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I want to minimize $x^t P x + q^t x$ subject to the following constraint:

For all $b \in B$, $|x^b| \le C \sum_{b' \in B} |x^{b'}|$

where $B = {1, ..., n}$ and $x^b$ is the $b$th component of the $n$-dimensional column vector $x$. $C$ is some positive constant which, to avoid triviality, should satisfy $1/|B| \le C \le 1$.

The only way I know how to do this is to do $2^{|B|}$ optimizations over the convex cone given by:

For all $b \in B$, $x^b \ge 0$ and $x^b \le C \sum_{b' \in B} x^{b'}$

and its reflections. Is there a more efficient way to solve this problem?

For my purposes let's say $C = 1/5$ and $n = 100$. I'm not sure I have much choice in the structure of $P$ and $q$, so an efficient solution for general $P$ and $q$ is desirable.

(Perhaps an approximate solution is much easier to find. Help with that would be appreciated too.)

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    Peter, can you explain how to ensure $y$ is minimized by using the $q^t x$? It's the whole function $q^t x$ that's minimized so I don't understand you can ensure that $y$ actually comes out as $|x|$.2011-07-26
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    This looks suspiciously like portfolio allocation with a constraint that the portfolio can't be dominated by any one asset. You might have more success with a constraint that's not as easy to interpret, but is more mathematically tractable. For example, it's easy to solve the equivalent problem with $|x^b| $\forall b$ (use convex optimization) and even easier if you have a penalty term of the form $\lambda\| x\|^2$ (use calculus).2012-07-28

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