Geodesic
The Baillon-Haddad Theorem Revisited
Journal of convex analysis, Tome 17 (2010) no. 3, pp. 781-787 Cet article a éte moissonné depuis la source Heldermann Verlag

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J.-B. Baillon and G. Haddad ["Quelque propriétés des opérateurs angle-bornés et n-cycliquement monotones", Israel J. Math. 26 (1977) 137--150] proved that if the gradient of a convex and continously differentiable function is nonexpansive, then it is actually firmly nonexpansive. This result, which has become known as the Baillon-Haddad theorem, has found many applications in optimization and numerical functional analysis. In this note, we propose short alternative proofs of this result and strengthen its conclusion.
Classification : 47H09, 90C25, 26A51, 26B25, 46C05, 47H05, 52A41
Mots-clés : Backward-backward splitting, Bregman distance, cocoercivity, convex function, Dunn property, firmly nonexpansive, forward-backward splitting, gradient, inverse strongly monotone, Moreau envelope, proximal mapping, proximity operator 
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     title = {The {Baillon-Haddad} {Theorem} {Revisited}},
     journal = {Journal of convex analysis},
     pages = {781--787},
     year = {2010},
     volume = {17},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2010_17_3_JCA_2010_17_3_a5/}
}
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H. H. Bauschke; P. L. Combettes. The Baillon-Haddad Theorem Revisited. Journal of convex analysis, Tome 17 (2010) no. 3, pp. 781-787. http://geodesic.mathdoc.fr/item/JCA_2010_17_3_JCA_2010_17_3_a5/