The tuple $LaTeX: (x^*, u^*, v^*)$ in $LaTeX: X \times \mathbb{R}^m \times \mathbb{R}^M$, with $LaTeX: u^* \ge 0$, is a saddlepoint of of the Lagrangian $LaTeX: L$ if, and only if, the strong duality properties hold:
1. $LaTeX: x^* \in \arg\max \{L(x, u^* v^*): x \in X\}$
2. $LaTeX: g(x^*) \le 0$ and $LaTeX: h(x^*) = 0$ ($LaTeX: x^*$ is feasible)
3. $LaTeX: (u^*)^T g(x^*) = 0$ ($LaTeX: x^*$ and $LaTeX: u^*$ satisfy complementary slackness)