Lagrange multiplier vs KKT Suppose task 1: maximize $f(x, y)$ subject to $g(x, y) = 0$ and $h(x,y) = 0$ Suppose task 2: maximize $f(x, y)$ subject to $g(x, y) \geqslant 0$ and $h(x,y) = 0$ According to wiki for the first task: $\Lambda (x,y, \lambda, \mu) = f(x,y) - \lambda g(x,y) - \mu h(x,y)$ And for the second **the same**. Why is it so? Am I miss something? Maybe some boarder constraints are different? Please clearify...

**@Aaron** in the comment above actually answered the question: For the KKT conditions (task 2) there are some other conditions. The multiplier λ≤0 and satisfies λg(x,y)=0 meaning that either λ=0 or g(x,y)=0. The last condition is sometimes called complementary slackness and adds a new element to the inequality problems (look at linear programming to understand it better). – Aaron Aug 28 at 16:19

**My additional comment:**

$λ < 0$ means that $\
abla f(x,y)$ and $\
abla g(x,y)$ are in opposite direction, so fence ($g(x,y) \ge 0$ constraint) actually prevents $f(x,y)$ from increase. $λ = 0$ means that $g(x,y) \ge 0$ is over-satisfied and $f(x,y)$ may be used to increased in the $\
abla f(x,y)$ direction.