 Solution

Let $r(u)=\Vert u-g{\Vert }_2^2$ and fix the reference noise budget $\sigma$. Define the convex perturbation functionPositive $z$ increases the allowed squared noise level. The feasible epigraph set $r(u)\le \sigma +z$ is convex and closed, so this is a proper lower semicontinuous jointly convex perturbation. Writing the inequality indicator as a supremum over its multiplier givesThe Lagrange dual function is $d(\lambda )={\inf }_{u}L(u,\lambda )$ for $\lambda \ge 0$. In the signed convex conjugate convention of part (a), $y=-\lambda$; positive $y$ gives dual objective $-\infty$ because the perturbation can be made arbitrarily large.

The feasible ball is nonempty and compact, so lower semicontinuity of total variation gives a primal minimizer. For $\sigma >0$, $u=g$ satisfies $r(g)=0<\sigma$, providing the Slater condition. Total variation is finite everywhere in the given discretization and hence continuous. Strong duality and dual attainment give a finite optimal ${\lambda }^{\ast }\ge 0$. The pair satisfies the Karush-Kuhn-Tucker conditionsEquivalently,Thus the set of saddle points is nonempty. Compactness handles primal attainment, while strict feasibility supplies a finite multiplier; these are distinct steps. An inactive constraint can yield ${\lambda }^{\ast }=0$. At $\sigma =0$, feasibility still forces $u=g$, but this strict-feasibility argument does not apply.