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Exact and approximate results for convex envelopes of special structured functions over simplices

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  • M. Locatelli

    (Università di Parma Parco Area delle Scienze)

Abstract

In this paper we describe how to derive the convex envelope of a function f over the n-dimensional unit simplex $$\Delta _n$$ Δ n at different levels of detail, depending on the properties of function f, by starting from its definition as the supremum of all the affine underestimators of f over $$\Delta _n$$ Δ n . At the first level we are able to derive the closed-form formula of the convex envelope. At the second level we are able to derive the exact value of the convex envelope at some point $${\mathbf{x}}\in \Delta _n$$ x ∈ Δ n , and a supporting hyperplane of the convex envelope itself at the same point, by solving a suitable convex optimization problem. Finally, at the third level we are able to derive an underestimating value which differs from the exact value of the convex envelope at some point $${\mathbf{x}}\in \Delta _n$$ x ∈ Δ n by at most a given threshold $$\delta $$ δ . The underestimation is obtained by solving a suitable LP problem and may lead also to a convex piecewise linear underestimator of f.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:2:d:10.1007_s10898-021-01112-0
    DOI: 10.1007/s10898-021-01112-0
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    References listed on IDEAS

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    1. 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.
    2. Martin Ballerstein & Dennis Michaels, 2014. "Extended formulations for convex envelopes," Journal of Global Optimization, Springer, vol. 60(2), pages 217-238, October.
    3. Rida Laraki & Jean-Bernard Lasserre, 2008. "Computing uniform convex approximations for convex envelopes and convex hulls," Post-Print hal-00243009, HAL.
    4. Marco Locatelli, 2020. "Convex envelope of bivariate cubic functions over rectangular regions," Journal of Global Optimization, Springer, vol. 76(1), pages 1-24, January.
    5. Marco Locatelli, 2016. "Non polyhedral convex envelopes for 1-convex functions," Journal of Global Optimization, Springer, vol. 65(4), pages 637-655, August.
    Full references (including those not matched with items on IDEAS)

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