$\min\{-\sum_{j=1}^n\log{x_j}: q(x)\leq0\};\ q(x)=\frac{1}{2}x^TQx+b^Tx+c $
Where Q is a positive definite matrix, $b\in R^n, c\in R$
So far I have:
$L(x,\lambda) = -\sum_j \log x_j + \lambda(\frac{1}{2}x^TQx+b^Tx+c)$
And, since first I should look for $\inf_{x}\{L(x,\lambda)\}$:
$\frac{\partial L}{\partial x_j} = -\frac{1}{x_j} + \lambda((x_j\sum_i Q_{ij})+b_j)=0$
How should I proceed to get the dual problem?