Calculate KKT point - simple example

Question

Problem: $\min φ(x)$ subject to $l ≤ x ≤ u$ - how do I find the KKT point?

I reformulated the problem so that we now min φ(x) subject to $x-l ≥ 0$ and $-x+u ≥ 0$, which means the lagrangian $L = φ(x) - λ_1(x - l) - λ_2(-x + u)$. How do I proceed from here?

Answer 1

Once you have formed your Lagrangian $\mathcal{L}(x, \lambda)$, you need to write the KKT conditions:

$\frac{\partial \mathcal{L}}{\partial x} = 0$ (stationarity)
$x - l \ge 0$, $u - x \ge 0$ (primal feasibility)
$\lambda_1 (x - l) = 0$, $\lambda_2 (u - x) = 0$ (complementarity)
$\lambda \ge 0$ (dual feasibility)