 Solution

Let ${\hat{u}}_{\alpha }$ minimize the nonsquared-residual objective. Comparison with $u^{†}$ givesThe source subgradient inequality givesConsequently,Thus forevery $0<\alpha <{\alpha }_0$ forces $A{\hat{u}}_{\alpha }=f$. The original comparison then gives $J({\hat{u}}_{\alpha })\le J(u^{†})$, so ${\hat{u}}_{\alpha }$ is itself $J$-minimizing. If $J$ is strictly convex, its restriction to the affine solution set has at most one minimizer, hence ${\hat{u}}_{\alpha }=u^{†}$. This is an exact penalty method.