I want to find a matrix $A\in\mathbb{R}^{n\times n}$ and vector $x\in\mathbb{R}^{n}$ satisfying: \begin{align} Ax & \leq b,\\ Ac & = d, \end{align} with $b\in\mathbb{R}^{n},c\in\mathbb{R}^{n}$ and $d\in\mathbb{R}^{n}$ known vectors.
My idea is to find a $J\in\mathbb{R}^{n}$ that satisfies: \begin{align} J & \leq b,\\ Ac & = d,\\ A&\text{ invertible} \end{align} Then $x = A^{-1}J$.
1) Is there a better way of solving my original problem?
2) If my idea is good, how do you enforce that the matrix $A$ is invertible in the second problem?