Monotone operator

An operator $F$ on an inner-product space is monotone whenfor every $v,w$. The gradient of every differentiable convex function is monotone.

A mapping $T$ on a normed vector space is nonexpansive when $\Vert T(v)-T(w)\Vert \le \Vert v-w\Vert$ for every $v,w$.

A mapping $T$ on an inner-product space is firmly nonexpansive whenEvery firmly nonexpansive mapping is nonexpansive, and the resolvent of a maximal monotone operator is firmly nonexpansive.

The proximal point algorithm seeks a zero of a monotone operator $F$ by repeatedly applying its resolvent:For $F=\partial f$, this is iteration of a proximal operator.

For two proximal maps $P_f,P_h$, define reflected maps $R_f=2P_f-I$ and $R_h=2P_h-I$. The Douglas--Rachford fixed-point map isBecause reflected proximal maps are nonexpansive, $T$ is firmly nonexpansive.

Finding a point in ${\bigcap }_{j=1}^{\ell }{C}_j$ is equivalent to intersecting the product $C_1\times \cdots \times C_{\ell }$ with the diagonal subspace in $({\mathbb{R}}^n{)}^{\ell }$. Projection onto the product is componentwise, while projection onto the diagonal replaces every component by their average.

Given a positive-definite matrix $M$, the preconditioned proximal point map isIt is firmly nonexpansive in the weighted inner product $⟨v,w{⟩}_M=⟨v,Mw⟩$ whenever $F$ is monotone.