Mathematical Programming

, Volume 103, Issue 2, pp 207–224 | Cite as

Convex envelopes for edge-concave functions

  • Clifford A. Meyer
  • Christodoulos A. Floudas


Deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions. In this paper we describe the structure of the polyhedral convex envelopes of edge-concave functions over polyhedral domains using geometric arguments. An algorithm for computing the facets of the convex envelope over hyperrectangles in ℝ3 is described. Sufficient conditions are described under which the convex envelope of a sum of edge-concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions.


Optimization Algorithm Global Optimization Mathematical Method Global Optimization Algorithm Polyhedral Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8 (2), 273–286 (1983)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont, Massachusetts, 2003Google Scholar
  3. 3.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1993Google Scholar
  4. 4.
    Cottle, R.W.: Minimal triangulations of the 4-cube. Discrete Math. 40, 25–29 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Crama, Y.: Concave extensions for nonlinear 0-1 maximization problems. Math. Program. 61, 53–60 (1993)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Floudas, C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications. Kluwer Academic Publishers, 2000Google Scholar
  7. 7.
    Haiman, M.: A simple and relatively efficient triangulation of the n-cube. Discrete Comput. Geom. 6, 287–289 (1991)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Horst, R., Tuy, H.: Global optimization: deterministic approaches. Springer-Verlag, Berlin, 1993. 2nd. rev. editionGoogle Scholar
  9. 9.
    Hughes, R.B., Anderson, M.R.: Simplexity of the cube. Discrete Mathematics. 158, 99–150 (1999)CrossRefGoogle Scholar
  10. 10.
    Hughs, R.B.: Lower bounds on cube simplexity. Discrete Math. 133, 123–138 (1994)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lee, C.W.: Triangulating the d-cube. In: J.W. Goodman (ed.), Discrete Geometry and Convexity. Volume 440 of Ann. N.Y. Acad. Sci., 1985, pp. 205–212Google Scholar
  12. 12.
    Mara, P.S.: Triangulations for the cube. J. Comb. Theory Ser. A. 20, 170–176 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I – convex underestimating problems. Math. Program. 10, 147–175 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Meyer, C.A.: Convex Relaxations and Convex Envelopes for Deterministic Global Optimization. PhD thesis, Princeton University, Princeton, New Jersey, 2004Google Scholar
  15. 15.
    Meyer, C.A., Floudas, C.A.: Convex envelopes of trilinear monomials with mixed sign domains. In: C.A. Floudas, P.M. Pardalos (eds.), Frontiers in Global Optimization, Dordrecht, 2003. Kluwer Academic Publishers, pp. 327–352Google Scholar
  16. 16.
    Meyer, C.A., Floudas, C.A. Convex envelopes of trilinear monomials with positive or negative domains. J. Global Optim. 29, 125–155 (2004)Google Scholar
  17. 17.
    Rikun, A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10, 425–437 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Sallee, J.F.: A triangulation of the n-cube. Discrete Math. 40, 81–86 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Mathematica Vietnamica 22, 245–270 (1997)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Sherali, H.D., Adam, W. P.: A Reformulation-Linearization Technique for Solving Discrete and Continous Nonconvex Problems. Volume 31, Kluwer Academic Publishers, Dordrecht, 1999Google Scholar
  21. 21.
    Sherali, H.D., Alameddine, A.: A new reformulation-linearization technique for bilinear programming problems. J. Global Optim. 2, 379–410 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Smith, W.D.: A lower bound on the simplexity of the n-cube via hyperbolic volumes. Europ. J. Combinatorics 21, 131–137 (2000)CrossRefzbMATHGoogle Scholar
  23. 23.
    Tardella, F.: On the existence of polyhedral convex envelopes. In: C.A. Floudas, P.M. Pardalos (eds.), Frontiers in Global Optimization, Dordrecht, 2003. Kluwer Academic Publishers, pp. 563–574Google Scholar
  24. 24.
    Tawarmalani, M., Ahmed, S., Sahinidis, N.V.: Product disaggregation in global optimization and relaxations of rational programs. Optimization and Engineering 3, 281–303 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and convex envelopes of lower semi-continuous functions. Math. Program. 2, 247–263 (2002)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Ziegler, G.M.: Lectures on Polytopes. Volume 152 of Graduate Texts in Mathematics. Springer Verlag, 1994Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Chemical EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations