= 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> uses
$$
T=(M+F)^{-1}M,\qquad M(u-Tu)\in F(Tu).
$$
Maximal monotonicity guarantees that $T$ is defined everywhere. It is a <firmly nonexpansive mapping> in the weighted <inner product> $\langle x,y\rangle_M=\langle x,My\rangle$.
Indeed, if $x=Tu$ and $y=Tv$, <operator monotonicity> gives
$$
0\leq\langle M(u-x)-M(v-y),x-y\rangle=\langle u-v,x-y\rangle_M-\|x-y\|_M^2.
$$
This proves the firm bound and single-valuedness wherever the inverse exists; maximality supplies the full domain. With mere monotonicity, existence can fail. On $\mathbb R$, $F(x)=\operatorname{sign}x$ 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.
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