Advertisement

On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

Conference paper
  • 496 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12125)

Abstract

Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.

Keywords

Surrogate relaxation MINLP Nonconvex optimization 

Notes

Acknowledgements

We gratefully acknowledge support from the Research Campus MODAL (BMBF Grant 05M14ZAM) and the Institute for Data Valorization (IVADO) through an IVADO Postdoctoral Fellowship.

References

  1. 1.
    MUMPS: Multifrontal massively parallel sparse direct solver. http://mumps.enseeiht.fr
  2. 2.
    Achterberg, T.: Constraint integer programming. Ph.D. thesis, Technische Universität Berlin (2007).  https://doi.org/10.14279/depositonce-1634. URN:nbn:de:kobv:83-opus-16117
  3. 3.
    Achterberg, T., Wunderling, R.: Mixed integer programming: analyzing 12 years of progress. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 449–481. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38189-8_18CrossRefzbMATHGoogle Scholar
  4. 4.
    van Ackooij, W., Frangioni, A., de Oliveira, W.: Inexact stabilized benders’ decomposition approaches with application to chance-constrained problems with finite support. Comput. Optim. Appl. 65(3), 637–669 (2016).  https://doi.org/10.1007/s10589-016-9851-zMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alidaee, B.: Zero duality gap in surrogate constraint optimization: a concise review of models. Eur. J. Oper. Res. 232(2), 241–248 (2014).  https://doi.org/10.1016/j.ejor.2013.04.023MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Amor, H.M.B., Desrosiers, J., Frangioni, A.: On the choice of explicit stabilizing terms in column generation. Discrete Appl. Math. 157(6), 1167–1184 (2009).  https://doi.org/10.1016/j.dam.2008.06.021MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Balas, E.: Discrete programming by the filter method. Oper. Res. 15(5), 915–957 (1967).  https://doi.org/10.1287/opre.15.5.915MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discrete Appl. Math. 89(1–3), 3–44 (1998).  https://doi.org/10.1016/s0166-218x(98)00136-xMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Banerjee, K.: Generalized Lagrange multipliers in dynamic programming. Ph.D. thesis, University of California, Berkeley (1971)Google Scholar
  10. 10.
    Bonami, P., Lodi, A., Tramontani, A., Wiese, S.: On mathematical programming with indicator constraints. Math. Program. 151(1), 191–223 (2015).  https://doi.org/10.1007/s10107-015-0891-4MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    COIN-OR: CppAD, a package for differentiation of C++ algorithms. http://www.coin-or.org/CppAD
  12. 12.
    COIN-OR: Ipopt, Interior point optimizer. http://www.coin-or.org/Ipopt
  13. 13.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods. Society for Industrial and Applied Mathematics, Philadelphia (2000).  https://doi.org/10.1137/1.9780898719857CrossRefzbMATHGoogle Scholar
  14. 14.
    Djerdjour, M., Mathur, K., Salkin, H.M.: A surrogate relaxation based algorithm for a general quadratic multi-dimensional knapsack problem. Oper. Res. Lett. 7(5), 253–258 (1988).  https://doi.org/10.1016/0167-6377(88)90041-7MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dyer, M.E.: Calculating surrogate constraints. Math. Program. 19(1), 255–278 (1980).  https://doi.org/10.1007/bf01581647MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fisher, M., Lageweg, B., Lenstra, J., Kan, A.: Surrogate duality relaxation for job shop scheduling. Discrete Appl. Math. 5(1), 65–75 (1983).  https://doi.org/10.1016/0166-218x(83)90016-1CrossRefzbMATHGoogle Scholar
  17. 17.
    Gavish, B., Pirkul, H.: Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality. Math. Program. 31(1), 78–105 (1985).  https://doi.org/10.1007/bf02591863MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Geoffrion, A.M.: Implicit enumeration using an imbedded linear program. Technical report, May 1967.  https://doi.org/10.21236/ad0655444
  19. 19.
    Glover, F.: A multiphase-dual algorithm for the zero-one integer programming problem. Oper. Res. 13(6), 879–919 (1965).  https://doi.org/10.1287/opre.13.6.879CrossRefzbMATHGoogle Scholar
  20. 20.
    Glover, F.: Surrogate constraints. Oper. Res. 16(4), 741–749 (1968).  https://doi.org/10.1287/opre.16.4.741MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Glover, F.: Surrogate constraint duality in mathematical programming. Oper. Res. 23(3), 434–451 (1975).  https://doi.org/10.1287/opre.23.3.434MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Glover, F.: Heuristics for integer programming using surrogate constraints. Decis. Sci. 8(1), 156–166 (1977).  https://doi.org/10.1111/j.1540-5915.1977.tb01074.xCrossRefGoogle Scholar
  23. 23.
    Glover, F.: Tutorial on surrogate constraint approaches for optimization in graphs. J. Heuristics 9(3), 175–227 (2003).  https://doi.org/10.1023/a:1023721723676CrossRefzbMATHGoogle Scholar
  24. 24.
    Gomory, R.E.: An algorithm for the mixed integer problem. Technical report. P-1885, The RAND Corporation, June 1960Google Scholar
  25. 25.
    Greenberg, H.J., Pierskalla, W.P.: Surrogate mathematical programming. Oper. Res. 18(5), 924–939 (1970).  https://doi.org/10.1287/opre.18.5.924MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Grossmann, I.E., Sahinidis, N.V.: Special issue on mixed integer programmingand its application to engineering, part I. Optim. Eng. 3(4), 52–76 (2002)Google Scholar
  27. 27.
    Hendel, G.: Empirical analysis of solving phases in mixed integer programming. Master’s thesis, Technische Universität Berlin, August 2014. URN:nbn:de: http://nbn-resolving.de/urn:nbn:de:0297-zib-54270
  28. 28.
    Horst, R., Tuy, H.: Global Optimization. Springer, Berlin Heidelberg (1996). DOI:  https://doi.org/10.1007/978-3-662-03199-5
  29. 29.
    ILOG, I.: ILOG CPLEX: High-performance software for mathematical programming and optimization. http://www.ilog.com/products/cplex/
  30. 30.
    Kelley Jr., J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960).  https://doi.org/10.1137/0108053MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Junttila, T., Kaski, P.: Bliss: a tool for computing automorphism groups and canonical labelings of graphs. (2012). http://www.tcs.hut.fi/Software/bliss/
  32. 32.
    Karwan, M.H.: Surrogate constraint duality and extensions in integer programming. Ph.D. thesis, Georgia Institute of Technology, January 1976Google Scholar
  33. 33.
    Karwan, M.H., Rardin, R.L.: Some relationships between Lagrangian and surrogate duality in integer programming. Math. Program. 17(1), 320–334 (1979).  https://doi.org/10.1007/bf01588253MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Karwan, M.H., Rardin, R.L.: Surrogate dual multiplier search procedures in integer programming. Oper. Res. 32(1), 52–69 (1984).  https://doi.org/10.1287/opre.32.1.52MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kim, S.L., Kim, S.: Exact algorithm for the surrogate dual of an integer programming problem: subgradient method approach. J. Optim. Theory Appl. 96(2), 363–375 (1998).  https://doi.org/10.1023/a:1022622231801MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part i – convex underestimating problems. Math. Program. 10(1), 147–175 (1976).  https://doi.org/10.1007/bf01580665CrossRefzbMATHGoogle Scholar
  37. 37.
    du Merle, O., Villeneuve, D., Desrosiers, J., Hansen, P.: Stabilized column generation. Discrete Math. 194(1–3), 229–237 (1999).  https://doi.org/10.1016/s0012-365x(98)00213-1MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
  39. 39.
    Nakagawa, Y.: An improved surrogate constraints method for separable nonlinear integer programming. J. Oper. Res. Soc. Jpn 46(2), 145–163 (2003).  https://doi.org/10.15807/jorsj.46.145MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Narciso, M.G., Lorena, L.A.N.: Lagrangean/surrogate relaxation for generalized assignment problems. Eur. J. Oper. Res. 114(1), 165–177 (1999).  https://doi.org/10.1016/s0377-2217(98)00038-1CrossRefzbMATHGoogle Scholar
  41. 41.
    Nemhauser, G.L., Wolsey, L.A.: A recursive procedure to generate all cuts for 0–1 mixed integer programs. Math. Program. 46(1–3), 379–390 (1990).  https://doi.org/10.1007/bf01585752MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Quesada, I., Grossmann, I.E.: A global optimization algorithm for linear fractional and bilinear programs. J. Glob. Optim. 6(1), 39–76 (1995).  https://doi.org/10.1007/bf01106605MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ryoo, H., Sahinidis, N.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19(5), 551–566 (1995).  https://doi.org/10.1016/0098-1354(94)00097-2CrossRefGoogle Scholar
  44. 44.
    Sarin, S., Karwan, M.H., Rardin, R.L.: A new surrogate dual multiplier search procedure. Naval Res. Logistics 34(3), 431–450 (1987).  https://doi.org/10.1002/1520-6750(198706)34:3<431::aid-nav3220340309>3.0.co;2-pMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    SCIP - Solving Constraint Integer Programs. http://scip.zib.de
  46. 46.
    Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Springer, New York (1999).  https://doi.org/10.1007/978-1-4757-4388-3CrossRefzbMATHGoogle Scholar
  47. 47.
    Sherali, H.D., Fraticelli, B.M.P.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Glob. Optim. 22(1/4), 233–261 (2002).  https://doi.org/10.1023/a:1013819515732MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Templeman, A.B., Xingsi, L.: A maximum entropy approach to constrained non-linear programming. Eng. Optim. 12(3), 191–205 (1987).  https://doi.org/10.1080/03052158708941094CrossRefGoogle Scholar
  49. 49.
    Vielma, J.P.: Small and strong formulations for unions of convex sets from the Cayley embedding. Math. Program. 177(1–2), 21–53 (2018).  https://doi.org/10.1007/s10107-018-1258-4MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Vigerske, S.: Decomposition in multistage stochastic programming and a constraint integer programming approach to mixed-integer nonlinear programming. Ph.D. thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II (2013). URN:nbn:de:kobv:11-100208240
  51. 51.
    Vigerske, S., Gleixner, A.: SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework. Optim. Methods Softw. 33(3), 563–593 (2017).  https://doi.org/10.1080/10556788.2017.1335312MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2005).  https://doi.org/10.1007/s10107-004-0559-yMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Xingsi, L.: An aggregate constraint method for non-linear programming. J. Oper. Res. Soc. 42(11), 1003–1010 (1991).  https://doi.org/10.1057/jors.1991.190CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Universidad de O’HigginsRancaguaChile
  3. 3.Polytechnique MontréalMontréalCanada

Personalised recommendations