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Neal Parikh coauthored an exposition of the proximal point algorithm with Stephen Boyd. A proximal operator replaces a difficult nonsmooth minimization step with a quadratic-regularized subproblem. Their account connects this operation with convex projection, resolvents and splitting methods, allowing nonsmooth objectives and constraints to be treated within a common algorithmic framework.^[ref]

The maximizer defining $g^{\ast }(x)$ is ${\mathrm{prox}}_{tf}(x)$, soConsequentlyThus the proximal point algorithm for $f$ is ordinary gradient descent with step $t$ on its smooth Moreau envelope.

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.