= Solution
Let $\widehat u_\alpha$ minimize the nonsquared-residual objective. Comparison with $u^\dagger$ gives
$$
\|A\widehat u_\alpha-f\|_Y
+\alpha[J(\widehat u_\alpha)-J(u^\dagger)]\leq0.
$$
The source subgradient inequality gives
$$
J(\widehat u_\alpha)-J(u^\dagger)
\geq\langle w^\dagger,A\widehat u_\alpha-f\rangle.
$$
Consequently,
$$
0\geq
\|A\widehat u_\alpha-f\|_Y
+\alpha\langle w^\dagger,A\widehat u_\alpha-f\rangle
\geq
(1-\alpha\|w^\dagger\|_{Y^*})
\|A\widehat u_\alpha-f\|_Y.
$$
Thus for
$$
\boxed{\alpha_0=
\begin{cases}
\|w^\dagger\|_{Y^*}^{-1},&w^\dagger\ne0,\\
+\infty,&w^\dagger=0,
\end{cases}}
$$
every $0<\alpha<\alpha_0$ forces $A\widehat u_\alpha=f$. The original comparison then gives $J(\widehat u_\alpha)\leq J(u^\dagger)$, so $\widehat u_\alpha$ is itself $J$-minimizing. If $J$ is <strictly convex>, its restriction to the affine solution set has at most one minimizer, hence $\widehat u_\alpha=u^\dagger$. This is an <exact penalty method>.
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