As an alternative to traditional integer programming (IP), decision diagrams (DDs) provide a new solution technology for discrete problems exploiting their combinatorial structure and dynamic programming representation. While the literature mainly focuses on the competitive aspects of DDs as a stand-alone solver, we investigate their complementary role by introducing IP techniques that can be derived from DDs and used in conjunction with IP to enhance the overall performance. This perspective allows for studying problems with more general structure than those typically modeled via recursive formulations. In particular, we develop linear programming and subgradient-type methods to generate valid inequalities for the convex hull of the feasible region described by DDs. For convex IPs, these cutting planes dominate the so-called linearized cuts used in the outer approximation framework. These cutting planes can also be derived for nonconvex IPs, which leads to a generalization of the outer approximation framework. Computational experiments show significant optimality gap improvement for integer nonlinear programs over the traditional cutting plane methods employed in the state-of-the-art solvers.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Andersen, H.R., Hadz̆ić, T., Hooker, J.N., Tiedemann, P.: A constraint store based on multivalued decision diagrams. In: Bessiére, C. (ed.) Principles and Practice of Constraint Programming-CP 2007, vol. 4741, pp. 118–132. Springer, New York (2007)
Balas, E.: Disjunctive programming. Ann. Discrete Math. 5, 3–51 (1979)
Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Discrete Math. 6, 466–486 (1985)
Behle, M.: Binary decision diagrams and integer programming. PhD thesis, Max Planck Institute for Computer Science (2007)
Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex minlp. Optim. Methods Softw. 24, 597–634 (2009)
Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numberica 22, 1–131 (2013)
Bergman, D., Cire, A.A.: Discrete nonlinear optimization by state-space decompositions. Manag. Sci. 64, 4700–4720 (2018)
Bergman, D., Cire, A.A., van Hoeve, W.J., Hooker, J.: Optimization bounds from binary decision diagrams. INFORMS J. Comput. 26, 253–268 (2013)
Bergman, D., Cire, A.A., van Hoeve, W.J., Hooker, J.: Decision Diagrams for Optimization. Springer, New York (2016a)
Bergman, D., Cire, A.A., van Hoeve, W.J., Hooker, J.: Discrete optimization with decision diagrams. INFORMS J. Comput. 28, 47–66 (2016b)
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)
Bixby, R.E.: A brief history of linear and mixed integer programming computation. Documenta Mathematica, 107–121 (2012)
Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154. Springer, New York (2012a)
Bonami, P., Linderoth, J.T., Lodi, A.: Disjunctive cuts for mixed integer nonlinear programming problems. Technical Report (2012b)
Boukouvalaa, F., Misener, R., Floudas, C.A.: Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO. Eur. J. Oper. Res. 252, 701–727 (2016)
Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17, 97–106 (2012)
Ciré, A.A., van Hoeve, W.J.: Multivalued decision diagrams for sequencing problems. Oper. Res. 61, 1411–1428 (2013)
Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, New York (2014)
Dean, J.: Pricing policies for new products. Harv. Bus. Rev. 54, 141–153 (1976)
Dunning, I., Huchette, J., Lubin, M.: Jump: a modeling language for mathematical optimization. SIAM Rev. 59(2), 295–320 (2017). https://doi.org/10.1137/15M1020575
Duran, M.A., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)
Eppstein, D.: Quasiconvex programming. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, vol. 52, pp. 287–331. MSRI Publications, Cambridge (2005)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)
Geoffrion, A.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1985)
Hadz̆ić, T., Hooker, J.N.: Discrete global optimization with binary decision diagrams. In: Workshop on Global Optimization: Integrating Convexity, Optimization, Logic Programming, and Computational Algebraic Geormetry (GICOLAG) (2006)
Hemmecke, R., Köppe, M., Lee, J., Weismantel, R.: Nonlinear integer programming. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 561–618. Springer, New York (2009)
Holdershaw, J., Gendall, P., Garland, R.: The widespread use of odd pricing in the retail sector. Mark. Bull. 8, 53–58 (1997)
Kilinc, M., Linderoth, J., Luedtke, J.: Lift-and-project cuts for convex mixed integer nonlinear programs. Math. Program. Comput. 9, 499–526 (2017)
Kronqvist, J., Lundell, A., Westerlund, T.: The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64, 249–272 (2016)
Kronqvist, J., Bernal, D., Lundell, A., Grossmann, I.: A review and comparison of solvers for convex MINLP. Optim. Eng. 20, 397–455 (2017)
Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Syst. Tech. J. 38, 985–999 (1959)
Lozano, L., Smith, J.C.: A binary decision diagram based algorithm for solving a class of binary two-stage stochastic programs. Mathematical Programming pp 1–24 (2018)
Nagle, T., Hogan, J., Zale, J.: The Strategy and Tactics of Pricing: A Guide to Growing More Profitably. Prentice Hall, London (2011)
Quesada, I., Grossmann, I.E.: An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)
St-Aubin, R., Hoey, J., Boutilier, C.: Approximation policy construction using decision diagrams. In: Proceedings of Conference on Neural Information Processing Systems, Nantes, France, pp. 1089–1095 (2000)
Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Dordrecht (2002)
Tjandraatmadja, C., van Hoeve, W.J.: Target cuts from relaxed decision diagrams. INFORMS J. Comput. 31, 285–301 (2019)
Vigerske, S.: A library of mixed-integer and continuous nonlinear programming instances. https://minlplib.org (2019)
Wegener, I.: Branching Programs and Binary Decision Diagrams: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2000)
Westerlund, T., Pettersson, F.: A cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, s131–s136 (1995)
We thank Nick Sahinidis and Michael Bussieck for providing GAMS license for our experiments. We also thank the anonymous referees and the Associate Editor for their helpful comments that contributed to improving the paper.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Davarnia, D., van Hoeve, W. Outer approximation for integer nonlinear programs via decision diagrams. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01475-4
- Decision diagrams
- Cutting planes
- Outer approximation
- Dynamic programming
- Integer nonlinear programs
Mathematics Subject Classification