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Journal of Global Optimization

, Volume 10, Issue 4, pp 425–437 | Cite as

A Convex Envelope Formula for Multilinear Functions

  • Anatoliy D. Rikun
Article

Abstract

Convex envelopes of multilinear functions on a unit hypercube arepolyhedral. This well-known fact makes the convex envelopeapproximation very useful in the linearization of non-linear 0–1programming problems and in global bilinear optimization. This paperpresents necessary and sufficient conditions for a convex envelope to be apolyhedral function and illustrates how these conditions may be used inconstructing of convex envelopes. The main result of the paper is a simpleanalytical formula, which defines some faces of the convex envelope of amultilinear function. This formula proves to be a generalization of the wellknown convex envelope formula for multilinear monomial functions.

Nonlinear 0–1 optimization linearization convex envelope concave extension bilinear programming global optimization 

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References

  1. 1.
    F.A. Al-Khayyal and J.E. Falk, Jointly constrained biconvex programming. Mathematics of Operations Research 8 (1983), 272–286.Google Scholar
  2. 2.
    E. Balas and J.B. Mazzola, Nonlinear 0–1 programming: I. Linearization techniques, Mathematical Programming 30 (1984), 1–21.Google Scholar
  3. 3.
    E. Balas and J.B. Mazzola, Nonlinear 0–1 programming: II. Dominance relations and algorithms, Mathematical Programming 30 (1984), 22–45.Google Scholar
  4. 4.
    Y. Crama, Concave extension for nonlinear 0–1 maximization problems, Mathematical Programming 61 (1993), 53–60.Google Scholar
  5. 5.
    Y. Crama, Recognition problems for polynomials in 0–1 variables, Mathematical Programming 44 (1989), 139–155.Google Scholar
  6. 6.
    F. Glover and E. Woolsey, Converting the 0–1 polynomial programming problem to a linear 0–1 program, Operations Research 22 (1974), 180–182.Google Scholar
  7. 7.
    P.L. Hammer, P. Hansen and B. Simeone, Roof duality, complementation and persistency in quadratic 0–1 optimization, Mathematical Programming 28 (1984), 121–155.Google Scholar
  8. 8.
    P.L. Hammer and S. Rudeanu, Boolean Methods in Operations Research and Related Areas (Springer, Berlin, New York, 1968).Google Scholar
  9. 9.
    P. Hansen and B. Simeone, On the equivalence of paved duality and standard linearization in nonlinear 0–1 optimization, Discrete Applied Mathematics 29 (1990), 187–193.Google Scholar
  10. 10.
    R. Horst and H. Tuy, Global Optimization, Deterministic Approaches (Springer-Verlag, Berlin, 1993).Google Scholar
  11. 11.
    A. Ioffe and V. Tihomirov, Teoria extremal’nih zadach (Theory of optimization problems), Moscow, Nauka (Science), 1974 (in Russian).Google Scholar
  12. 12.
    M. Minoux Programmation mathematique. Theorie et algorithmes dunod, Bordas et C.N.E.T.-E.N.S.T., Paris, 1989.Google Scholar
  13. 13.
    P. Pardalos, On the passage from local to global optimization, In: J.R. Birge and K.G. Murty, Mathematical Programming: The State of the Art (The University of Michigan, 1994, p. 220–247).Google Scholar
  14. 14.
    A.D. Rikun, Branch-and-Bound Method in Solution of Some Non-Convex Mathematical Programming Problems in Water Pollution Control .Ph.D. Thesis, USSR Academy of Sci. Computer Center, Moscow, 1982.Google Scholar
  15. 15.
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  16. 16.
    H.D. Sherali and A. Alameddine, An Explicit Characterization of the Convex Envelope of a Bivariate Bilinear Function over Special Polytopes, Annals of Operations Research, Computational Methods in Global Optimization, Eds. P.M. Pardalos and J.B. Rosen, Vol. 25, pp. 197–210, 1990.Google Scholar
  17. 17.
    H.D. Sherali and C.H. Tuncbilek, A Global Optimization Algorithm for Polynomial Programming Problems Using a Reformulation-Linearization Technique, Journal of Global Optimization 2 (1992), 101–112.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Anatoliy D. Rikun
    • 1
  1. 1.East Coast Product GroupHamdenU.S.A.

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