtml.

  • Smith, E., & Pantelides, C. (1999). A symbolic reformulation/spatial branch and bound algorithm for the global optimization of nonconvex MINLPs. Computers & Chemical Engineering, 23, 457–478.

    Article  Google Scholar 

  • Tawarmalani, M., & Sahinidis, N. (2004). Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Mathematical Programming, 99, 563–591.

    Article  Google Scholar 

  • Vigerske, S. (2012). Decomposition in multistage stochastic programming and a constraint integer programming approach to mixed-integer nonlinear programming. PhD Thesis, Humboldt-Universität zu Berlin.

  • Westerlund, T., & Pettersson, F. (1995). A cutting plane method for solving convex MINLP problems. Computers & Chemical Engineering, 19, S131–S136.

    Article  Google Scholar 

  • Westerlund, T., & Pörn, R. (2002). Solving pseudo-convex mixed integer problems by cutting plane techniques. Optimization and Engineering, 3, 253–280.

    Article  Google Scholar 

  • Westerlund, T., Skrifvars, H., Harjunkoski, I., & Pörn, R. (1998). An extended cutting plane method for solving a class of non-convex MINLP problems. Computers & Chemical Engineering, 22, 357–365.

    Article  Google Scholar 

  • XML-RPC. URL http://www.xmlrpc.com.

  • XPRESS. URL http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx.

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Acknowledgements

We are grateful to Silvano Martello for precious suggestions. Thanks are also due to Stefan Vigerske for useful comments and discussions.

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Correspondence to Andrea Lodi.

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This is an updated version of the paper that appeared in 4OR, 9(4), 329–349 (2011).

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D’Ambrosio, C., Lodi, A. Mixed integer nonlinear programming tools: an updated practical overview. Ann Oper Res 204, 301–320 (2013). https://doi.org/10.1007/s10479-012-1272-5

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