# Caratheodory conditions

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:

y_0 \nabla f(x) - y^T \nabla h(x) = 0.

$

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.