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Convex envelopes of bivariate functions through the solution of KKT systems

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  • Marco Locatelli

    (Università di Parma)

Abstract

In this paper we exploit a slight variant of a result previously proved in Locatelli and Schoen (Math Program 144:65–91, 2014) to define a procedure which delivers the convex envelope of some bivariate functions over polytopes. The procedure is based on the solution of a KKT system and simplifies the derivation of the convex envelope with respect to previously proposed techniques. The procedure is applied to derive the convex envelope of the bilinear function xy over any polytope, and the convex envelope of functions $$x^n y^m$$ x n y m over boxes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:2:d:10.1007_s10898-018-0626-1
    DOI: 10.1007/s10898-018-0626-1
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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. 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.
    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.
    4. Rida Laraki & Jean-Bernard Lasserre, 2008. "Computing uniform convex approximations for convex envelopes and convex hulls," Post-Print hal-00243009, HAL.
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    Cited by:

    1. M. Locatelli, 2022. "Exact and approximate results for convex envelopes of special structured functions over simplices," Journal of Global Optimization, Springer, vol. 83(2), pages 201-220, June.
    2. Marco Locatelli, 2020. "Convex envelope of bivariate cubic functions over rectangular regions," Journal of Global Optimization, Springer, vol. 76(1), pages 1-24, January.
    3. Ralf Lenz & Felipe Serrano, 2022. "Tight Convex Relaxations for the Expansion Planning Problem," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 325-352, July.

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