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Write $A_1={\mathrm{Div}}_1$ and $A_2={\mathrm{Div}}_2$, with Euclidean adjoints $A_1^{\ast },A_2^{\ast }$ determined by the chosen discretization and boundary conditions. Take the usual positive total generalized variation weights ${\alpha }_1,{\alpha }_2$. Introduce a primal variable $w\in {\mathbb{R}}^{n\times 2}$ throughThe support function of the row-ball product is a sum of row norms. Convex duality gives the equivalent augmented saddle problemThe equality follows by dualizing the row-ball constraint. With positive radii, $v=0$ strictly satisfies both row constraints, supplying the finite-dimensional qualification for this splitting. The original feasible dual set is compact and nonempty, and the quadratic primal term is coercive; saddle points exist. The TGV divergence splitting avoids the difficult projection onto $\{v:v\in C,A_{1}v\in D\}$.

Use the Chambolle–Pock algorithm. Choose $\tau ,\sigma >0$ with $\tau \sigma \Vert K{\Vert }^2<1$; the sufficient bound $\tau \sigma (\Vert A_2{\Vert }^2+\Vert A_1{\Vert }^2)<1$ is convenient. Initialize $u^0,w^0,v^0$ and ${\bar{u}}^0=u^0$, ${\bar{w}}^0=w^0$. For $k=0,1,\dots$, computeThe dual update is a Euclidean projection onto a convex set onto $C$; the two primal updates are the quadratic proximal operator and radial soft thresholding. Their signs follow from $K^{\ast }v=(A_{2}v,-A_{1}v)$. All substeps are closed form, and the standard finite-dimensional primal-dual convergence result applies to this saddle problem with the stated step-size condition. The iterates $v^k$ satisfy $C$; the additional $D$ constraint is enforced through the splitting at convergence, not claimed for every intermediate iterate. If a weight is zero, the corresponding row projection or support-function proximal step is interpreted directly rather than by division by zero.

For the Euclidean norm, rotational symmetry and minimizing the radial objective give the radial soft thresholding formulaIndeed, if $r=\Vert x{\Vert }_2$, an optimal $u$ is a nonnegative scalar multiple of $x$; minimizing $s+(r-s{)}^2/(2\tau )$ over $s\ge 0$ gives $s=(r-\tau {)}_{+}$. The gradient of a Moreau envelope is consequentlyAt $x=0$ this is $\{0\}$, and the two expressions agree when $\Vert x{\Vert }_2=\tau$. The Moreau envelope of the Euclidean norm itself isThus a quadratic core replaces the nondifferentiable tip while the outer gradient remains the normalized radial direction. The Euclidean norm has an unbounded effective domain, unlike the earlier bounded-domain hypothesis; the preceding argument explicitly shows that this restriction is unnecessary here.

For convex optimization, this Moreau–Yosida regularisation permits gradient descent and other methods for functions with a Lipschitz gradient, using a gradient Lipschitz constant $1/\tau$. It approximates the norm uniformly: $0\le \Vert x{\Vert }_2-f_{\tau }(x)\le \tau /2$. Smaller $\tau$ improves approximation but makes the permitted gradient steps smaller. Minimizing $f_{\tau }$ alone preserves the minimizers and minimum value of $f$; replacing one term in a larger objective can shift the minimizer, so the smoothing parameter controls that approximation error.

Again write $z=x-\alpha \nabla f(x)$ and $\tau =\alpha \lambda$, and let $z_{\ell }$ denote its $\ell$th block of $k$ coordinates. The block mixed norm in this part makes the proximal operator separate over blocks. For a single block, its minimiser solvesAt a nonzero $u$, the gradient of the Euclidean norm is $u/\Vert u\Vert$. The subgradient optimality condition therefore becomesThus $u$ points in the direction of $z_{\ell }$ and $\Vert z_{\ell }\Vert =\Vert u\Vert +\tau$. This yields $u=(1-\tau /\Vert z_{\ell }\Vert )z_{\ell }$ when $\Vert z_{\ell }\Vert >\tau$.

At zero, the subdifferential of the Euclidean norm is the closed Euclidean unit ball: the subgradient inequality is $q^{T}w\le \Vert w\Vert$ for every $w$, which is equivalent to $\Vert q\Vert \le 1$ by the Cauchy-Schwarz inequality and the choice $w=q$. Hence zero solves the subgradient optimality condition exactly when $\Vert z_{\ell }\Vert \le \tau$. This also covers $z_{\ell }=0$ and avoids division by zero.

The resulting radial soft thresholding isIf $y_{\ell }$ is the corresponding block of $x$, the required proximal gradient mapping isWhole blocks can become zero, while nonzero blocks retain their direction after the gradient step. For a block of size one, radial soft thresholding agrees with the soft-thresholding operator in part (a).

For a quadratic data term and divergence maps $A_1,A_2$, the dual constraint $A_{1}v\in D$ can be split through ${\delta }_D(A_{1}v)={\sup }_w[⟨w,A_{1}v⟩-{\delta }_D^{\ast }(w)]$. For a product of Euclidean row balls of radius $\beta$, its support function is ${\delta }_D^{\ast }(w)=\beta {\sum }_i\Vert w_i{\Vert }_2$. The resulting saddle coupling is $K(u,w)=A_2^{\ast }u-A_1^{\ast }w$. The Chambolle–Pock algorithm then uses a row-ball projection, a quadratic proximal operator and radial soft thresholding, with no projection onto an intersection involving a divergence operator.