Mathematical Programming Computation

, Volume 7, Issue 1, pp 1–37 | Cite as

Global optimization of nonconvex problems with multilinear intermediates

  • Xiaowei Bao
  • Aida Khajavirad
  • Nikolaos V. Sahinidis
  • Mohit Tawarmalani
Full Length Paper

Abstract

We consider global optimization of nonconvex problems containing multilinear functions. It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP). When used as cutting planes, these facets can significantly enhance the quality of conventional relaxations in general-purpose global solvers. However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function. To cope with this growth, we propose a graph decomposition scheme that exploits the structure of a multilinear function to decompose it to lower-dimensional components, for which the aforementioned LP can be solved very efficiently by employing a customized simplex algorithm. We embed this cutting plane generation strategy at every node of the branch-and-reduce global solver BARON, and carry out an extensive computational study on quadratically constrained quadratic problems, multilinear problems, and polynomial optimization problems. Results show that the proposed multilinear cuts enable BARON to solve many more problems to global optimality and lead to an average 60 % CPU time reduction.

Keywords

Multilinear functions Global optimization Convex envelope Polyhedral relaxations 

Mathematics Subject Classification

90C26 90C57 65K05 

References

  1. 1.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bao, X., Sahinidis, N.V., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs. Optim. Methods Softw. 24, 485–504 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Belotti, P.: COUENNE: A User’s Manual. Lehigh University, Technical report (2009)Google Scholar
  4. 4.
    Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientific, UK (1997)Google Scholar
  5. 5.
    Cafieri, S., Lee, J., Liberti, L.: On convex relaxations of quadrilinear terms. J. Glob. Optim. 47, 661–685 (2010)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Crama, Y.: Recognition problems in polynomials in \(0{-}1\) programming. Math. Program. 44, 139–155 (1989)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Crama, Y.: Concave extensions for nonlinear \(0{-}1\) maximization problems. Math. Program. 61, 53–60 (1993)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dolan, E., More, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci. 15, 550–569 (1969)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete problems. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 47–63. ACM, New York (1974)Google Scholar
  11. 11.
    Gill, P.E., Murray, W., Saunders, M.A.: User’s Guide for SNOPT 7.2.4: A FORTRAN Package for Large-Scale Nonlinear Programming. Technical report, University of California, San Diego and Stanford University, CA (2008)Google Scholar
  12. 12.
    Gray, F.: Pulse code communication. U.S. Patent No. 2,632,058 (1953)Google Scholar
  13. 13.
    Hendrickson, B., Leland, R.: A multilevel algorithm for partitioning graphs. In: Proceedings of the 1995 ACM/IEEE Conference on Supercomputing, p. 28. ACM, New York (1995)Google Scholar
  14. 14.
    Hopcroft, J., Tarjan, R.: Efficient algorithms for graph manipulation. Commun. ACM 16, 372–378 (1973)CrossRefGoogle Scholar
  15. 15.
  16. 16.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392 (1999)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49, 291–307 (1970)CrossRefMATHGoogle Scholar
  18. 18.
    Lin, Y., Schrage, L.: The global solver in the LINDO API. Optim. Methods Softw. 24, 657–668 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Luedtke, J., Namazifar, M., Linderoth, J.T.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136, 325–351 (2012)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Meyer, C.A., Floudas, C.A.: Trilinear monomials with positive or negative domains: facets of the convex and concave envelopes. In: Floudas, C.A., Pardolos, P.M. (eds.) Frontiers in Global Optimization, vol. 103, pp. 327–352. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  22. 22.
    Meyer, C.A., Floudas, C.A.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Glob. Optim. 29, 125–155 (2004)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207–224 (2005)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Murtagh, B.A., Saunders, M.A.: MINOS 5.5 User’s Guide. Technical Report SOL 83–20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, CA (1995)Google Scholar
  25. 25.
    Namazifar, M.: Strong Relaxations and Computations for Multilinear Programming. PhD thesis, Department of Industrial and Systems Engineering, University of Wisconsin–Madison (2011)Google Scholar
  26. 26.
    Rikun, A.D.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10, 425–437 (1997)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19, 403–424 (2001)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8, 201–205 (1996)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Sahinidis, N.V., Tawarmalani, M.: BARON 10.3: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2012)Google Scholar
  30. 30.
    Saunders, M.A.: LUMOD: Fortran software for updating dense LU factors. http://www.stanford.edu/group/SOL/software/lumod.html
  31. 31.
    Sherali, H.D.: A constructive proof of the representation theorem for polyhedral set based on fundamental definitions. Am. J. Math. Manag. Sci. 7, 253–270 (1987)MATHMathSciNetGoogle Scholar
  32. 32.
    Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22, 245–270 (1997)MATHMathSciNetGoogle Scholar
  33. 33.
    Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363–375 (2008)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Tawarmalani, M.: Inclusion certificates and simultaneous convexification of functions (2010). http://www.optimization-online.org/DB_HTML/2010/09/2722.html
  35. 35.
    Tawarmalani, M., Richard, J.-P., Chung, K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124, 481–512 (2010)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Tawarmalani, M., Richard, J.-P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. (2012). doi:10.1007/s10107-012-0581-4 Google Scholar
  37. 37.
    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)CrossRefGoogle Scholar
  38. 38.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • Xiaowei Bao
    • 1
  • Aida Khajavirad
    • 2
  • Nikolaos V. Sahinidis
    • 3
  • Mohit Tawarmalani
    • 4
  1. 1.IBMSan Francisco Bay AreaUSA
  2. 2.Business Analytics and Mathematical SciencesIBM T. J. Watson Research CenterYorktown HeightsUSA
  3. 3.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA
  4. 4.Krannert School of ManagementPurdue UniversityWest LafayetteUSA

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