What should be closed form for $\iota_C^{\star}(\mathbf{u})$ if $C$ is closed and bounded and convex? ($\iota_C(x)=0$ for $x\in C$ and $\infty$ otherwise).
And if $f$ is closed and convex, can $\min_{\mathbf{u}} f^{\star}(\mathbf{u}) +\iota_C^{\star}(\mathbf{u})$ be further simplified. Thank you.