I'm having trouble solving one of the possible cases that arise when solving the KKT conditions of the following problem:
We have the following optimization problem in $ \mathrm x \in \mathbb R^n$, with $\mathrm Q$ being an $n\times n$ positive definite matrix, $\mathrm p \in \mathbb R^n$,$\mathrm b \in \mathbb R^n$ and $\mathrm c \in \mathbb R$ .
$$\begin{array}{ll} \text{minimize} & \mathrm x^{\top} \mathrm Q \mathrm x + 2 \mathrm p^{\top} \mathrm x \\ \text{subject to} & \mathrm x^{\top} \mathrm Q \mathrm x + 2 \mathrm b^{\top}\mathrm x \leq \mathrm c \end{array}$$
Assigning multiplier $\mu \in \mathbb R$ to the constraint, I get the following KKT system:
$$ \nabla f(x) + \mu \nabla g(x) = 0 \iff 2 \mathrm Q \mathrm x + 2 \mathrm p + \mu \mathrm Q \mathrm x + 2 \mu \mathrm b = 0 $$ $$ \mathrm x^{\top} \mathrm Q \mathrm x + 2 \mathrm b^{\top} \mathrm x \leq \mathrm c $$ $$ \mu \geq 0 $$ $$ \mu(\mathrm x^{\top} \mathrm Q \mathrm x + 2 \mathrm b^{\top}\mathrm x - \mathrm c) = 0 $$
I have to consider 2 different cases: either $\mu = 0$ or $\mathrm x^{\top} \mathrm Q \mathrm x + 2 \mathrm b^{\top} \mathrm x - \mathrm c = 0$.
The first one is rather easy. If $\mu = 0$ then the first equation is reduced to
$$ \nabla f(x) = 0 \iff 2 \mathrm Q \mathrm x + 2 \mathrm p = 0 \iff \mathrm x = - \mathrm Q^{-1} \mathrm p $$
Plugging $\mathrm x $ in the second equation gives us the condition we have to check in order to validate this minimizer. $$ (- \mathrm Q^{-1} \mathrm p)^{\top} \mathrm Q(- \mathrm Q^{-1} \mathrm p) + 2 \mathrm b^{\top} (- \mathrm Q^{-1} \mathrm p) \leq \mathrm c $$
which, after manipulation and completing the square can be written as
$$ \frac{(\mathrm p - \mathrm b )^{\top}}{\sqrt{c + \mathrm b^{\top} \mathrm Q \mathrm b }} \mathrm Q^{-1} \frac{(\mathrm p - \mathrm b )}{\sqrt{c + \mathrm b^{\top} \mathrm Q \mathrm b }} \leq 1 $$
This means that when this condition is met, we can plug $x = - \mathrm Q^{-1} \mathrm p$ into the objective function to get the respective minimum. That is, $- \mathrm p^{\top}\mathrm Q^{-1} \mathrm p$.
My difficulty is the case when $\mathrm x^{\top} \mathrm Q \mathrm x + 2 \mathrm b^{\top} \mathrm x - \mathrm c = 0$. If I follow the same reasoning I get
$$ \mathrm x = \mathrm Q^{-1} (\frac{-2 \mathrm p - 2 \mu \mathrm b}{2 + \mu}) $$ But since $\mathrm x$ depends on $\mu$ I can't advance.