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Relaxations of factorable functions with convex-transformable intermediates

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Abstract

We propose to strengthen standard factorable relaxations of global optimization problems through the use of functional transformations of intermediate expressions. In particular, we exploit convex transformability of the component functions of factorable programs as a tool in the generation of bounds. We define suitable forms of transforming functions and assess, theoretically and computationally, the sharpness of the resulting relaxations in comparison to existing schemes.

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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  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Debreu, G.: Least concave utility functions. J. Math. Econ. 3, 121–129 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fenchel, W.: Convex Cones, Sets, and Functions (Mimeographed Lecture Notes). Princeton University Press (1953)

  6. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Kannai, Y.: Concavifiability and construction of concave utility functions. J. Math. Econ. 4, 1–56 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program. (2011). doi:10.1007/s10107-011-0496-5

  10. Khajavirad, A., Sahinidis, N.V.: Convex envelopes of products of convex and component-wise concave functions. J. Glob. Optim. 52, 391–409 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Maranas, C.D., Floudas, C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21, 351–369 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Prékopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. 34, 335–343 (1973)

    MATH  Google Scholar 

  20. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton (1970)

  21. Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19, 403–424 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Sahinidis, N.V., Tawarmalani, M.: BARON 9.3: Global Optimization of Mixed-Integer Nonlinear Programs. User’s Manual (2011)

  24. Schaible, S.: Quasiconvex optimization in general real linear spaces. Zeitschrift fur Operations Research 16, 205–213 (1972)

    MATH  MathSciNet  Google Scholar 

  25. Schaible, S.: Maximization of quasiconcave quotients and products of finitely many functional. Cahiers du Centre d’Etudes de Recherche Opérationelle 16, 45–53 (1974)

    MATH  MathSciNet  Google Scholar 

  26. Schaible, S., Ziemba, W.T.: Generalized Concavity in Optimization and Economics. Academic Press, London (1981)

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. 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

  30. 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  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank two anonymous referees for comments and suggestions that improved the quality of this manuscript.

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Authors and Affiliations

  1. Mathematical Sciences, IBM T. J. Watson Research Center, Yorktown Heights, NY, USA

    Aida Khajavirad

  2. Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

    Jeremy J. Michalek

  3. Department of Engineering and Public Policy, Carnegie Mellon University, Pittsburgh, PA, USA

    Jeremy J. Michalek

  4. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

    Nikolaos V. Sahinidis

Authors
  1. Aida Khajavirad
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  2. Jeremy J. Michalek
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  3. Nikolaos V. Sahinidis
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Corresponding author

Correspondence to Nikolaos V. Sahinidis.

Additional information

This research was supported in part by National Science Foundation award CMII-1030168.

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Khajavirad, A., Michalek, J.J. & Sahinidis, N.V. Relaxations of factorable functions with convex-transformable intermediates. Math. Program. 144, 107–140 (2014). https://doi.org/10.1007/s10107-012-0618-8

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  • DOI: https://doi.org/10.1007/s10107-012-0618-8

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