I am trying to optimize the following problem
$$\text{ min }\text{log}|\sum_{i=1}^{d}c_{ii}I^{(i)}|+s^{T}c$$
subject to
$$\sum_{i=q+1}^{d}c_{ii}-\sum_{i=1}^{q}c_{ii}-t_2\ge 0$$
$$\sum_{i=q+1}^{d}c_{ii}+\sum_{i=1}^{q}c_{ii}-t_1\le 0$$
$$\sum_{i=1}^{d}c_{ii}I^{(i)}\succ0$$
where $I^{(i)}$ is a diagonal matrix with all zero entries except for a 1 at diagonal entry $i$ and the optimization is over $c$.
I need to write it as the following form
$$\text{ min }\text{log}|G(x)|+c^{T}x$$
$$G(x)>0$$ $$F(X)\ge0$$
where $G=G_0+\sum_{i=1}^{d}x_i G_i$ and $F=F_0+\sum_{i=1}^{d}x_i F_i$ are combinations of $F_i$ and $G_i$ in order to use determinant maximization with linear inequalities. I came up with a representation but when I attempt the Newton Rhapson algorithm it does not converge.