Fritz John conditions

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These extend Carathéodory's conditions to deal with inequality constraints:

Suppose LaTeX: x^* is in LaTeX: \arg\!\max\{f(x): x \in \mathbb{R}^n, \; g(x) \le 0\}, where LaTeX: f and LaTeX: g are in smooth. Then, there exists multipliers LaTeX: (y_0, y) \ge 0, not all zero, such that

y_0 \nabla f(x) - y^T \nabla g(x) = 0 \mbox{ and } y^Tg(x) = 0.

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