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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 $(A)$?

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    $\begingroup$ Generally, No. You might need to do this iteratively. Also it depends on exactly how you combine $x^*$ and $P_{C_2}$ (indeed, there are many different ways). If you just alternate, it might no converge to global optimum of original problem. You might need some clever averaging / interpolation. For example, consider the case when $f(x) = \|x-a\|^2$ for some $a \not\in C_1 \cup C_2$. The problem then corresponds to projection onto the intersection of convex sets. $\endgroup$
    – dohmatob
    Commented Apr 22 at 12:10
  • $\begingroup$ Is there a condition on the convex set, in which it must converge? $\endgroup$
    – ABIM
    Commented Apr 22 at 14:36

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