Given a strong (not only strict) convex function $f: \mathbb{R}^n\to\mathbb{R}$. On such problems, stochastic gradient decent (SGD) has a convergence rate of $O(1/T)$, where $T$ is the number of iterations [1]. How is the convergence rate affected if a constrained is added to the problem and the projected subgradient method is used to solve it?
E.g. \begin{align} \text{minimize } &f(x) \text{ subject to } \\ &x\in C, \end{align} where $C$ is a convex set? The projected subgradient method uses the iteration
$x^{(k+1)} = P \left(x^{(k)} - \alpha_k g^{(k)} \right)$
where $P$ is projection on $C$ and $g^{(k)}$ is any subgradient of $f$ at $x^{(k)}$. Is the convergence rate affected by the projection? If yes, are there better approaches to keep the convergence rate of $O(1/T)$?