The Baillon-Haddad Theorem Revisited

In 1977, Baillon and Haddad proved that if the gradient of a convex and continuously dierentiable 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.

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