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Global optimization of nonconvex problems with convex-transformable intermediates

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Abstract

This paper addresses the global optimization of problems which contain convex-transformable functions. We present algorithms for identification of convex-transformable functions in general nonconvex problems, and introduce a new class of cutting planes based on recently developed relaxations for convex-transformable functions. These cutting planes correspond to the supporting hyperplanes of these convex relaxations. We integrate our recognition and cutting plane generation algorithms into the global solver BARON, and test our implementation by conducting numerical experiments on a large collection of nonconvex problems. Results demonstrate that the proposed implementation accelerates the convergence speed of the branch-and-bound algorithm, by significantly reducing computational time, number of nodes in the search tree, and required memory.

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References

  1. Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum Press, New York (1988)

    Book  MATH  Google Scholar 

  3. Ayotte-Sauv, E.: NRC Library, Personal Communication (2016)

  4. Bao, X., Khajavirad, A., Sahinidis, N.V., Tawarmalani, M.: Global optimization of nonconvex problems with multilinear intermediates. Math. Program. Comput. 7, 1–37 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao, X., Sahinidis, N.V., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs. Optim. Methods Softw. 24, 485–504 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berthold, T., Gamrath, G., Hendel, G., Heinz, S., Koch, T., Pfetsch, M., Vigerske, S., Waniek, R., Winkler, M., Wolter, K.: SCIP 3.2, User’s Manual (2016)

  8. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  9. Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib—a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15, 114–119 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. GLOBAL Library. http://www.gamsworld.org/global/globallib.htm. Accessed 20 June 2017

  12. Grossmann, I.E.: Mixed-integer programming approach for the synthesis of integrated process flowsheets. Comput. Chem. Eng. 9, 463–482 (1985)

    Article  Google Scholar 

  13. Hunting, M.: AIMMS Library, Personal Communication (2016)

  14. Jach, M., Michaels, D., Weismantel, R.: The convex envelope of \((n-1)\)-convex functions. SIAM J. Optim. 19, 1451–1466 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Khajavirad, A., Michalek, J.J.: A deterministic Lagrangian-based global optimization approach for quasiseparable nonconvex mixed-integer nonlinear programs. J. Mech. Des. 131, 051009 (2009)

    Article  Google Scholar 

  16. Khajavirad, A., Michalek, J.J., Sahinidis, N.V.: Relaxations of factorable functions with convex-transformable intermediates. Math. Program. 144, 107–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khajavirad, A., Sahinidis, N.V.: Convex envelopes of products of convex and component-wise concave functions. J. Glob. Optim. 52, 391–409 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program. 137, 371–408 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khajavirad, A., Sahinidis, N.V.: A hybrid LP/NLP paradigm for global optimization relaxations. Mathematical Programming Computation, Under review after minor revision (2017)

  20. Li, H., Tsai, J., Floudas, C.A.: Convex underestimation for posynomial functions of positive variables. Optim. Lett. 2, 333–340 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, Y., Schrage, L.: The global solver in the LINDO API. Optim. Methods Softw. 24, 657–668 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lu, H., Li, H., Gounaris, C.E., Floudas, C.A.: Convex relaxation for solving posynomial programs. J. Glob. Optim. 46, 147–154 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lundell, A., Westerlund, J., Westerlund, T.: Some transformation techniques with applications in global optimization. J. Glob. Optim. 43, 391–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lundell, A., Westerlund, T.: Convex underestimation strategies for signomial functions. Optim. Methods Softw. 24, 505–522 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I-Convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207–224 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Misener, R., Floudas, ChA: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59, 503–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Misener, R., Gounaris, C.E., Floudas, C.A.: Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emissions constraints. Comput. Chem. Eng. 34, 1432–1456 (2010)

    Article  Google Scholar 

  30. Polisetty, P.K., Gatzke, E.P., Voit, E.O.: Yield optimization of regulated metabolic systems using deterministic branch-and-reduce methods. Biotechnol. Bioeng. 99, 1154–1169 (2008)

    Article  Google Scholar 

  31. Princeton Library. http://www.gamsworld.org/performance/princetonlib/princetonlib.htm. Accessed 20 June 2017

  32. Rikun, A.D.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10, 425–437 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8, 201–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sahinidis, N.V.: BARON 15.9.22: global optimization of mixed-integer nonlinear programs, User’s Manual (2015)

  35. Sahinidis, N.V., Tawarmalani, M., Yu, M.: Design of alternative refrigerants via global optimization. AIChE J. 49, 1761–1775 (2003)

    Article  Google Scholar 

  36. Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22, 245–270 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Sherali, H.D., Wang, H.: Global optimization of nonconvex factorable programming problems. Math. Program. 89, 459–478 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tawarmalani, M., Richard, J.-P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. (2012). https://doi.org/10.1007/s10107-012-0581-4

    MATH  Google Scholar 

  39. Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel techniques for constructing convex envelopes of nonlinear functions. J. Glob. Optim. 20, 137–158 (2001)

    Article  MATH  Google Scholar 

  40. Tawarmalani, M., Sahinidis, N.V.: Convex extensions and convex envelopes of l.s.c. functions. Math. Program. 93, 247–263 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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)

    Book  MATH  Google Scholar 

  42. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, USA

    Carlos J. Nohra & Nikolaos V. Sahinidis

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  1. Carlos J. Nohra
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  2. Nikolaos V. Sahinidis
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Correspondence to Nikolaos V. Sahinidis.

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Nohra, C.J., Sahinidis, N.V. Global optimization of nonconvex problems with convex-transformable intermediates. J Glob Optim 72, 255–276 (2018). https://doi.org/10.1007/s10898-018-0631-4

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  • DOI: https://doi.org/10.1007/s10898-018-0631-4

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