I'm trying to solve an optimization problem in Matlab. The expressions you will find below. Problem is it is all matrices, and I have no idea which solver to use for that. $w$ is of size $n \times 1$, $\mu_{BL}$ is of size $1 \times n$, $\lambda$ a scalar/constant, and $\Sigma$ an $n \times n$ matrix.

I need to solve for $w$. (Portfolio optimization problem). Quadproc I tried but our function is more complex here. I'm clueless. Maybe anyone can help me out.

Thanks in advance.

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The First one is really simple (Quadratic Function):

$$ \arg \min_{w} {\mu}_{BL}^{T} w - \lambda w {\Sigma}_{BL} w $$

Its minimum (Assuming $ {\Sigma}_{BL} $ is NSD or $ \lambda < 0 $) is given by:

$$ w = \frac{1}{2 \lambda} {\Sigma}_{BL}^{-1} {\mu}_{BL} $$


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