Editing Lagrange multiplier (section)

== Summary and rationale ==
The basic idea is to convert a constrained problem into a form such that the [[derivative test]] of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the '''Lagrangian function''' or Lagrangian.<ref>{{cite book |first1=Brian |last1=Beavis |first2=Ian M. |last2=Dobbs |chapter=Static Optimization |title=Optimization and Stability Theory for Economic Analysis |location=New York |publisher=Cambridge University Press |year=1990 |isbn=0-521-33605-8 |page=40 |chapter-url=https://books.google.com/books?id=L7HMACFgnXMC&pg=PA40 }}</ref> In the general case, the Lagrangian is defined as

<math display="block">\mathcal{L}(x, \lambda) \equiv f(x) + \langle \lambda, g(x)\rangle</math>

for functions <math>f, g</math>; the notation <math>\langle \cdot, \cdot \rangle</math> denotes an [[inner product]]. The value <math>\lambda</math> is called the '''Lagrange multiplier'''.

In simple cases, where the inner product is defined as the [[dot product]], the Lagrangian is

<math display="block">\mathcal{L}(x, \lambda) \equiv f(x) + \lambda\cdot g(x)</math>

The method can be summarized as follows: in order to find the maximum or minimum of a function <math>f </math> subject to the equality constraint <math>g(x) = 0</math>, find the [[stationary point]]s of <math>\mathcal{L} </math> considered as a function of <math>x </math> and the Lagrange multiplier <math>\lambda ~</math>. This means that all [[partial derivative]]s should be zero, including the partial derivative with respect to <math>\lambda ~</math>.<ref>{{cite book |first1=Murray H. |last1=Protter |author-link=Murray H. Protter |first2=Charles B. Jr. |last2=Morrey |author-link2=Charles B. Morrey Jr. |year=1985 |title=Intermediate Calculus |place=New York, NY |publisher=Springer |edition=2nd |page=267 |isbn=0-387-96058-9 }}</ref>
{{block indent | em = 1.5 | text = <math> \frac{ \partial \mathcal{L} }{\partial x} = 0 </math> {{pad|2em}} and {{pad|2em}} <math> \frac{\ \partial \mathcal{L}\ }{\partial \lambda} = 0\ ;</math>}}
or equivalently
{{block indent | em = 1.5 | text = <math> \frac{ \partial f(x) }{\partial x} + \lambda\cdot \frac{ \partial g(x) }{\partial x} = 0 </math> {{pad|2em}} and {{pad|2em}}<math> g(x) = 0 ~.</math>}}

The solution corresponding to the original [[constrained optimization]] is always a [[saddle point]] of the Lagrangian function,<ref name=Walsh1975>{{cite book |first=G.R. |last=Walsh |year=1975 |title=Methods of Optimization |place=New York, NY |publisher=John Wiley & Sons |isbn=0-471-91922-5 |chapter=Saddle-point Property of Lagrangian Function |pages=39–44 |chapter-url=https://books.google.com/books?id=K0EZAQAAIAAJ&pg=PA39 }}</ref><ref>{{cite journal |first=Dan |last=Kalman |year=2009 |title=Leveling with Lagrange: An alternate view of constrained optimization |journal=[[Mathematics Magazine]] |volume=82 |issue=3 |pages=186–196 |doi=10.1080/0025570X.2009.11953617 |jstor=27765899 |s2cid=121070192 }}</ref> which can be identified among the stationary points from the [[Definiteness of a matrix|definiteness]] of the [[Bordered Hessian|bordered Hessian matrix]].<ref name=SilberbergSuen>{{cite book |first1=Eugene |last1=Silberberg |first2=Wing |last2=Suen |year=2001 |title=The Structure of Economics: A Mathematical Analysis |location=Boston |publisher=Irwin McGraw-Hill |edition=Third |isbn=0-07-234352-4 |pages=134–141 }}</ref>

The great advantage of this method is that it allows the optimization to be solved without explicit [[Parametrization (geometry)|parameterization]] in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems. Further, the method of Lagrange multipliers is generalized by the [[Karush–Kuhn–Tucker conditions]], which can also take into account inequality constraints of the form <math>h(\mathbf{x}) \leq c </math> for a given constant <math>c </math>.