On a generalized Baillon-Haddad theorem for convex functions on Hilbert space (Q2789230)

Let \(\mathcal H\) be a Hilbert space. An operator \(T:\mathcal H\to \mathcal H\) is called nonexpansive (firmly nonexpansive) if \(\| Tx-Ty\| \leq\| x-y\| \) (resp., \( \| Tx-Ty\| ^2\leq\langle Tx-Ty,x-y\rangle\)) for all \(x,y\in X\). It is known that the projection operator \(P_C\) on a closed convex subset of \(\mathcal H\) is firmly nonexpansive. The operator \(T\) is called convergent if the sequence of iterations \(x^k=Tx^{k-1},\, k\in\mathbb N\), is convergent for every starting vector \(x^0\in \mathcal H\), and \(\alpha\)-averaged if it is of the form \((1-\alpha)\mathrm{Id}+\alpha N\), for some \(\alpha\in (0,1)\) and a nonexpansive operator \(N\). It is easy to see that \(T\) is firmly nonexpansive iff it is 1/2-averaged, and so it is convergent, since, by the Krasnoselskiĭ-Mann theorem, any averaged operator is convergent.NEWLINENEWLINEA result of \textit{J.-B. Baillon} and \textit{G. Haddad} [Isr. J. Math. 26, 137--150 (1977; Zbl 0352.47023)] says that, if a convex function \(f:\mathcal H\to \mathbb R\) is Gâteaux differentiable on \(\mathcal H\) with \(T=\nabla f\) nonexpansive, then \(f\) is Fréchet differentiable and \(T\) is firmly nonexpansive. Baillon and Haddad obtained the result as a consequence of a more general theorem on \(n\)-cyclically monotone operators in Banach spaces. \textit{H.~H. Bauschke} and \textit{P.~L. Combettes} [J. Convex Anal. 17, No. 3--4, 781--787 (2010; Zbl 1208.47046)] gave a new proof and an extension of the Baillon-Haddad result. Due to the convergence property, the Baillon-Haddad result provides an important link between optimization and fixed point iteration.NEWLINENEWLINEThe author of this paper gives several conditions equivalent to those from the Baillon-Haddad theorem, providing at the same time an elementary proof, based only on fundamental properties of convex differentiable functions. The exact result is the following one (Theorem 1.2). Let \(\mathcal H\) be a Hilbert space and \(f:\mathcal H\to\mathbb R\) a convex and Gâteaux differentiable function. Then the following are equivalent: {\parindent=8mmNEWLINENEWLINE\begin{itemize}\item[1.] the function \(F(x)=\frac12\| x\| ^2-f(x)\) is convex;NEWLINENEWLINE\item[2.] the inequality \(\; \frac12\| x-z\| ^2\geq f(z)-f(x)-\langle \nabla f(x),z-x\rangle\;\) holds or all \(x,z\in \mathcal H\);NEWLINENEWLINE\item[3.] \(T=\nabla f\) is firmly nonexpansive;NEWLINENEWLINE\item[4.] \(f\) is Fréchet differentiable and \(T=\nabla f\) is nonexpansive.NEWLINENEWLINE\end{itemize}}