Lagrangian saddlepoint equivalence

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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)

See the supplement on Lagrangian Saddle Point Equivalence for further information.

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