Lagrange multiplier associated to an active inequality constraint

Question

Why is the Lagrange multiplier associated to an active inequality constraint is positive. How can we see this from the KKT conditions?

Answer 1

It is not true that the Lagrange multiplier, $\lambda$, associated with constraint $g(x) \le 0$ which is active (i.e., satisfies $g(x) = 0$) is necessarily positive at an optimum.

Presuming a constraint qualification holds, as does the required continuous differentiability, and therefore that the KKT conditions are necessary for optimality, it is true that complementary slackness must hold at an optimum. That is, $\lambda g(x) = 0$. Complementary slackness does not force $g(x) = 0$ (constraint is active) to require $\lambda = 0$. However, if the constraint is inactive,. i.e. $g(x) < 0$, then $\lambda$ must equal $0$ at an optimum, otherwise, complementary slackness would be violated.