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An affine minorant of a function $f$ on ${\mathbb{R}}^n$ is an affine function $x\mapsto a\cdot x+b$ satisfying $a\cdot x+b\le f(x)$ everywhere. Every proper convex function that is lower semicontinuous on ${\mathbb{R}}^n$ has an affine minorant: separate a point strictly below its closed epigraph by a hyperplane. The finiteness of $f$ somewhere makes the separating vertical coefficient strictly positive, so the separator can be solved for height.

This supplies a useful existence estimate. For $\tau >0$ and $v\in {\mathbb{R}}^n$,The proximal objective is therefore coercive and lower semicontinuous; in finite dimensions it attains a minimum. Its strong convexity makes that minimizer unique, giving a well-defined proximal operator.^[ref] The finite-dimensional attainment argument should not be applied to arbitrary infinite-dimensional normed spaces merely from boundedness of a minimizing sequence.

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]

For a differentiable convex function $f$ on a Euclidean space with $L$-Lipschitz gradient, $L>0$, one form of the Nesterov accelerated gradient method usesstarting with $x_{-1}=x_0$. The first extrapolation is zero. Each later step uses the previous displacement to form an extrapolated point $y_k$, then evaluates the gradient there.^[ref]

If a global minimizer $x_{\ast }$ exists with $\Vert x_0-x_{\ast }\Vert \le R$, the objective error is $O(L{R}^2/k^2)$. Reaching error $\delta >0$ therefore takes $O(\sqrt{L{R}^2/\delta })$ iterations. The parameter schedule and convex smoothness hypotheses are part of that guarantee.

The same extrapolation can be combined with a proximal gradient method step to handle an additional nonsmooth convex term.

Let $C$ be a nonempty closed convex set in a real finite-dimensional Euclidean space, and let $F$ be a differentiable convex function. A projected gradient descent step isThe Euclidean projection onto a convex set is single-valued under these assumptions and restores feasibility after the unconstrained gradient descent step. Closedness matters: the nearest point to zero in the convex set $(0,1)$ does not exist.

This is the proximal gradient method with nonsmooth term equal to the indicator functional of a constraint set, whose proximal operator is ${\Pi }_C$. If $\nabla F$ is $L$-Lipschitz continuous, a standard choice $0<\eta \le 1/L$ gives the usual convex objective-error bound relative to an existing minimizer.^[ref]

For $C=[0,1]$, ${\Pi }_C(z)=\max (0,\min (z,1))$. Minimizing $F(x)=(x-2{)}^2/2$ from $x_0=0$ with $\eta =1$ gives $x_1={\Pi }_C(2)=1$, the constrained optimum.