The Baillon-Haddad Theorem Revisited

TL;DR

This paper revisits the Baillon-Haddad theorem, providing new, concise proofs and strengthening its conclusions, which are fundamental in optimization and functional analysis.

Contribution

It offers alternative, shorter proofs of the theorem and enhances its conclusions, advancing theoretical understanding in convex analysis.

Findings

Provided concise alternative proofs of the Baillon-Haddad theorem.

Strengthened the theorem's conclusion, broadening its applicability.

Enhanced theoretical foundations for optimization methods.

Abstract

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