Monotone-operator theory removes the distinction between gradients, subgradients, normal cones, and equilibrium maps. Optimization becomes the problem of finding a zero of a set-valued operator, and many algorithms become fixed-point iterations of its resolvent.
Monotonicity and maximality
Definition. A set-valued operator \(T:\mathbb R^d\rightrightarrows\mathbb R^d\) is monotone if \[ \langle u-v,x-y\rangle\ge0 \quad\text{whenever }u\in T(x),\ v\in T(y). \] It is maximal monotone if its graph is not properly contained in the graph of another monotone operator.
Convex subdifferentials are monotone. If \(f\) is proper and convex, then \(\partial f\) is monotone. If \(f\) is also closed, \(\partial f\) is maximal monotone.
Proof. For \(u\in\partial f(x)\) and \(v\in\partial f(y)\), \[ f(y)\ge f(x)+\langle u,y-x\rangle,\qquad f(x)\ge f(y)+\langle v,x-y\rangle. \] Adding gives \(\langle u-v,x-y\rangle\ge0\). For maximality, suppose \((x,u)\) is monotonically related to the graph of \(\partial f\). Minimize \(z\mapsto f(z)-\langle u,z\rangle+\frac12\|z-x\|^2\). Its minimizer \(p\) satisfies \(u+x-p\in\partial f(p)\). Monotonicity of \((x,u)\) with this graph point gives \(-\|x-p\|^2\ge0\), hence \(p=x\) and \(u\in\partial f(x)\).
Variational inequalities as zero problems
Definition. Given a closed convex set \(C\) and a map \(F:C\to\mathbb R^d\), the problem \(\operatorname{VI}(C,F)\) asks for \(x^\star\in C\) such that \[ \langle F(x^\star),x-x^\star\rangle\ge0 \qquad(x\in C). \]
Normal-cone formulation. \(x^\star\) solves \(\operatorname{VI}(C,F)\) if and only if \[ 0\in F(x^\star)+N_C(x^\star). \] When \(F=\nabla f\) for convex \(f\), this is exactly constrained minimization. A general monotone \(F\) can describe games and equilibria that have no scalar potential.
Resolvents are firmly nonexpansive
Minty resolvent theorem. If \(T\) is maximal monotone and \(\eta>0\), then \[ J_{\eta T}=(I+\eta T)^{-1} \] is defined on all of \(\mathbb R^d\), is single-valued, and satisfies \[ \|J_{\eta T}x-J_{\eta T}y\|^2 \le\langle J_{\eta T}x-J_{\eta T}y,x-y\rangle. \]
Proof. Surjectivity of \(I+\eta T\) is Minty's characterization of maximal monotonicity in finite dimensions. Let \(p=J_{\eta T}x\) and \(q=J_{\eta T}y\). Then \((x-p)/\eta\in T(p)\) and \((y-q)/\eta\in T(q)\). Monotonicity gives \[ \langle(x-p)-(y-q),p-q\rangle\ge0, \] which rearranges to firm nonexpansiveness. It also proves uniqueness.
For \(T=\partial f\), the resolvent is \(\operatorname{prox}_{\eta f}\). For \(T=N_C\), it is projection onto \(C\).
The proximal-point iteration
Fej?r convergence. Suppose \(T\) is maximal monotone and \(T^{-1}(0)\ne\varnothing\). The iteration \[ x_{k+1}=J_{\eta T}x_k \] satisfies, for every zero \(x^\star\), \[ \|x_{k+1}-x^\star\|^2 \le\|x_k-x^\star\|^2-\|x_{k+1}-x_k\|^2. \] In finite dimensions, \(x_k\) converges to a zero of \(T\).
Proof. Since \(J_{\eta T}x^\star=x^\star\), firm nonexpansiveness gives \[ \|x_{k+1}-x^\star\|^2 \le\langle x_{k+1}-x^\star,x_k-x^\star\rangle. \] Apply \(2\langle a,b\rangle=\|a\|^2+\|b\|^2-\|a-b\|^2\) and rearrange. Distances to the zero set decrease and the increments tend to zero. Every cluster point is a zero by closedness of the graph; Fej?r monotonicity then forces the whole finite-dimensional sequence to converge.
Forward-backward and extragradient splitting
For \(0\in A(x)+B(x)\), with \(A\) maximal monotone and \(B\) sufficiently cocoercive, forward-backward splitting is \[ x_{k+1}=J_{\eta A}(x_k-\eta Bx_k). \] Proximal gradient is the case \(A=\partial g\), \(B=\nabla f\). For merely monotone Lipschitz \(B\), a single forward step may fail; the extragradient correction \[ y_k=P_C(x_k-\eta F(x_k)),\qquad x_{k+1}=P_C(x_k-\eta F(y_k)) \] evaluates the field at the predicted point and restores convergence for monotone variational inequalities.