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A new technique to derive tight convex underestimators (sometimes envelopes)

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  • M. Locatelli

    (Università di Parma)

Abstract

The convex envelope value for a given function f over a region X at some point $$\textbf{x}\in X$$ x ∈ X can be derived by searching for the largest value at that point among affine underestimators of f over X. This can be computed by solving a maximin problem, whose exact computation, however, may be a hard task. In this paper we show that by relaxation of the inner minimization problem, duality, and, in particular, by an enlargement of the class of underestimators (thus, not only affine ones) an easier derivation of good convex understimating functions, which can also be proved to be convex envelopes in some cases, is possible. The proposed approach is mainly applied to the derivation of convex underestimators (in fact, in some cases, convex envelopes) in the quadratic case. However, some results are also presented for polynomial, ratio of polynomials, and some other separable functions over regions defined by similarly defined separable functions.

Suggested Citation

  • M. Locatelli, 2024. "A new technique to derive tight convex underestimators (sometimes envelopes)," Computational Optimization and Applications, Springer, vol. 87(2), pages 475-499, March.
    • Handle: RePEc:spr:coopap:v:87:y:2024:i:2:d:10.1007_s10589-023-00534-8
    DOI: 10.1007/s10589-023-00534-8
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    References listed on IDEAS

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    1. Ben-Tal, A. & den Hertog, D. & Laurent, M., 2011. "Hidden Convexity in Partially Separable Optimization," Other publications TiSEM 56b82c13-ee8f-4072-be97-f, Tilburg University, School of Economics and Management.
    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.
    3. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    4. Rida Laraki & Jean-Bernard Lasserre, 2008. "Computing uniform convex approximations for convex envelopes and convex hulls," Post-Print hal-00243009, HAL.
    5. Ben-Tal, A. & den Hertog, D. & Laurent, M., 2011. "Hidden Convexity in Partially Separable Optimization," Discussion Paper 2011-070, Tilburg University, Center for Economic Research.
    6. Marco Locatelli, 2016. "Non polyhedral convex envelopes for 1-convex functions," Journal of Global Optimization, Springer, vol. 65(4), pages 637-655, August.
    7. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    Full references (including those not matched with items on IDEAS)

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