Take the following optimization problem:
$$\max_{x\in X}f(x) \text{ s.t. } g(x)\geq 0, h(x)=0$$ for $X$ an open subset of $\mathbb R^n, f:X\to \mathbb R, g:X\to \mathbb R^k,h:X\to\mathbb R^m$.
The linearly independent constraint qualification (LICQ) is said to hold at a point when the gradients of all the binding constraint functions at the point are linearly independent. My understanding is LICQ guarantees the Karush Kuhn Tucker (KKT) conditions are met at a maximizer. I see why we focus on binding constraints; by complementary slackness, those that don't bind are annihilated by a zero multiplier in the KKT conditions. For simplicity, take the case of two constraints ($k=m=1$), and let's assume both bind. Then the KKT conditions are
$$\nabla f(x)+\lambda \nabla g(x)+\mu \nabla h(x)=0,g(x)=h(x)=0,\lambda\geq 0,\mu\geq 0.$$
Writing the first equation in matrix form gives
$$\left[\begin{array}{cc} \nabla g(x) & \nabla h(x)\end{array}\right]\left[\begin{array}{c} \lambda\\ \mu \end{array}\right]=-\nabla f(x)$$
but how does linearly independent $\nabla g(x),\nabla h(x)$ guarantee KKT conditions are met at a maximizer?
