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In [[mathematical optimization]], the '''method of Lagrange multipliers''' is a strategy for finding the local [[maxima and minima]] of a [[function (mathematics)|function]] subject to [[constraint (mathematics)|equality constraints]] (i.e., subject to the condition that one or more [[equation]]s have to be satisfied exactly by the chosen values of the variables).<ref>{{cite book |last=Hoffmann |first=Laurence D. |last2=Bradley |first2=Gerald L. |title=Calculus for Business, Economics, and the Social and Life Sciences |location= |publisher= |year=2004 |edition=8th |pages=575–588 |isbn=0-07-242432-X }}</ref> 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. Once [[stationary point]]s have been identified from the first-order necessary conditions, the [[Definiteness of a matrix|definiteness]] of the [[Bordered Hessian|bordered Hessian matrix]] determines whether those points are maxima, minima, or [[saddle point]]s.<ref name="SilberbergSuen">{{cite book |first=Eugene |last=Silberberg |first2=Wing |last2=Suen |title=The Structure of Economics : A Mathematical Analysis |location=Boston |publisher=Irwin McGraw-Hill |edition=Third |year=2001 |isbn=0-07-234352-4 |pages=134–141 }}</ref>

The Lagrange multiplier theorem roughly states that at any [[stationary point]] of the function that also satisfies the equality constraints, 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 |authorlink=David Luenberger |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |year=1969 |isbn= |pages=188–189 }}</ref> 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'''.<ref>{{cite book |first=Brian |last=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 |chapterurl=https://books.google.com/books?id=L7HMACFgnXMC&pg=PA40 }}</ref>

The great advantage of this method is that it allows the optimization to be solved without explicit [[parameterization]] in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems. The method can be summarized as follows: in order to find the stationary points of a function <math>f(x)</math> subjected to the equality constraint <math>g(x) = 0</math>, form the Lagrangian function
:<math>\mathcal{L}(x, \lambda) = f(x) + \lambda g(x)</math>
and find the stationary points of <math>\mathcal{L}</math> considered as a function of <math>x</math> and the Lagrange multiplier <math>\lambda</math>.<ref>{{cite book |first=Murray H. |last=Protter |authorlink=Murray H. Protter |first2=Charles B., Jr. |last2=Morrey |authorlink2=Charles B. Morrey Jr. |title=Intermediate Calculus |location=New York |publisher=Springer |edition=2nd |year=1985 |page=267 |isbn=0-387-96058-9 }}</ref> The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function.<ref>{{cite journal |first=Dan |last=Kalman |title=Leveling with Lagrange: An Alternate View of Constrained Optimization |journal=[[Mathematics Magazine]] |volume=82 |issue=3 |year=2009 |pages=186–196 |jstor=27765899 }}</ref>

== Single constraint ==
[[Image:LagrangeMultipliers2D.svg|thumb|right|300px|Figure 1: The red curve shows the constraint <span style="color: DarkRed">{{math|''g''(''x'', ''y'') {{=}} ''c''}}</span>. The blue curves are contours of <span style="color:DarkSlateBlue">{{math|''f''(''x'', ''y'')}}</span>. The point where the red constraint tangentially touches a blue contour is the maximum of <span style="color:DarkSlateBlue">{{math|''f''(''x'', ''y'')}}</span> along the constraint, since {{math|''d''<sub>1</sub> > ''d''<sub>2</sub>}}.]]

For the case of only one constraint and only two choice variables (as exemplified in Figure 1), consider the [[optimization problem]]

:<math> \text{maximize}\ f(x,y) </math>
:<math> \text{subject to:}\ g(x,y) = 0 </math>

(Sometimes an additive constant is shown separately rather than being included in <math>g</math>, in which case the constraint is written <math> g(x,y)=c </math>, as in Figure 1.) We assume that both <math> f </math> and <math> g </math> have continuous first [[partial derivative]]s.  We introduce a new variable (<math> \lambda </math>) called a '''Lagrange multiplier''' (or '''Lagrange undetermined multiplier''') and study the '''Lagrange function''' (or '''Lagrangian''' or '''Lagrangian expression''') defined by

:<math> \mathcal{L}(x,y,\lambda) = f(x,y) - \lambda g(x,y),</math>

where the <math> \lambda </math> term may be either added or subtracted. If <math> f(x_0, y_0) </math> is a maximum of <math> f(x,y) </math> for the original constrained problem, then there exists <math> \lambda_0 </math> such that (<math> x_0, y_0, \lambda_0 </math>) is a [[stationary point]] for the Lagrange function (stationary points are those points where the first partial derivatives of <math>\mathcal{L}</math> are zero). Also, it must be assumed that <math>\nabla g \ne 0.</math> However, not all stationary points yield a solution of the original problem, as the method of Lagrange multipliers yields only a [[necessary condition]] for optimality in constrained problems.<ref>{{cite book  | last = Bertsekas  | first = Dimitri P.| authorlink = Dimitri P. Bertsekas | title = Nonlinear Programming| edition=  Second  | publisher = Athena Scientific | year = 1999  | location = Cambridge, MA.  | isbn = 1-886529-00-0 |ref=harv}}</ref><ref>{{springer | id=Lagrange_multipliers | title=Lagrange multipliers| first=I.B. | last=Vapnyarskii }}.</ref><ref>{{cite book|last=Lasdon|first=Leon&nbsp;S.|title=Optimization Theory for Large Systems|publisher=Dover |location=Mineola, New York|year=2002 |edition=Reprint of the 1970 Macmillan |mr=1888251 |isbn=0-486-41999-1 }}</ref><ref>{{cite book|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|chapter=XII&nbsp;Abstract duality for practitioners|title=Convex analysis and minimization algorithms, Volume&nbsp;II: Advanced theory and bundle methods|series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]| volume=306 |publisher=Springer-Verlag |location=Berlin|year=1993| isbn=3-540-56852-2|mr=1295240|authorlink2=Claude Lemaréchal|ref=harv|pages=136–193 (and Bibliographical comments on pp.&nbsp;334–335)}}</ref><ref>{{Cite book |title=Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May&nbsp;15–19,&nbsp;2000 |last=Lemaréchal |first=Claude |publisher=Springer-Verlag |year=2001 |isbn=3-540-42877-1 |editor-last=Jünger |editor-first=Michael |series=Lecture Notes in Computer Science |volume=2241 |location=Berlin |pages=112–156 |chapter=Lagrangian relaxation |doi=10.1007/3-540-45586-8_4 |mr=1900016 |author-link=Claude Lemaréchal |editor-last2=Naddef |editor-first2=Denis}}</ref> Sufficient conditions for a minimum or maximum [[Bordered Hessian|also exist]], but if a particular [[candidate solution]] satisfies the sufficient conditions, it is only guaranteed that that solution is the best one ''locally'' – that is, it is better than any permissible nearby points. The ''global'' optimum can be found by comparing the values of the original objective function at the points satisfying the necessary and locally sufficient conditions.

The method of Lagrange multipliers relies on the intuition that at a maximum, <math> f(x,y) </math> cannot be increasing in the direction of any such neighboring point that also has <math> g=0 </math>. If it were, we could walk along <math>g=0 </math> to get higher, meaning that the starting point wasn't actually the maximum.

We can visualize [[Contour line|contour]]s of <math> f </math> given by <math> f(x,y)=d </math> for various values of <math> d </math>, and the contour of <math> g </math> given by <math> g(x,y) = c </math>.

Suppose we walk along the contour line with <math> g=c </math>. We are interested in finding points where <math> f </math> does not change as we walk, since these points might be maxima.

There are two ways this could happen: 

# We could be following a contour line of <math> f </math>, since by definition <math> f </math> does not change as we walk along its contour lines. This would mean that the contour lines of <math> f </math> and <math> g </math> are parallel here.
# We have reached a "level" part of <math> f </math>, meaning that <math> f </math> does not change in any direction.

To check the first possibility (we are following a contour line of <math> f </math>), notice that since the [[gradient]] of a function is perpendicular to the contour lines, the contour lines of <math> f </math> and <math> g </math>  are parallel if and only if the gradients of <math> f </math> and <math> g </math> are parallel. Thus we want points (<math> x,y </math>) where <math> g(x,y) = c </math> and

:<math>\nabla_{x,y} f = \lambda \, \nabla_{x,y} g,</math>

for some <math> \lambda </math>

where

:<math> \nabla_{x,y} f= \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right), \qquad \nabla_{x,y} g= \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right)</math>

are the respective gradients. The constant <math> \lambda </math> is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. This constant is called the Lagrange multiplier. (In some conventions <math> \lambda </math> is preceded by a minus sign).

Notice that this method also solves the second possibility, that <math> f </math> is level: if <math> f </math> is level, then its gradient is zero, and setting <math> \lambda = 0 </math> is a solution regardless of <math>\nabla_{x,y}g</math>.

To incorporate these conditions into one equation, we introduce an auxiliary function

:<math> \mathcal{L}(x,y,\lambda) = f(x,y) - \lambda g(x,y), </math>

and solve

:<math> \nabla_{x,y,\lambda} \mathcal{L}(x , y, \lambda)=0. </math>

Note that this amounts to solving three equations in three unknowns. This is the method of Lagrange multipliers. Note that <math>\nabla_{\lambda} \mathcal{L}(x , y, \lambda)=0</math> implies <math> g(x,y) = 0 </math>. To summarize
:<math>
\nabla_{x,y,\lambda} \mathcal{L}(x , y, \lambda)=0 \iff 
\begin{cases}
\nabla_{x,y} f(x , y) = \lambda \, \nabla_{x,y} g(x , y)\\
 g(x,y) = 0
\end{cases}</math>

The method generalizes readily to functions on <math>n</math> variables

:<math> \nabla_{x_1, \dots, x_n,\lambda} \mathcal{L}(x_1, \dots, x_n, \lambda)=0</math>

which amounts to solving <math>n+1</math> equations in <math>n+1</math> unknowns.

The constrained extrema of <math> f </math> are ''[[Critical point (mathematics)|critical points]]'' of the Lagrangian <math>\mathcal{L}</math>, but they are not necessarily ''local extrema'' of <math>\mathcal{L}</math> (see [[#Example 2|Example 2]] below).

One may [[Hamiltonian mechanics#As a reformulation of Lagrangian mechanics|reformulate the Lagrangian]] as a [[Hamiltonian mechanics|Hamiltonian]], in which case the solutions are local minima for the Hamiltonian. This is done in [[optimal control]] theory, in the form of [[Pontryagin's minimum principle]].

The fact that solutions of the Lagrangian are not necessarily extrema also poses difficulties for numerical optimization. This can be addressed by computing the ''magnitude'' of the gradient, as the zeros of the magnitude are necessarily local minima, as illustrated in the [[#Example 4: Numerical optimization|numerical optimization example]].

== Multiple constraints ==
[[Image:As wiki lgm parab.svg|thumb|right|300px|Figure 2: A paraboloid constrained along two intersecting lines.]]
[[Image:As wiki lgm levelsets.svg|thumb|right|300px|Figure 3: Contour map of Figure 2.]]
The method of Lagrange multipliers can be extended to solve problems with multiple constraints using a similar argument. Consider a [[paraboloid]] subject to two line constraints that intersect at a single point. As the only feasible solution, this point is obviously a constrained extremum. However, the [[level set]] of <math>f</math> is clearly not parallel to either constraint at the intersection point (see Figure 3); instead, it is a linear combination of the two constraints' gradients. In the case of multiple constraints, that will be what we seek in general: the method of Lagrange seeks points not at which the gradient of <math>f</math> is multiple of any single constraint's gradient necessarily, but in which it is a linear combination of all the constraints' gradients.

Concretely, suppose we have <math>M</math> constraints and are walking along the set of points satisfying <math>g_i(\mathbf{x})=0, i=1, \dots, M</math>. Every point <math>\mathbf{x}</math> on the contour of a given constraint function <math>g_i</math> has a space of allowable directions: the space of vectors perpendicular to <math>\nabla g_i(\mathbf{x})</math>.  The set of directions that are allowed by all constraints is thus the space of directions perpendicular to all of the constraints' gradients.  Denote this space of allowable moves by <math>A</math> and denote the span of the constraints' gradients by <math>S</math>.  Then <math>A = S^{\perp}</math>, the space of vectors perpendicular to every element of <math>S</math>.

We are still interested in finding points where <math>f</math> does not change as we walk, since these points might be (constrained) extrema. We therefore seek <math>\mathbf{x}</math> such that any allowable direction of movement away from <math>\mathbf{x}</math> is perpendicular to <math>\nabla f(\mathbf{x})</math> (otherwise we could increase <math>f</math> by moving along that allowable direction).  In other words, <math>\nabla f(\mathbf{x}) \in A^{\perp} = S</math>.  Thus there are scalars <math> \lambda_1, \lambda_2,....\lambda_M </math> such that 

: <math>\nabla f(\mathbf{x}) = \sum_{k=1}^M  \lambda_k \, \nabla g_k (\mathbf{x})  \quad \iff \quad \nabla f(\mathbf{x}) - \sum_{k=1}^M {\lambda_k \nabla g_k (\mathbf{x})} = 0.</math>

These scalars are the Lagrange multipliers. We now have <math>M</math> of them, one for every constraint.

As before, we introduce an auxiliary function

: <math>\mathcal{L}\left( x_1,\ldots , x_n, \lambda_1, \ldots, \lambda _M \right) = f\left( x_1, \ldots, x_n \right) - \sum\limits_{k=1}^M {\lambda_k g_k\left( x_1, \ldots , x_n \right)}</math>

and solve

:<math> \nabla_{x_1, \ldots , x_n, \lambda_1, \ldots, \lambda _M} \mathcal{L}(x_1, \ldots , x_n, \lambda_1, \ldots, \lambda _M)=0 \iff 
\begin{cases}
\nabla f(\mathbf{x}) - \sum_{k=1}^M {\lambda_k \, \nabla g_k (\mathbf{x})} = 0\\
 g_1(\mathbf{x}) = \cdots = g_M(\mathbf{x}) = 0
\end{cases}</math>

which amounts to solving <math>n+M</math> equations in <math>n+M</math> unknowns.

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>.

==Modern formulation via differentiable manifolds==

The problem of finding the local maxima and minima subject to constraints can be generalized to finding local maxima and minima on a [[differentiable manifold]] <math>M</math>.<ref>{{cite book|last1=Lafontaine|first1=Jacques|title=An Introduction to Differential Manifolds|publisher=Springer|isbn=9783319207353|page=70|url=https://books.google.com/books?id=KNhJCgAAQBAJ&pg=PA70 |year=2015  |ref=harv }}</ref> In what follows, it is not necessary that <math>M</math> be a Euclidean space, or even a Riemannian manifold. All appearances of the gradient <math>\nabla</math> (which depends on a choice of Riemannian metric) can be replaced with the [[exterior derivative]] <math>d</math>.

===Single constraint===
Let <math>M</math> be a [[smooth manifold]] of dimension <math>m</math>. Suppose that we wish to find the stationary points <math>x</math> of a smooth function <math>f:M\to\R</math> when restricted to the submanifold <math>N</math> defined by <math>g(x)=0,</math> where <math>g:M\to\R</math> is a smooth function for which 0 is a [[regular value]] of <math>g</math>.

Let <math>df</math> and <math>dg</math> be the [[exterior derivative]]s. Stationarity for the restriction <math>f|_{N}</math> at <math>x\in N</math> means <math>d(f|_N)_x=0.</math> Equivalently, the kernel <math>\ker(df_x)</math> contains <math>T_x N=\ker(dg_x).</math> In other words, <math>df_x</math> and <math>dg_x</math> are proportional vectors. For this it is necessary and sufficient that the following system of <math>m(m-1)/2</math> equations holds:

:<math>df_x \wedge dg_x = 0 \in \Lambda^2(T^{*}_x M)</math>

where <math>\wedge</math> denotes the [[exterior algebra|exterior product]]. The stationary points <math>x</math> are the solutions of the above system of equations plus the constraint <math>g(x)=0.</math> Note that the <math>\tfrac{1}{2} m(m-1)</math> equations are not independent, since the left-hand side of the equation belongs to the subvariety of <math>\Lambda^{2}(T^*_x M)</math> consisting of [[exterior algebra|decomposable elements]].

In this formulation, it is not necessary to explicitly find the Lagrange multiplier, a number <math>\lambda</math> such that <math>df_x = \lambda \, dg_x.</math>

===Multiple constraints===
Let <math>M</math> and <math>f</math> be as in the above section regarding the case of a single constraint. Rather than the function <math>g</math> described there, now consider a smooth function <math>G:M\to \R^p (p>1),</math> with component functions <math>g_i: M \to\R,</math> for which <math>0\in\R^p</math> is a [[regular value]]. Let <math>N</math> be the submanifold of <math>M</math> defined by <math>G(x)=0.</math>

<math>x</math> is a stationary point of <math>f|_{N}</math> if and only if <math>\ker(df_x)</math> contains <math>\ker(dG_x)</math>. For convenience let <math>L_x=df_x</math> and <math>K_x=dG_x,</math> where <math>dG</math> denotes the tangent map or Jacobian <math>TM\to T\R^p.</math> The subspace <math>\ker(K_x)</math> has dimension smaller than that of <math>\ker(L_x)</math>, namely <math>\dim(\ker(L_x))=n-1</math> and <math>\dim(\ker(K_x))=n-p.</math> <math>\ker(K_x)</math> belongs to <math>\ker(L_x)</math> if and only if <math>L_x \in T^*_x M</math> belongs to the image of <math>K^*_x:\R^{p*}\to T^*_x M.</math> Computationally speaking, the condition is that <math>L_x</math> belongs to the row space of the matrix of <math>K_x,</math> or equivalently the column space of the matrix of <math>K^*_x</math> (the transpose). If <math>\omega_x \in \Lambda^{p}(T^*_x M)</math> denotes the exterior product of the columns of the matrix of <math>K^*_x,</math> the stationary condition for <math>f|_{N}</math> at <math>x</math> becomes

:<math>L_x \wedge \omega_x = 0 \in \Lambda^{p+1} \left (T^*_x M \right )</math>

Once again, in this formulation it is not necessary to explicitly find the Lagrange multipliers, the numbers <math>\lambda_1,\ldots,\lambda_p</math> such that 

:<math>df_x = \sum_{i=1}^p \lambda_i d(g_i)_x.</math>

==Interpretation of the Lagrange multipliers==
Often the Lagrange multipliers have an interpretation as some quantity of interest.  For example, if the Lagrangian expression is

: <math>
\begin{align}
& \mathcal{L}(x_1, x_2, \ldots;\lambda_1, \lambda_2, \ldots) \\[4pt]
= {} & f(x_1, x_2, \ldots) + \lambda_1(c_1-g_1(x_1, x_2, \ldots))+\lambda_2(c_2-g_2(x_1, x_2, \dots))+\cdots
\end{align}
</math>

then

:<math>\frac{\partial \mathcal{L}}{\partial c_k} = \lambda_k.</math>

So, {{math|''λ<sub>k</sub>''}} is the rate of change of the quantity being optimized as a function of the constraint parameter.
As examples, in [[Lagrangian mechanics]] the equations of motion are derived by finding stationary points of the [[Action (physics)|action]], the time integral of the difference between kinetic and potential energy.  Thus, the force on a particle due to a scalar potential, {{math|''F'' {{=}} −∇''V''}}, can be interpreted as a Lagrange multiplier determining the change in action (transfer of potential to kinetic energy) following a variation in the particle's constrained trajectory.  
In control theory this is formulated instead as [[costate equations]].

Moreover, by the [[envelope theorem]] the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: if we denote values at the optimum with an asterisk, then it can be shown that

:<math>\frac{\text{d} f(x_1^*(c_1, c_2, \dots), x_2^*(c_1, c_2, \dots), \dots)}{\text{d} c_k} = \lambda_k^*.</math>

For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context {{math|''λ''*}} is the [[marginal cost]] of the constraint, and is referred to as the [[shadow price]].<ref>{{cite book |first=Avinash K. |last=Dixit |authorlink=Avinash Dixit |chapter=Shadow Prices |title=Optimization in Economic Theory |location=New York |publisher=Oxford University Press |edition=2nd |year=1990 |pages=40–54 |isbn=0-19-877210-6 |chapterurl=https://books.google.com/books?id=dHrsHz0VocUC&pg=PA40 }}</ref>

==Sufficient conditions==
{{Main|Bordered Hessian}}

Sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors (determinants of upper-left-justified sub-matrices) of the bordered [[Hessian matrix]] of second derivatives of the Lagrangian expression.<ref name="SilberbergSuen" /><ref>{{cite book|authorlink=Alpha Chiang |last=Chiang |first= Alpha C. |title=Fundamental Methods of Mathematical Economics |url=https://archive.org/details/fundamentalmetho0000chia_h4v2 |url-access=registration |publisher=McGraw-Hill |edition=Third |date=1984 |page=[https://archive.org/details/fundamentalmetho0000chia_h4v2/page/386 386] |isbn=0-07-010813-7 }}</ref>

== Examples ==

=== Example 1===
[[Image:Lagrange very simple.svg|thumb|right|300px|Illustration of the constrained optimization problem]]

====Example 1a====

Suppose we wish to maximize <math>f(x,y)=x+y</math> subject to the constraint <math>x^2+y^2=1</math>. The [[Candidate solution|feasible set]] is the unit circle, and the [[level set]]s of {{mvar|f}} are diagonal lines (with slope −1), so we can see graphically that the maximum occurs at <math>\left(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2}\right)</math>, and that the minimum occurs at <math>\left(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2}\right)</math>.

For the method of Lagrange multipliers, the constraint is

:<math>g(x,y)=x^2+y^2-1,</math>

hence

:<math>\begin{align} 
\mathcal{L}(x, y, \lambda) &= f(x,y) + \lambda \cdot g(x,y) \\[4pt]
&= x+y +  \lambda (x^2 + y^2 - 1).
\end{align}</math>

Now we can calculate the gradient:

:<math>\begin{align}
\nabla_{x,y,\lambda} \mathcal{L}(x , y, \lambda) &= \left( \frac{\partial \mathcal{L}}{\partial x}, \frac{\partial \mathcal{L}}{\partial y}, \frac{\partial \mathcal{L}}{\partial \lambda} \right ) \\[4pt]
&= \left ( 1  + 2 \lambda x, 1 + 2 \lambda y, x^2 + y^2 -1 \right)
\end{align}</math>

and therefore:

:<math>\nabla_{x,y,\lambda} \mathcal{L}(x , y, \lambda)=0 \quad \Leftrightarrow \quad \begin{cases} 1 + 2 \lambda x = 0 \\ 1 + 2 \lambda y = 0 \\ x^2 + y^2 -1  = 0 \end{cases}</math>

Notice that the last equation is the original constraint.

The first two equations yield

:<math>x= y = - \frac{1}{2\lambda}, \qquad \lambda \neq 0.</math>

By substituting into the last equation we have:

:<math>\frac{1}{4\lambda^2}+\frac{1}{4\lambda^2} - 1=0, </math>

so

:<math>\lambda = \pm \frac{1}{\sqrt{2}},</math>

which implies that the stationary points of <math>\mathcal{L}</math> are

:<math>\left(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2}, -\tfrac{1}{\sqrt{2}}\right), \qquad \left(-\tfrac{\sqrt{2}}{2}, -\tfrac{\sqrt{2}}{2}, \tfrac{1}{\sqrt{2}}\right).</math>

Evaluating the objective function {{mvar|f}} at these points yields

:<math>f\left(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2}\right)=\sqrt{2}, \qquad  f\left(-\tfrac{\sqrt{2}}{2}, -\tfrac{\sqrt{2}}{2}\right)=-\sqrt{2}.</math>

Thus the constrained maximum is <math>\sqrt{2}</math> and the constrained minimum is <math>-\sqrt{2}</math>.

====Example 1b====

Now we modify the objective function of Example 1a so that we minimize <math>f(x,y)=(x+y)^2</math> instead of <math>f(x,y)=x+y,</math> again along the circle <math>g(x,y)=x^2+y^2-1=0</math>.  Now the level sets of ''f'' are still lines of slope −1, and the points on the circle tangent to these level sets are again <math>(\sqrt{2}/2,\sqrt{2}/2)</math> and <math>(-\sqrt{2}/2,-\sqrt{2}/2)</math>. These tangency points are maxima of&nbsp;''f''.

On the other hand, the minima occur on the level set for ''f''&nbsp;=&nbsp;0 (since by its construction ''f'' cannot take negative values), at <math>(\sqrt{2}/2,-\sqrt{2}/2)</math> and <math> (-\sqrt{2}/2, \sqrt{2}/2)</math>, where the level curves of ''f'' are not tangent to the constraint. The condition that <math>\nabla f=\lambda \, \nabla g</math> correctly identifies all four points as extrema; the minima are characterized in particular by <math>\lambda =0.</math>

=== Example 2 ===
[[Image:Lagrange simple.svg|thumb|right|300px| Illustration of the constrained optimization problem]]

In this example we will deal with some more strenuous calculations, but it is still a single constraint problem.

Suppose we want to find the maximum values of

:<math> f(x, y) = x^2 y</math>

with the condition that the {{mvar|x}} and {{mvar|y}} coordinates lie on the circle around the origin with radius {{radic|3}}, that is, subject to the constraint

:<math> g(x,y) = x^2 + y^2 - 3 = 0.</math>

As there is just a single constraint, we will use only one multiplier, say {{mvar|λ}}.

The constraint {{math|''g''(''x'', ''y'')}} is identically zero on the circle of radius {{radic|3}}. See that any multiple of {{math|''g''(''x'', ''y'')}} may be added to {{math|''f''(''x'',&nbsp;''y'')}} leaving {{math|''f''(''x'',&nbsp;''y'')}} unchanged in the region of interest (on the circle where our original constraint is satisfied).

Apply the ordinary Langrange multiplier method. Let:

:<math>\begin{align}
\mathcal{L}(x, y, \lambda) &= f(x,y) + \lambda \cdot g(x, y) \\
&= x^2y +  \lambda (x^2 + y^2 - 3)
\end{align}</math>

Now we can calculate the gradient:

:<math>\begin{align}
\nabla_{x,y,\lambda} \mathcal{L}(x , y, \lambda) &= \left ( \frac{\partial \mathcal{L}}{\partial x}, \frac{\partial \mathcal{L}}{\partial y}, \frac{\partial \mathcal{L}}{\partial \lambda} \right ) \\
&= \left ( 2 x y + 2 \lambda x, x^2 + 2 \lambda y, x^2 + y^2 -3 \right )
\end{align}</math>

And therefore:

:<math>\nabla_{x,y,\lambda} \mathcal{L}(x , y, \lambda)=0 \quad \Leftrightarrow \quad \begin{cases} 2 x y + 2 \lambda x = 0 \\ x^2 + 2 \lambda y = 0 \\ x^2 + y^2 - 3 = 0 \end{cases} \quad \Leftrightarrow \quad \begin{cases} x (y + \lambda) = 0 & \text{(i)} \\ x^2 = -2 \lambda y & \text{(ii)} \\ x^2 + y^2 = 3 & \text{(iii)} \end{cases} </math>

Notice that (iii) is just the original constraint. (i) implies {{math|''x'' {{=}} 0}} ''or'' {{math|''λ'' {{=}} −''y''}}. If {{math|''x'' {{=}} 0}} then <math>y = \pm \sqrt{3}</math> by (iii) and consequently  {{math|''λ'' {{=}} 0}} from (ii). If {{math|''λ'' {{=}} −''y''}}, substituting in (ii) we get {{math|''x''<sup>2</sup> {{=}} 2''y''<sup>2</sup>}}. Substituting this in (iii) and solving for {{mvar|y}} gives {{math|''y'' {{=}} ±1}}. Thus there are six critical points of <math>\mathcal{L}</math>:

:<math> (\sqrt{2},1,-1); \quad (-\sqrt{2},1,-1); \quad (\sqrt{2},-1,1); \quad (-\sqrt{2},-1,1); \quad (0,\sqrt{3}, 0); \quad (0,-\sqrt{3}, 0). </math>

Evaluating the objective at these points, we find that

:<math> f(\pm\sqrt{2},1) = 2; \quad f(\pm\sqrt{2},-1) = -2; \quad f(0,\pm \sqrt{3})=0. </math>

Therefore, the  objective function attains the [[global maximum]] (subject to the constraints) at <math>(\pm\sqrt{2},1)</math> and the [[global minimum]] at <math>(\pm\sqrt{2},-1).</math> The point <math>(0,\sqrt{3})</math> is a [[local minimum]] of ''f'' and <math>(0,-\sqrt{3})</math> is a [[local maximum]] of ''f'', as may be determined by consideration of the [[Hessian (mathematics)#Bordered Hessian|Hessian matrix]] of <math>\mathcal{L}(x,y,0)</math>.

Note that while <math>(\sqrt{2}, 1, -1)</math> is a critical point of <math>\mathcal{L}</math>, it is not a local extremum of <math>\mathcal{L}.</math> We have

:<math>\mathcal{L} \left (\sqrt{2} + \varepsilon, 1, -1 + \delta \right ) = 2 + \delta \left( \varepsilon^2 + \left (2\sqrt{2} \right)\varepsilon \right ).</math>

Given any neighbourhood of <math>(\sqrt{2}, 1, -1)</math>, we can choose a small positive <math>\varepsilon</math> and a small <math>\delta</math> of either sign to get <math>\mathcal{L}</math> values both greater and less than <math>2</math>. This can also be seen from the fact that the Hessian matrix of <math>\mathcal{L}</math> evaluated at this point (or indeed at any of the critical points) is an [[indefinite matrix]]. Each of the critical points of <math>\mathcal{L}</math> is a [[saddle point]] of <math>\mathcal{L}.</math>

=== Example 3: [[Entropy]] ===
Suppose we wish to find the [[Probability distribution#Discrete probability distribution|discrete probability distribution]] on the points <math>\{p_1, p_2, \ldots, p_n\}</math> with maximal [[information entropy]]. This is the same as saying that we wish to find the [[Principle of maximum entropy|least structured]] probability distribution on the points <math>\{p_1, p_2, \cdots, p_n\}</math>. In other words, we wish to maximize the [[Shannon entropy]] equation:

:<math>f(p_1,p_2,\ldots,p_n) = -\sum_{j=1}^n p_j\log_2 p_j.</math>

For this to be a probability distribution the sum of the probabilities <math> p_i </math> at each point <math>x_i</math> must equal 1, so our constraint is:

:<math>g(p_1,p_2,\ldots,p_n)=\sum_{j=1}^n p_j = 1.</math>

We use Lagrange multipliers to find the point of maximum entropy, <math>\vec{p}^{\,*}</math>, across all discrete probability distributions <math>\vec{p}</math> on <math>\{x_1,x_2, \ldots, x_n\}</math>. We require that:

:<math>\left.\frac{\partial}{\partial \vec{p}}(f+\lambda (g-1))\right|_{\vec{p}=\vec{p}^{\,*}}=0,</math>

which gives a system of {{mvar|n}} equations, <math> k = 1,\ldots,n</math>, such that:

:<math>\left.\frac{\partial}{\partial p_k}\left\{-\left (\sum_{j=1}^n p_j \log_2 p_j \right ) + \lambda \left(\sum_{j=1}^n p_j - 1\right) \right\}\right|_{p_k=p^*_k} = 0.</math>

Carrying out the differentiation of these {{mvar|n}} equations, we get

:<math>-\left(\frac{1}{\ln 2}+\log_2 p^*_k \right)  + \lambda = 0.</math>

This shows that all <math>p^*_k</math> are equal (because they depend on {{mvar|λ}} only). By using the constraint

:<math>\sum_j p_j =1,</math>

we find

:<math>p^*_k = \frac{1}{n}.</math>

Hence, the uniform distribution is the distribution with the greatest entropy, among distributions on {{mvar|n}} points.

=== Example 4: Numerical optimization ===
[[Image:lagnum1.png|thumb|right|300px|Lagrange multipliers cause the critical points to occur at saddle points.]]

[[Image:lagnum2.png|thumb|right|300px|The magnitude of the gradient can be used to force the critical points to occur at local minima.]]

The critical points of Lagrangians occur at [[saddle point]]s, rather than at local maxima (or minima).<ref name="Heath2005">{{cite book|first=Michael T.|last= Heath|authorlink=Michael Heath (computer scientist)|title=Scientific Computing: An Introductory Survey|url=https://books.google.com/books?id=gwBrMAEACAAJ|pages=203|year=2005|publisher=McGraw-Hill|isbn=978-0-07-124489-3|ref=harv}}</ref> Unfortunately, many numerical optimization techniques, such as [[hill climbing]], [[gradient descent]], some of the [[quasi-Newton method]]s, among others, are designed to find local maxima (or minima) and not saddle points. For this reason, one must either modify the formulation to ensure that it's a minimization problem (for example, by extremizing the square of the [[gradient]] of the Lagrangian as below), or else use an optimization technique that finds [[stationary points]] (such as [[Newton's method in optimization|Newton's method]] without an extremum seeking [[line search]]) and not necessarily extrema.

As a simple example, consider the problem of finding the value of {{mvar|x}} that minimizes <math>f(x)=x^2</math>, constrained such that <math>x^2=1</math>. (This problem is somewhat pathological because there are only two values that satisfy this constraint, but it is useful for illustration purposes because the corresponding unconstrained function can be visualized in three dimensions.)

Using Lagrange multipliers, this problem can be converted into an unconstrained optimization problem:

:<math>\mathcal{L}(x,\lambda)=x^2+\lambda(x^2-1).</math>

The two critical points occur at saddle points where {{math|''x'' {{=}} 1}} and {{math|''x'' {{=}} −1}}.

In order to solve this problem with a numerical optimization technique, we must first transform this problem such that the critical points occur at local minima. This is done by computing the magnitude of the gradient of the unconstrained optimization problem.

First, we compute the partial derivative of the unconstrained problem with respect to each variable:

:<math>
\begin{align}
& \frac{\partial \mathcal{L}}{\partial x}=2x+2x\lambda \\[5pt]
& \frac{\partial \mathcal{L}}{\partial \lambda}=x^2-1.
\end{align}
</math>

If the target function is not easily differentiable, the differential with respect to each variable can be approximated as

: <math>
\begin{align}
\frac{\partial \mathcal{L}}{\partial x}\approx\frac{\mathcal{L}(x+\varepsilon,\lambda)-\mathcal{L}(x,\lambda)}{\varepsilon}, \\[5pt]
\frac{\partial \mathcal{L}}{\partial \lambda}\approx\frac{\mathcal{L}(x,\lambda+\varepsilon)-\mathcal{L}(x,\lambda)}{\varepsilon},
\end{align}
</math>

where <math>\varepsilon</math> is a small value.

Next, we compute the magnitude of the gradient, which is the square root of the sum of the squares of the partial derivatives:

: <math>
\begin{align}
h(x,\lambda) & = \sqrt{(2x+2x\lambda)^2+(x^2-1)^2} \\[4pt]
& \approx\sqrt{\left(\frac{\mathcal{L}(x+\varepsilon,\lambda)-\mathcal{L}(x,\lambda)}{\varepsilon}\right)^2+\left(\frac{\mathcal{L}(x,\lambda+\varepsilon)-\mathcal{L}(x,\lambda)}{\varepsilon}\right)^2}.
\end{align}
</math>

(Since magnitude is always non-negative, optimizing over the squared-magnitude is equivalent to optimizing over the magnitude. Thus, the <nowiki>''</nowiki>square root" may be omitted from these equations with no expected difference in the results of optimization.)

The critical points of {{mvar|h}} occur at {{math|''x'' {{=}} 1}} and {{math|''x'' {{=}} −1}}, just as in <math>\mathcal{L}</math>. Unlike the critical points in <math>\mathcal{L}</math>, however, the critical points in {{mvar|h}} occur at local minima, so numerical optimization techniques can be used to find them.

==Applications==

===Control theory===
In [[optimal control]] theory, the Lagrange multipliers are interpreted as [[costate]] variables, and Lagrange multipliers are reformulated as the minimization of the [[Hamiltonian (control theory)|Hamiltonian]], in [[Pontryagin's minimum principle]].

===Nonlinear programming===
The Lagrange multiplier method has several generalizations. In [[nonlinear programming]] there are several multiplier rules, e.g. the Carathéodory–John Multiplier Rule and the Convex Multiplier Rule, for inequality constraints.<ref>{{Cite journal |last=Pourciau |first=Bruce H. |date=1980 |title=Modern multiplier rules |url=http://www.maa.org/programs/maa-awards/writing-awards/modern-multiplier-rules |journal=[[American Mathematical Monthly]] |volume=87 |issue=6 |pages=433–452 |doi=10.2307/2320250 |jstor=2320250}}</ref>

==See also==
{{div col}}
* [[Adjustment of observations]]
* [[Duality (optimization)|Duality]]
* [[Gittins index]]
* [[Karush–Kuhn–Tucker conditions]]: generalization of the method of Lagrange multipliers
* [[Lagrange multipliers on Banach spaces]]: another generalization of the method of Lagrange multipliers
* [[Lagrange multiplier test]] in maximum likelihood estimation
* [[Lagrangian relaxation]]
{{colend}}

==References==
{{Reflist|30em}}

==Further reading==
*{{cite book |first=Brian |last=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 |pages=32–72 |chapterurl=https://books.google.com/books?id=L7HMACFgnXMC&pg=PA32 }}
*{{cite book |first=Dimitri P. |last=Bertsekas |authorlink=Dimitri Bertsekas |title=Constrained Optimization and Lagrange Multiplier Methods |location=New York |publisher=Academic Press |year=1982 |isbn=0-12-093480-9 }}
*{{cite book |first=Gordon S. G. |last=Beveridge |first2=Robert S. |last2=Schechter |chapter=Lagrangian Multipliers |title=Optimization: Theory and Practice |location=New York |publisher=McGraw-Hill |year=1970 |pages=244–259 |isbn=0-07-005128-3 |chapterurl=https://books.google.com/books?id=TfhVXlWtOPQC&pg=PA244 }}
*{{cite book |first=Brian R. |last=Binger |first2=Elizabeth |last2=Hoffman |chapter=Constrained Optimization |title=Microeconomics with Calculus |location=Reading |publisher=Addison-Wesley |edition=2nd |year=1998 |pages=56–91 |isbn=0-321-01225-9 }}
*{{cite book |first=Michael |last=Carter |chapter=Equality Constraints |title=Foundations of Mathematical Economics |location=Cambridge |publisher=MIT Press |year=2001 |pages=516–549 |isbn=0-262-53192-5 |chapterurl=https://books.google.com/books?id=KysvrGGfzq0C&pg=PA516 }}
*{{cite book |first=Magnus R. |last=Hestenes |authorlink=Magnus Hestenes |chapter=Minima of functions subject to equality constraints |title=Calculus of Variations and Optimal Control Theory |location=New York |publisher=Wiley |year=1966 |pages=29–34 |isbn= }}
*{{cite book |first=C. Ray |last=Wylie |first2=Louis C. |last2=Barrett |chapter=The Extrema of Integrals under Constraint |title=Advanced Engineering Mathematics |location=New York |publisher=McGraw-Hill |edition=Sixth |year=1995 |pages=1096–1103 |isbn=0-07-072206-4 }}

==External links==
{{wikibooks|Calculus optimization methods|Lagrange multipliers}}
Exposition
*[http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html Conceptual introduction] (plus a brief discussion of Lagrange multipliers in the [[calculus of variations]] as used in physics)
*[http://ece.k-state.edu/people/faculty/carpenter/documents/lagrange.pdf Lagrange Multipliers for Quadratic Forms With Linear Constraints] by Kenneth H. Carpenter
For additional text and interactive applets
*[http://www.umiacs.umd.edu/~resnik/ling848_fa2004/lagrange.html Simple explanation with an example of governments using taxes as Lagrange multipliers]
*[http://nlp.cs.berkeley.edu/tutorials/lagrange-multipliers.pdf Lagrange Multipliers without Permanent Scarring] Explanation with focus on the intuition by Dan Klein
*[http://demonstrations.wolfram.com/GeometricRepresentationOfMethodOfLagrangeMultipliers Geometric Representation of Method of Lagrange Multipliers] Provides compelling insight in 2 dimensions that at a minimizing point, the direction of steepest descent must be perpendicular to the tangent of the constraint curve at that point. [Needs InternetExplorer/Firefox/Safari] ''Mathematica'' demonstration by Shashi Sathyanarayana
*[http://ocw.mit.edu/ans7870/18/18.02/f07/tools/LagrangeMultipliersTwoVariables.html Applet]
*[http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-13-lagrange-multipliers/ MIT OpenCourseware Video Lecture on Lagrange Multipliers from Multivariable Calculus course]
*[http://www.athenasc.com/NLP_Slides.pdf Slides accompanying Bertsekas's nonlinear optimization text], with details on Lagrange multipliers (lectures 11 and 12)
*[http://www-mtl.mit.edu/Courses/6.050/2004/unit9/wyatt.apr.7.pdf Geometric idea behind Lagrange multipliers]
*[http://matlab.cheme.cmu.edu/2011/12/24/using-lagrange-multipliers-in-optimization/ MATLAB example of using Lagrange Multipliers in Optimization]

[[Category:Multivariable calculus]]
[[Category:Mathematical optimization]]
[[Category:Mathematical and quantitative methods (economics)]]