Assume that $\mathcal{A}$ is a convex set of square matrices of order $n$ and the equation \begin{equation} Ax=b, \end{equation} has a single unique solution $x>\bf{0}$ for any $A\in \mathcal{A}$, where $b$ is a fixed vector.
I'm trying to minimize the sum of $x$: \begin{equation} \min_{A\in\mathcal{A},x\in\mathbb{R}^n} \{x_1 + x_2 + \dots + x_n : Ax=b \} \end{equation} The set $\mathcal{A}$ is such that the problem can equivalently be stated as: \begin{equation} \min_{\mu\in\mathbb{R}^m,x\in\mathbb{R}^n} \{x_1 + x_2 + \dots + x_n : \mu_i (r_i^Tx)=b_i \; (i=1,\ldots,m), \;Q\mu>q \} \end{equation}
Are there some known ways to do this?
Maybe I can use a modification of the simplex method?
Thanks for any ideas and information.