I have the following convex optimization problem. I would like to ask is there any efficient way to solve it in Python? Can I use CVXOPT package? If so, any detailed instruction? Thanks a lot.
$$ \min_{T\in[0, \infty]^{10}}\sum_{i=1}^{10}T_ia_i $$ subject to $$ x^{\top}_j(\sum_{i=1}^{10}T_i x_ix_i^{\top})^{-1}x_j\leq b_j, \text{for all} \ j\in[10]. $$ Here, $\{a_1,\ldots, a_{10}\}\subset \mathbb R$ and $\{x_1, \ldots, x_{10}\}\subset\mathbb R^5$. 10 and 5 are just generic constants. If the inverse does not exist, we could replace it by pseudo inverse.