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Convex envelope of bivariate cubic functions over rectangular regions

  • Marco LocatelliEmail author
Article
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Abstract

In recent years many papers have derived polyhedral and non-polyhedral convex envelopes for different classes of functions. Except for the univariate cases, all these classes of functions share the property that the generating set of their convex envelope is a subset of the border of the region over which the envelope is computed. In this paper we derive the convex envelope over a rectangular region for a class of functions which does not have this property, namely the class of bivariate cubic functions without univariate third-degree terms.

Keywords

Convex envelope Generating set Cubic functions 

Notes

Acknowledgements

The author is extremely grateful to an anonymous reviewer for his/her careful reading and for the very detailed comments, which considerably helped to improve the paper with respect to its original version.

References

  1. 1.
    Ballerstein, M., Michaels, D.: Extended formulations for convex envelopes. J. Glob. Optim. 60, 217–238 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gounaris, C., Floudas, C.A.: Tight convex underestimators for \(C^2\)-continuous problems: I. Univariate functions. J. Glob. Optim. 42, 51–67 (2008)CrossRefGoogle Scholar
  3. 3.
    Jach, M., Michaels, D., Weismantel, R.: The convex envelope of \((n-1)\)-convex functions. SIAM J. Optim. 19(3), 1451–1466 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Khajavirad, A., Sahinidis, N.V.: Convex envelopes of products of convex and component-wise concave functions. J. Glob. Optim. 52, 391–409 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program. 137, 371–408 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim. 25, 157–168 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Laraki, R., Lasserre, J.B.: Computing uniform convex approximations for convex envelopes and convex hulls. J. Convex Anal. 15(3), 635–654 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Locatelli, M.: Convex envelopes of bivariate functions through the solution of KKT systems. J. Glob. Optim. 72(2), 277–303 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Locatelli, M.: Non polyhedral convex envelopes for 1-convex functions. J. Glob. Optim. 65(4), 637–655 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs. I. Convex underestimating problems. Math. Program. 10, 147–175 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207–224 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Misener, R., Floudas, C.A.: Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations. Math. Program. 136(1), 155–182 (2012) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rikun, A.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10, 425–437 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51, 569–606 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim. 20, 137–158 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tardella, F.: On the existence of polyhedral convex envelopes. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 563–574. Kluwer, Dordrecht (2003)Google Scholar
  17. 17.
    Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363–375 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tawarmalani, M., Richard, J.-P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138, 531–577 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria e ArchitetturaUniversità di ParmaParmaItaly

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