KKT Conditions for NLP

Question

How may I state the KKT conditions for minimize $f(x) = ax^2$ subject to $Ax \leq b$, $x$ unrestricted?

Answer 1

The Lagrangian is $L(x,z) = c^Tx + z^T(Ax - b)$. So \begin{equation} \nabla_x L(x,z) = c + A^T z. \end{equation} The KKT conditions are:

1. $Ax \leq b$ (primal feasibility)
2. $A^T z + c = 0$
3. $z \geq 0$ (dual feasibility)
4. $z^T(b - Ax) = 0$ (complementary slackness)