 # Constraint qualification

Conditions on the constraint functions ( $LaTeX: g$ and $LaTeX: h$) that are sufficient to make the Lagrange Multiplier Rule valid. Here is an example to illustrate what can go wrong: $LaTeX: \max x : x^2 \le 0$.
Since $LaTeX: x=0$ is the only feasible solution, it is optimal. The Lagrange Multiplier Rule requires that there exist $LaTeX: u$ for which $LaTeX: f' - u^T g' = 0$, but $LaTeX: f' = 1$ and $LaTeX: g'= 0$, so no such $LaTeX: u$ exists. Slater used this example in illustrating his interiority condition. The classical qualification, given by Lagrange's multiplier theorem without inequality constraints, is that $LaTeX: \nabla h(x)$ have full row rank, which stems from the Implicit Function Theorem. Another constraint qualification is that all constraint functions be affine (even with redundant constraints). Each constraint qualification gives a sufficient condition for the Lagrange Multiplier Theorem to be valid. A constraint qualification is necessary if it must hold in order to guarantee that the Lagrange Multiplier Rule is valid for all smooth $LaTeX: f$ having optimal value at $LaTeX: x$.