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Extended formulations for convex envelopes

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  • Martin Ballerstein
  • Dennis Michaels

Abstract

In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables. These functions account for more than 30 % of all nonlinearities in common benchmark libraries. To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope based on the Reformulation–Linearization-Technique introduced by Sherali and Adams (SIAM J Discret Math 3(3):411–430, 1990 ). Computational results are reported showing that the extended formulation strategy is a useful tool in global optimization. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Martin Ballerstein & Dennis Michaels, 2014. "Extended formulations for convex envelopes," Journal of Global Optimization, Springer, vol. 60(2), pages 217-238, October.
    • Handle: RePEc:spr:jglopt:v:60:y:2014:i:2:p:217-238
    DOI: 10.1007/s10898-013-0104-8
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    References listed on IDEAS

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    1. Warren Adams & Hanif Sherali, 2005. "A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems," Annals of Operations Research, Springer, vol. 140(1), pages 21-47, November.
    2. Sonia Cafieri & Jon Lee & Leo Liberti, 2010. "On convex relaxations of quadrilinear terms," Journal of Global Optimization, Springer, vol. 47(4), pages 661-685, August.
    3. Hanif Sherali & Evrim Dalkiran & Leo Liberti, 2012. "Reduced RLT representations for nonconvex polynomial programming problems," Journal of Global Optimization, Springer, vol. 52(3), pages 447-469, March.
    4. Hanif Sherali & Evrim Dalkiran & Jitamitra Desai, 2012. "Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts," Computational Optimization and Applications, Springer, vol. 52(2), pages 483-506, June.
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    Cited by:

    1. Marco Locatelli, 2020. "Convex envelope of bivariate cubic functions over rectangular regions," Journal of Global Optimization, Springer, vol. 76(1), pages 1-24, January.
    2. 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.
    3. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.

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