Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex.
I want to optimize $$ \tag{(A)} \min_{x\in C_1\cap C_2}\, f(x). $$

I currently have the following at my disposal:

1. I can compute the orthogonal projection operator $P_{C_2}:H\to C_2$ in closed-form.
2. I can solve the following problem in closed-form $$ \tag{(B)} x^{\star}:=\operatorname{argmin}_{x\in C_1}\, f(x). $$

Is it possible to use $P_{C_2}$ and $x^{\star}$ to obtain a solution to $(A)$?

If not, can be meaningfully bound how far can the optimal value of $P_{C_2}(x^{\star})$ be from an optimizer of $(A)$?