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.)