 Solution

Put $d_i(u)=((G^{(1)}u{)}_i,(G^{(2)}u{)}_i)$. The image penalty is discrete isotropic total variation ${\sum }_i\Vert d_i(u){\Vert }_2$; the fidelity term is the L1 norm $\Vert u-f{\Vert }_1$. Introduce scalars $v_i,t_i$ and minimize the linear objectivesubject to the following affine residuals being cone-feasible:where ${\mathcal{L}}_3=\{(t,a,b):t\ge \sqrt{a^2+b^2}\}$ is the Lorentz cone. Every component displayed is affine in $(u,v,t)$, so stacking them gives exactly the requested $Ax-b\in K$ form, withThe first two inequalities imply $v_i\ge |u_i-f_i|$, so separate nonnegativity constraints for $v_i$ are unnecessary. The cone constraints imply $t_i\ge \Vert d_i(u){\Vert }_2$. Every lifted feasible objective bounds the original objective from above; choosing $v_i=|u_i-f_i|$ and $t_i=\Vert d_i(u){\Vert }_2$ gives equality. Since both objective weights are positive, an optimum uses these tight values. Thus the lift is an exact second-order cone program for box-constrained TV-L1 denoising.

The product cone is proper, closed, convex and self-dual. The stacked $A$ has full column rank: the box rows recover $u$, the fidelity rows then recover $v$, and the first coordinates of the Lorentz blocks recover $t$. Strict feasibility is also explicit: choose $u_i=1/2$, then take $v_i>|u_i-f_i|$ and $t_i>\Vert d_i(u){\Vert }_2$.

For generic cone slack $(r,q)$, a canonical barrier ison $r_j>0$ and $q_{i0}>\sqrt{q_{i1}^2+q_{i2}^2}$. The positive sheet condition matters: positivity of the quadratic expression alone would also include the wrong Lorentz nappe.

Composing the barrier with the affine residual map givesIts domain includes the strict inequalities above. Each orthant coordinate contributes one to the self-concordant barrier parameter and each Lorentz block contributes two. Therefore the product-cone logarithmically homogeneous barrier hasThe affine composition $\Phi$ is the barrier used in the primal variables; it need not itself be logarithmically homogeneous in $(u,v,t)$ because the image data and box offsets are affine constants.