Editing Lagrange multiplier (section)

== Statement ==

The following is known as the Lagrange multiplier theorem.<ref>{{cite book |first=Angel |last={{nobr|de la Fuente}} |title=Mathematical Methods and Models for Economists |location=Cambridge |publisher=Cambridge University Press |year=2000 |doi=10.1017/CBO9780511810756 |isbn=978-0-521-58512-5 |page=[https://archive.org/details/mathematicalmeth00fuen/page/n288 285] |url=https://archive.org/details/mathematicalmeth00fuen |url-access=limited }}</ref>

Let <math> f: \mathbb{R}^n \to \mathbb{R} </math> be the [[objective function]] and let <math> g: \mathbb{R}^n \to \mathbb{R}^c </math> be the constraints function, both belonging to <math>C^1</math> (that is, having continuous first derivatives). Let <math> x_\star </math> be an optimal solution to the following optimization problem such that, for the matrix of partial derivatives <math> \Bigl[ \operatorname{D}g(x_\star) \Bigr]_{j,k} = \frac{\ \partial g_j\ }{\partial x_k}</math>, <math> \operatorname{rank} (\operatorname{D}g(x_\star)) = c \le n </math>:

<math display="block">\begin{align}
& \text{maximize } f(x) \\
& \text{subject to: } g(x) = 0
\end{align}</math>

Then there exists a unique Lagrange multiplier <math> \lambda_\star \in \mathbb{R}^c </math> such that <math> \operatorname{D}f(x_\star) = \lambda_\star^{\mathsf{T}}\operatorname{D}g(x_\star) ~.</math> (Note that this is a somewhat conventional thing where <math> \lambda_\star </math> is clearly treated as a column vector to ensure that the dimensions match. But, we might as well make it just a row vector without taking the transpose.){{Tone inline|date=April 2025}}

The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the [[gradient]] of the function (at that point) can be expressed as a [[linear combination]] of the gradients of the constraints (at that point), with the Lagrange multipliers acting as [[coefficient]]s.<ref>{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |year=1969 |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |pages=188–189 }}</ref> This is equivalent to saying that any direction perpendicular to all gradients of the constraints is also perpendicular to the gradient of the function. Or still, saying that the [[directional derivative]] of the function is {{math|0}} in every feasible direction.