Maximal monotone operators and resolvents

An operator A may assign a set of outputs to an input. Its graph contains all pairs (x,u) with u in A(x). It is monotone when the inner product of x−y with u−v is nonnegative for every two graph pairs. Empty output sets are allowed. Adding the two support inequalities proves that the subdifferential of a proper convex function is monotone. If the function is also lower semicontinuous on the Hilbert space, its subdifferential is maximal monotone. The maximality part is a substantive theorem, beyond the immediate two-inequality calculation.

Maximal means there is no strictly larger monotone graph containing the given graph. For example, the zero operator restricted to the open interval (0,1) is monotone but not maximal: adding (0,0), or extending zero to the whole line, preserves monotonicity. In contrast, the subdifferential of |x| includes the full vertical interval [−1,1] at zero. Omitting one of those admissible pairs would leave a monotone but nonmaximal graph. A steep single-valued derivative can be nonmaximal after a domain restriction, while a flat everywhere-defined zero operator is maximal.

Minty’s surjectivity theorem states that a monotone operator on a real Hilbert space is maximal monotone exactly when the range of I+λA is the whole space for a positive λ; equivalently this holds for every positive λ. Thus a maximal monotone operator has a resolvent output for every input. Monotonicity makes that output unique. Indeed, if z and w solve the relation for inputs x and y, substitute (x−z)/λ and (y−w)/λ into the monotonicity inequality and rearrange. This proves the firm nonexpansiveness bound displayed below.

Cauchy–Schwarz then gives ordinary nonexpansiveness: output distance cannot exceed input distance. This bound does not mean every repeated resolvent iteration converges to a zero when no zero exists. If A is the constant operator A(x)={1} on the line, its resolvent is x−λ and has no fixed point. When a zero does exist, fixed points of the resolvent are exactly those zeros. Convergence claims for repeated updates need their own assumptions; in infinite-dimensional Hilbert spaces one must also distinguish weak convergence from convergence in norm.

On $\mathbb R^2$, let R(x₁,x₂)=(−x₂,x₁), a ninety-degree rotation. It is monotone because the inner product of every vector with its rotation is zero. The matrix I+λR has determinant 1+λ² and is invertible for every positive λ, so Minty’s theorem gives maximal monotonicity. However, R is not the gradient of a convex potential: its nonsymmetric Jacobian cannot be a Hessian. More generally, subdifferentials satisfy cyclic monotonicity, a stronger condition than the two-point monotonicity used here.

For a proper lower semicontinuous convex f, the resolvent of ∂f is precisely prox_{λf}, by the proximal optimality relation. This unifies projection, nonsmooth minimization and implicit monotone updates, while preserving the distinction between their mathematical classes. Explicit gradient descent has a step-size stability restriction even for a scalar quadratic; the implicit resolvent of its monotone gradient is nonexpansive for every positive parameter. This stability is not free computation: evaluating a general resolvent may itself require solving a difficult nonlinear subproblem.