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With zero forward differences at the grid boundary, discrete isotropic total variation is the sum of the Euclidean lengths of the two-component forward gradients. Its dual constraint is the product of unit Euclidean balls, and minimizing $\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over that product yields $u_{\ast }=g-D^{\ast }{p}_{\ast }$. The exact difference adjoint operator is essential. A dual proximal gradient method projects $p+\tau D(g-D^{\ast }p)$ pointwise onto unit balls; $0<\tau \le 1/8$ is safe on an unscaled square grid.

The iterative soft-thresholding algorithm applies the proximal gradient method to $\Vert x{\Vert }_1+\frac{1}{2}\Vert Ax-y{\Vert }_2^2$:Here $S_{\tau }$ is coordinatewise soft thresholding. The objective is convex and coercive, so iterates converge to a minimizer in finite dimensions; uniqueness requires additional hypotheses.

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.

For a differentiable convex function $F$ with $L$-Lipschitz gradient and a proper lower-semicontinuous convex function $H$, the proximal gradient method isThe proximal operator is ${\mathrm{prox}}_{\tau H}(v)=\arg {\min }_z\{H(z)+\Vert z-v{\Vert }^2/(2\tau )\}$. To obtain an explicit iteration for discrete isotropic total variation, let $D=\nabla$ with the printed zero boundary differences andThe dual is minimization of $F(p)=\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over $\mathcal{P}$, and the primal minimizer is $u_{\ast }=g-D^{\ast }{p}_{\ast }$. Here $D^{\ast }=-\mathrm{div}$ must be the exact adjoint operator of these differences. For example,with the analogous formula in $j$ for $(D_y^{+}{)}^{\ast }$; for $N=1$ both operators are zero. Equivalently use $p_{0j}^1=p_{Nj}^1=p_{i0}^2=p_{iN}^2=0$ in the backward-difference expression, ignoring unused edge components.

The gradient of the dual objective is $D(D^{\ast }p-g)$, with Lipschitz constant $\Vert D{\Vert }^2\le 8$. Taking $H$ to be the indicator functional of a constraint set makes its proximal operator the Euclidean projection onto a convex set. Thus a valid explicit algorithm isThe bound follows by summing $(a-b{)}^2\le 2a^2+2b^2$ over grid differences. Finite-dimensional proximal gradient method convergence gives a dual minimizer and convergence of the recovered $u^k$ to the unique primal minimizer. Projection is pointwise in the dual; applying pointwise soft thresholding directly to $Du$ would not compute the primal TV proximal operator.

Take $F(x)=\frac{1}{2}\Vert Ax-y{\Vert }_2^2$, with $\nabla F(x)=A^T(Ax-y)$ and Lipschitz constant $L=\Vert A{\Vert }_{2\to 2}^2$, and $H(x)=\Vert x{\Vert }_1$. The proximal operator separates over coordinates. Solving the scalar subgradient optimality condition $0\in \partial |z|+(z-v)/\tau$ gives soft thresholding $S_{\tau }(v)=\mathrm{sgn}(v)(|v|-\tau {)}_{+}$. Therefore the iterative soft-thresholding algorithm isHere soft thresholding is applied coordinatewise. The objective is convex and coercive because of its ${\ell }^1$ term, so minimizers exist and the proximal gradient method converges to a minimizer. Uniqueness need not hold if $A$ has a nontrivial null space. If $A=0$, the unique minimizer is zero and any positive step size is admissible.

SetThe gradient has Lipschitz continuity with constantwhere the norm is the spectral norm. The proximal gradient method is thereforeFor $\lambda >0$, part d makes the second step explicit:where $C$ is the capped simplex; when $\lambda =0$, this proximal step is the identity.

A standard fixed choice is $0<\alpha \le 1/L$; the wider interval $0<\alpha <2/L$ also gives convergence under the usual forward-backward conditions. For a general convex objective, the function-value error is $O(1/r)$. If $A$ has full column rank, the quadratic term is strongly convex and an appropriate fixed step gives a linear convergence rate.

Put $z=x-\alpha \nabla f(x)$ and $\tau =\alpha \lambda$. The proximal operator for the L1 norm separates into independent scalar minimization problems:The subgradient optimality condition is $0\in p_i-z_i+\tau \partial |p_i|$. By the subgradient of the absolute value, a positive solution must satisfy $p_i=z_i-\tau >0$, and a negative solution must satisfy $p_i=z_i+\tau <0$. At zero, the subgradient optimality condition holds exactly when $|z_i|\le \tau$. The squared term makes the objective function a strictly convex function, so these cases give its unique minimiser.

Thus the proximal operator is the soft-thresholding operatorConsequently the requested proximal gradient mapping, rather than just the proximal operator, isThe corresponding proximal gradient method update is $x_i^{+}=S_{\alpha \lambda }(x_i-\alpha {\partial }_{i}f(x))$. Small coordinates are set exactly to zero. When $\lambda =0$, the soft-thresholding operator is the identity and the proximal gradient mapping reduces to $\nabla f(x)$.

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.

For a convex function $h$ with a well-defined proximal operator, and a differentiable function $f$, the proximal gradient mapping is the displacement of one proximal gradient method update divided by its positive step size $\alpha$. Writing $p=x-\alpha G_{\alpha }(x)$, the subgradient optimality condition gives $G_{\alpha }(x)-\nabla f(x)\in \partial h(p)$. For convex functions $f,h$, its zeros are precisely the global minimizers of $f+h$, by the subgradient optimality condition and subdifferential sum rule. When $h=0$, it is the ordinary gradient.

Target: convex-optimization.bigb, under Proximal gradient method.

These methods optimize a smooth loss plus a term with a tractable proximal operator, such as a sparsity regularizer or a convex-set indicator. The basic proximal gradient method combines a gradient step with a proximal step; projected gradient and accelerated variants are related members of the family.

For $\varphi =f+h$, with $f$ a convex function having an $L$-Lipschitz gradient and $h$ a proper convex function with a well-defined proximal operator, the proximal gradient method with step size $1/L$ satisfies the displayed bound whenever a global minimizer $x^{\ast }$ exists. The one-step subgradient inequality and descent lemma giveThe objective function values are nonincreasing. A telescoping sum over $j=0,\dots ,k-1$, followed by comparison of the last error with the arithmetic mean of the errors, proves the result. The bound controls the last objective function error without requiring a strongly convex function or a unique minimizer.

Target: convex-optimization.bigb, under Proximal gradient method.