The Baillon-Haddad Theorem Revisited

By Heinz H. Bauschke, Patrick L. Combettes. 2009

64 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/93c68b0833be7cedf77150302a387c4e6ae5b3d2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Modeling and Simulation

Bang Công Vu

In Numerical Analysis

Citations in the corpus

Firmly Nonexpansive Mappings and Maximally Monotone Operators: Correspondence and Duality (2011, background): “Since Prox f is the Fréchet gradient mapping of the continuous convex function f ∗ 1 2‖ · ‖2 : X → R, the Baillon-Haddad theorem (see [1] and also [4] for recent results) guarantees that β−1 Prox f is firmly nonexpansive.”
A regularization interpretation of the proximal point method for weakly convex functions (2020, background): “From Lemma 3.1 we infer the following property of the prox-operator, which in the literature is known as cocoercivity, and which can be derived as a consequence of the Baillon-Haddad Theorem [5, 6, 7] as φλ is ( 1 λ − c)-strongly convex, and ∇φ∗λ = Pλf(λ(·)), see Proposition 3.”
A splitting algorithm for dual monotone inclusions involving cocoercive operators (2011, background): “Since ∇h is µ−1-Lipschitz continuous, by the Baillon-Haddad Theorem [4, 5], it is µ-cocoercive.”