Let $x \in \mathbb{R}^N$ and let $K$ be a closed convex set in $\mathbb{R}^n$.
Let $$ \widehat{y} = \textrm{arg} \, \textrm{max} _{\,\,y \in K} \langle x,y \rangle,$$
where $\langle \cdot, \cdot \rangle$ denotes the usual inner-product.
A) Provide an iterative algorithm for computing $\widehat{y}$. Does it involve a projection onto the convex set?
Now let $$ \widehat{y} = \textrm{arg} \, \textrm{max} _{\,\,y \in K_1 \cap K_2} \langle x,y \rangle,$$ where $K_1$ and $K_2$ are two closed convex sets.
B) Is there an "successive projection" analog of A) to solve the above problem?