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Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes

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  • Marco Locatelli

    (Università di Parma)

Abstract

In this paper we show that the convex envelope over polytopes for a class of bivariate functions, including the bilinear and fractional functions as special cases, is characterized by a polyhedral subdivision of the polytopes, and is such that over each member of the subdivision the convex envelope has a given (although possibly only implicitly defined) functional form. For the bilinear and fractional case we show that there are three possible functional forms, which can be explicitly defined.

Suggested Citation

  • Marco Locatelli, 2016. "Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes," Journal of Global Optimization, Springer, vol. 66(4), pages 629-668, December.
    • Handle: RePEc:spr:jglopt:v:66:y:2016:i:4:d:10.1007_s10898-016-0418-4
    DOI: 10.1007/s10898-016-0418-4
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    References listed on IDEAS

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    1. Marco Locatelli, 2014. "A technique to derive the analytical form of convex envelopes for some bivariate functions," Journal of Global Optimization, Springer, vol. 59(2), pages 477-501, July.
    2. Rida Laraki & Jean-Bernard Lasserre, 2008. "Computing uniform convex approximations for convex envelopes and convex hulls," Post-Print hal-00243009, HAL.
    3. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
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    Cited by:

    1. Marcia Fampa & Jon Lee, 2021. "Convexification of bilinear forms through non-symmetric lifting," Journal of Global Optimization, Springer, vol. 80(2), pages 287-305, June.
    2. Marco Locatelli, 2018. "Convex envelopes of bivariate functions through the solution of KKT systems," Journal of Global Optimization, Springer, vol. 72(2), pages 277-303, October.

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