Optimization and Lagrange Multipliers
Theorem (Lagrange Multipliers)
Suppose \(U \subset {\mathbb R}^n\) is open. Let \(F : U \rightarrow {\mathbb R}^m\) (where \(m < n\)) be \(C^1\) and suppose that \(DF\) has full rank \(m\) at every point of \(U\). Consider the set
\[ \Sigma := \left\{x \in U \ : \ F(x) = c \right\}\]
for some fixed \(c\). If \(\Sigma\) is nonempty and if \(f\) is a \(C^1\) function on \(U\) which attains a maximum on \(\Sigma\), i.e., such that there exists \(x \in \Sigma\) such that \(f(x) = \sup_{y \in \Sigma} f(x)\), then there exists \(\lambda \in {\mathbb R}^n\) such that
\[ \frac{\partial f}{\partial x_i}(x) + \sum_{j=1}^m \lambda_j \frac{\partial F_j}{\partial x_i} (x) = 0 \tag{1}\]
for all \(i=1,\ldots,n\), where \(F_1,\ldots,F_m\) are the coordinates of \(F\), i.e., \(F(x) = (F_1(x),\ldots,F_m(x))\).