Given a linear function, I'm wondering how to get its maximal value under quadratic and boundary constraint.
Here is a more formally expression
$$\max \sum_{i=1}^N a_i x_i$$ $$s.t \quad \sum_{i=1}^N \lambda_i (x_i - \mu_i)^2 \le C$$ $$and \quad l\le x_i\le h, \quad\forall i={1,2,\dots,N}$$
where $N>0, a_i>0, \lambda_i>0, \mu_i, C>0, l>0, h>0$ are known.
when throwing the boundary constraint, i.e $l\le x_i\le h$ away, then it seems the problem becomes much easier, the target function gets its maximal value when $x$ is just on the boundary of the ellipsoid, so we may rewrite the quadratic inequality into an equal form, and finally gets $x_i$'s optimal value under a closed form solution.
But when adding the constraint for each $x_i\in[l,h]$, how can we solve this convex optimization problem in a efficient way? since $N$ might be a big integer, say millions or so.
Thank you for any suggestions.