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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?

user67350
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  • How is $K$ specified? (A closed convex set can be described by its support functional.) – copper.hat Mar 18 '13 at 19:27
  • As generally as you wish (regardless of any computational considerations). I'm looking to understand the structure of the algorithm and where/how projections come in. Thanks! – user67350 Mar 18 '13 at 19:29
  • What I am saying is the the support functional is almost synonymous with a closed convex set. – copper.hat Mar 18 '13 at 19:34

1 Answers1

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Given the Gradient or the Sub Gradient of $f$ are known (Which is easy in your case), a simple way is to use the Projected (Sub) Gradient:

$$ \boldsymbol{x}^{k + 1} \leftarrow {P}_{\mathcal{K}} \left( \boldsymbol{x}^{k} - \mu \nabla f \left( \boldsymbol{x}^{k} \right) \right) $$

In case $\mathcal{K} = \mathcal{K}_{1} \cap \mathcal{K}_{2}$ you need to follow Orthogonal Projection onto the Intersection of Convex Sets.
You may use the successive method if the sets are sub spaces.

Royi
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