Preconditioned proximal point algorithm

Let $M$ be a real symmetric positive-definite matrix and $F$ a maximal monotone operator on a finite-dimensional Euclidean space, possibly set-valued. The preconditioned proximal point algorithm usesMaximal monotonicity guarantees that $T$ is defined everywhere. It is a firmly nonexpansive mapping in the weighted inner product $⟨x,y{⟩}_M=⟨x,My⟩$.

Indeed, if $x=Tu$ and $y=Tv$, operator monotonicity givesThis proves the firm bound and single-valuedness wherever the inverse exists; maximality supplies the full domain. With mere monotonicity, existence can fail. On $\mathbb{R}$, $F(x)=\mathrm{sign}x$ with $F(0)=0$ is monotone, but $x+F(x)=1/2$ has no solution. Its maximal monotone completion instead assigns the interval $[-1,1]$ at zero.