Geodesic
Computing Uniform Convex Approximations for Convex Envelopes and Convex Hulls
Journal of convex analysis, Tome 15 (2008) no. 3, pp. 635-654

Voir la notice de l'article provenant de la source Heldermann Verlag

We provide a numerical procedure to compute uniform convex approximations $\{f_{r}\}$ of the convex envelope $\widehat{f}$ of a rational fraction $f$ defined on a compact basic semi-algebraic set $\mathbf{D}$. At each point $x$ of the convex hull $\mathbf{K}=\mathrm{co}(\mathbf{D})$, computing $f_{r}(x)$ reduces to solving a semidefinite program. We next characterize $\mathbf{K}$ in terms of the projection of a \textit{semi-infinite} LMI, and provide outer convex approximations $\{\mathbf{K}_{r}\}\downarrow \mathbf{K}$. Testing whether $x\notin \mathbf{K}$ reduces to solving finitely many semidefinite programs. 
@article{JCA_2008_15_3_JCA_2008_15_3_a12,
     author = {R. Laraki and J. B. Lasserre},
     title = {Computing {Uniform} {Convex} {Approximations} for {Convex} {Envelopes} and {Convex} {Hulls}},
     journal = {Journal of convex analysis},
     pages = {635--654},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {2008},
     url = {http://geodesic.mathdoc.fr/item/JCA_2008_15_3_JCA_2008_15_3_a12/}
}
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R. Laraki; J. B. Lasserre. Computing Uniform Convex Approximations for Convex Envelopes and Convex Hulls. Journal of convex analysis, Tome 15 (2008) no. 3, pp. 635-654. http://geodesic.mathdoc.fr/item/JCA_2008_15_3_JCA_2008_15_3_a12/