Caratheodory conditions

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For the classical Lagrange form, LaTeX: \min \{f(x): x \in \mathbb{R}^n, \; h(x) = 0 \}, where LaTeX: f and LaTeX: h are smooth, the following conditions are necessary for a feasible LaTeX: x to be optimal: there exists LaTeX: (y_0, y) \in \mathbb{R}^{m+1}\backslash \emptyset, called multipliers, such that

LaTeX: 
</p><p>y_0 \nabla f(x) - y^T \nabla h(x) = 0.
</p><p>

This reduces to the Lagrange Multiplier Rule when LaTeX: y_0 is not zero (divide by LaTeX: y_0), which must be the case if LaTeX: \nabla h(x) has full row rank.


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