Constraint qualification for linear constraints

Question

How to formulate conditions under which the LI CQ is fulfilled at arbitrary feasible point of the set of feasible solutions given by linear inequalities $S=(x\in\mathbb{R}^{n}:Ax\le b, x\ge 0)$? I know I need to show that gradients of constraints are linear independent, however, it depends on the number of active inequalities. I do not know how to count with this fact. Any idea, thank you?

Answer 1

LICQ is quite hard to show in this case since in order to show that LICQ is satisfied at every point, you need to show that for every possible active subset of the linear inequality constraints $Ax \le b$ and nonnegativity constraints $x \ge 0$, the gradients of the active constraints are linearly independent.

There are other constraint qualifications besides LICQ that you might use that could be much easier to establish. For example, you could use linear programming to determine whether Slater's constraint qualification was satisfied.