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I am trying to figure out how to solve:

$\min_U r_{p}$

where $r_{p}=\alpha^\intercal\omega$ and $U$ is a sphere centered at $\alpha$ with radius equal to $\chi|\alpha|$ . ( $\omega$ is a vector or weights and $\chi$ lies between 0 and 1.)

The authors end up with the following solution:

$\min_U r_{p}=\alpha^\intercal\omega-\chi|\alpha||\omega|$

The authors hint at Bayesian estimation but I am not familiar with it. Any ideas as to how they may have arrived at this would be great. Many thanks in advance.

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I don't understand the question. What is fixed and what are the decision variables? All of the variables you mention end up in the solution so these cannot have been optimized out. –  Noah Stein Jan 23 '13 at 13:26
    
Hi Noah. Thanks for taking the time to respond. We are letting $\alpha$ lie in an uncertainty region given by the sphere described above. $\omega$ is fixed. If you would like to understand this better, please have a look at the paper on this link: ssrn.com/abstract=1483412 (specifically page 6.) –  Geraldine Bailey Jan 23 '13 at 13:46
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