Explicit polyhedral approximation of the euclidean ball
RAIRO - Operations Research - Recherche Opérationnelle, Tome 44 (2010) no. 1, pp. 45-59.

We discuss the problem of computing points of I Rn whose convex hull contains the euclidean ball, and is contained in a small multiple of it. Given a polytope containing the euclidean ball, we introduce its successor obtained by intersection with all tangent spaces to the euclidean ball, whose normals point towards the vertices of the polytope. Starting from the L∞ ball, we discuss the computation of the two first successors, and give a complete analysis in the case when n=6.

DOI : 10.1051/ro/2010003
Classification : 90C05
Mots-clés : polyhedral approximation, convex hull, invariance by a group of transformations, canonical cuts, reduction 
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     title = {Explicit polyhedral approximation of the euclidean ball},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {45--59},
     publisher = {EDP-Sciences},
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     doi = {10.1051/ro/2010003},
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Frédéric Bonnans, J.; Lebelle, Marc. Explicit polyhedral approximation of the euclidean ball. RAIRO - Operations Research - Recherche Opérationnelle, Tome 44 (2010) no. 1, pp. 45-59. doi : 10.1051/ro/2010003. http://archive.numdam.org/articles/10.1051/ro/2010003/

[1] A. Schrijver, Theory of linear and integer programming. Wiley (1986). | Zbl

[2] C.B. Barber, D.P. Dobkin and H. Huhdanpaa, The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22 (1996) 469-483. | Zbl

[3] A. Ben-Tal and A. Nemirovski, On polyhedral approximations of the second-order cone. Math. Oper. Res. 26 (2001) 193-205. | Zbl

[4] J.F. Bonnans, Optimisation continue. Dunod, Paris (2006).

[5] F. Glineur, Computational experiments with a linear approximation of second-order cone optimization. Faculté Polytechnique de Mons (2000).

[6] J.B. Hiriart-Urruty and M. Pradel, Les boules. Quadrature (2004).

[7] G. Nemhauser and L. Wolsey, Integer and combinatorial optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York (1999). Reprint of the 1988 original, A Wiley-Interscience Publication. | Zbl

[8] G.L. Nemhauser, A.H.G. Rinnoy Kan and M.J. Todd, editors. Optimization, Handbooks in Operations Research and Management Science, Vol. 1, North-Holland, Amsterdam (1989). | Zbl

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