What is Lagrange multiplier: Definition and 75 Discussions

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. 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.The method can be summarized as follows: in order to find the maximum or minimum of a function


{\displaystyle f(x)}
subjected to the equality constraint


{\displaystyle g(x)=0}
, form the Lagrangian function




{\displaystyle {\mathcal {L}}(x,\lambda )=f(x)-\lambda g(x)}
and find the stationary points of


{\displaystyle {\mathcal {L}}}
considered as a function of


{\displaystyle x}
and the Lagrange multiplier


{\displaystyle \lambda }
. The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function, which can be identified among the stationary points from the definiteness of the bordered Hessian matrix.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. 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





{\displaystyle h(\mathbf {x} )\leq c}

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  1. deuteron

    Constraint force using Lagrangian Multipliers

    Consider the following setup where the bead can glide along the rod without friction, and the rod rotates with a constant angular velocity ##\omega##, and we want to find the constraint force using Lagrange multipliers. I chose the generalized coordinates ##q=\{r,\varphi\}## and the...
  2. F

    Stationary points classification using definiteness of the Lagrangian

    Hello, I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse. So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
  3. Addez123

    Can't get Lagrange multiplier to work in a single exercise

    So I understand the concept of lagrange multiplier but I fail at every single execise I encounter anyways. Because you always end up with unsolvable equations of x^3yzb3gh + 37y^38x^3 + k^5x = 0 Anways here's my stupid attempt: Instead of doing $$grad(f) + \lambda grad(g) = 0$$ I solve $$...
  4. ergospherical

    A Einbein as Lagrange Multiplier: How Does it Work?

    Let ##g_{\mu \nu}(x)## be a time-independent metric. A photon following a curve ##\Gamma## has action\begin{align*} I[x,e]= \dfrac{1}{2} \int_{\Gamma} e^{-1}(\lambda) g_{\mu \nu}(x)\dot{x}^{\mu} \dot{x}^{\nu} d\lambda \end{align*}with ##e(\lambda)## an independent function of ##\lambda## (an...
  5. hello_world30

    I Proving that ##\omega_0^2 < 2g/l ## for a simple pendulum.

    Here is the problem : A pendulum is composed of a mass m attached to a string of length l, which is suspended from a fixed point. When hanging at equilibrium, the pendulum is hit with a horizontal impulse that results in an initial angular velocity ω0. Show that if ω20 < 2g/l, the string will...
  6. K

    A Two equations of generalized forces

    Wikipedia article under generalized forces says Also we know that the generalized forces are defined as How can I derive the first equation from the second for a monogenic system ?
  7. T

    Small deviations from equilibrium and Lagrange multipliers

    According to the book "Principles of Statistical Mechanics" by Amnon Katz, page 123, ##\alpha## must be such that ##\exp ( -\alpha N ) ## can be expanded in powers of ##\alpha## with only the first order term kept. Is this the necessary and sufficient condition for small deviations from...
  8. agnimusayoti

    How to model a function of a box's volume using Lagrange multiplier methods

    I started to understand how to apply Lagrange multiplier methods. But, for problem like this, I have difficulty to build the function to describe the volume that will be maximized. For the second question, I know from the example (in ML Boas) that ##V=8xyz## becase (x,y,z) is in the 1st octant...
  9. A

    Finding the Local Max/Min of f(x,y) on C

    Hi everyone, I'm struggling with this problem: Let ##f(x,y) = \begin{cases} (x-y)\ln(y-x) & \text{if } y>x \\ 0 & \text{if } y\leq x \end{cases}## and let ##C=\{(x,y)\in \mathbb{R}^2|x^2+y^2=1\}## Then proof that ##max_Cf=1/e## and ##min_Cf=-(\ln2)/\sqrt2## My solution: I used Lagrange...
  10. A

    Determining force of constraint

    Homework Statement Consider a particle moving over the curve ##z=a-bx^2## under the force of gravity. If the particle starts from rest at point ##(0,0)## (I'm guessing it means point ##(0,a)##), tell if the particle ever separates from the curve; if yes, find the point at which it does...
  11. M

    I Optimizing fractions and Lagrange Multiplier

    Hi PF! When minimizing some fraction ##f(x)/g(x)## can we use Lagrange multipliers and say we are trying to optimize ##f## subject to the constraint ##g=1##? Thanks
  12. K

    I Solve Lagrange Multiplier Mystery: ∂Σ{Ni}/∂Nj = ∂N/∂Nj=0

    Hi, I have a question about lagrange multiplier Let's say we are given with the following constraints Σ{Ni}=N and Σ{NiEi}=total energy. N and total energy are constants by definition. if we take the derivative with respect to Nj, ∂Σ{Ni}/∂Nj=∂N/∂Nj where i=j, ∂Σ{Ni}/∂Nj=1 and ∂N/∂Nj = 0...
  13. N

    Maximum of entropy and Lagrange multiplier

    Hello, I have to find the density of probability which gives the maximum of the entropy with the following constraint\bar{x} = \int x\rho(x)dx \int \rho(x) dx = 1 the entropy is : S = -\int \rho(x) ln(\rho(x)) dx L = -\int \rho(x) ln(\rho(x)) dx - \lambda_1 ( \int \rho(x) dx -1 ) -...
  14. King_Silver

    I Lagrange Multiplier. Dealing with f(x,y) =xy^2

    Given a question like this: Findhe maximum and minimum of http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/eq0043M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/empty.gif subject to the constraint...
  15. mr.tea

    I Lagrange multipliers and critical points

    Hi, I have (probably) a fundamental problem understanding something related critical points and Lagrange multipliers. As we know, if a function assumes an extreme value in an interior point of some open set, then the gradient of the function is 0. Now, when dealing with constraint...
  16. Samia qureshi

    What is the Lagrange Multiplier

    Can anybody explain in simple and easy words "Lagrange Multiplier" What is it? and when it is used? i googled it but that was explained in much difficult words.
  17. S

    I Lagrange Multiplier where constraint is a rectangle

    Hello, How can I use Lagrange Multipliers to get the Extrema of a curve f(x,y) = x2+4y2-2x2y+4 over a rectangular region -1<=x<=1 and -1<=y<=1 ? Thanks
  18. a255c

    Lagrange optimization: cylinder and plane intersects,

    Homework Statement The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin. Homework Equations $f(x) = x^2 + y^2 + z^2$ $h(x) = x^2 + y^2 = 1$ $g(x) = x + z = 1$ The Attempt at a Solution $\langle 2x, 2y, 2z \rangle...
  19. H

    I Lagrange multiplier in Hamilton's and D'Alembert's principles

    Why do displaced paths need to satisfy the equations of constraint when using the method of Lagrange multiplier? I thought that with the multiplier, all the coordinates are free and hence should not be required to satisfy the equations of constraint. Source...
  20. Dethrone

    MHB Lagrange Multiplier Ellipsoid

    Use Lagrange multipliers to find $a,b,c$ so that the volume $V=\frac{4\pi}{3}abc$ of an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, passing through the point $(1,2,1)$ is as small as possible. I just need to make sure my setup is correct. $\triangledown...
  21. M

    Calculus of Variations & Lagrange Multiplier in n-dimensions

    extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to constraint $$g(\mathbf{y}, t) = 0 $$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha n_i(t) $$ $$\delta S = \int \sum_i \left(\frac{\partial\mathcal{L} }{\partial y_i} - \frac{d}{dt} \frac{\partial...
  22. kostoglotov

    Discontinuity of a constraint in Lagrange Method

    Homework Statement My question is quite specific, but I will include the entire question as laid out in the text Consider the problem of minimizing the function f(x,y) = x on the curve y^2 + x^4 -x^3 = 0 (a piriform). (a) Try using Lagrange Multipliers to solve the problem (b) Show that the...
  23. Coffee_

    A Lagrange multiplier approach to the catenary problem

    In general, when dealing with mechanics problems using a function ##f(q1,q2,...)=0## that represent constraints one is minimizing the action ##S## while adding a term to the Lagrangian of the not-independent coordinates ##L + \lambda f ##. One can show that this addition doesn't change the...
  24. Coffee_

    How can I know when the Lagrange multiplier is a constant?

    Consider a holonomic system where I have ##n## not independent variables and one constraint ##f(q1,q2,...,qN,t)=0##. One can rewrite the minimal action principle as: ##\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial q'_i} - \lambda \frac{\partial f}{\partial q_i} = 0...
  25. U

    Lagrange multiplier no solution or incorrect formulation

    1. The problem statement I'm stuck with this problem which does not yield a solution. I feel as if I'm not formulating it correctly. Here it is described below. I've also written down the equations as they're easier to be read (attachment) This is something that I was doing with batteries and...
  26. T

    Lagrange multiplier systems of equations -- Help please

    Homework Statement Hi guys I am new here and i really need help with this question. I've tried it multiple times but can't find all the critical points, help would be greatly appreciated. the question is as follows: Find the maximum and minimum values of w=4x-(1/2)y+(27/2)z on the surface...
  27. J

    Word problem using lagrange multiplier

    Homework Statement The Baraboo, Wisconsin plant of International Widget Co. uses aluminum, iron and magnesium to produce high-quality widgets. The quantity of widgets which may be produced using x tonnes of aluminum, y tonnes of iron and z tonnes of magnesium is Q(x,y,z) = xyz. the cost of raw...
  28. J

    Solving a system in five unknowns for lagrange multiplier

    Homework Statement I have to find the extrema of a given function with two constraints f(x,y,z) = x+y+z;x^2-y^2=1;2x+z=1 The Attempt at a Solution If I create a new function F then I have F(x,y,z,\lambda,\mu)=x+y+z-(x^2\lambda - y^2\lambda -\lambda) -(2x\mu + z\mu -...
  29. J

    Lagrangian mechanics, Lagrange multiplier.

    Homework Statement I've thought of a problem to help me with Lagrange multipliers but have got stuck. Consider a particle of mass m moving on a surface described by the curve y = x2, the particle is released from rest at t = 0 and a position x = l. I'm trying to work out the EOM's but have...
  30. M

    Lagrange Multiplier Problem

    Homework Statement Find the extrema of f(x, y) = x2−2xy+ 2y2, subject to the constraint x2 +y2 = 1.Homework Equations ∇f(x,y) = λg(x,y)The Attempt at a Solution This is the work I have thus far: Letting g(x,y) = x2+y2-1, We obtain the following three equations from the Lagrange Multiplier...
  31. B

    LaGrange Multiplier Problem

    Homework Statement Consider the intersection of the elliptic paraboloid Z = X2+4Y2 , and the cylinder X2+Y2= 1. Use Lagrange multipliers to find the highest, and lowest points on the curve of intersection.Homework Equations The gradient equations of both functions.The Attempt at a Solution I...
  32. B

    Solve Lagrange Multiplier Problem | f(X,Y,Z) = 2XY + 6YZ + 8XZ

    Homework Statement Minimize f(X, Y, Z) = 2XY + 6YZ + 8XZ subject to the constraint XYZ = 12. Homework Equations The gradients of the equations, and XYZ = 12. The Attempt at a Solution I have the gradients for both of the equations. ∇f = <2Y + 8Z, 2X + 6Z, 6Y + 8X> ∇g = <...
  33. C

    Optimizing Multivariate Function with Constraint: Lagrange Multiplier Troubles?

    Homework Statement Find extrema for f\left( x,y,z \right) ={ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 } under the constraint g\left( x,y,z \right) ={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=16Homework Equations (1) \nabla f=\lambda \nabla g (2) g\left( x,y,z \right) ={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=16...
  34. O

    Very important, Lagrange multiplier

    Guys, i would be really greatfull if someone help me with this because i really don't know how to deal with this math problem: Find the maximum and minimum values of f = x^(1/4) + y^(1/3) on the boundary of g = 4*x+ 6*y = 720. Please help me someone, i am desperate from this :(
  35. H

    Optimizing Multivariate Function with Lagrange Multiplier Method

    Homework Statement Find the stationary value of $$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$ Constraint: $$ u^2 + v^2 + w^2 = t^2 $$ Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.Homework...
  36. T

    Lagrange Multiplier Question

    Homework Statement Find the product of the maximal and the minimal values of the function z = x - 2y + 2xy in the region (x -1)2+(y + 1/2)2≤2 Homework Equations The Attempt at a Solution I have taken the partial derivatives and set-up the problem, but I am having difficulty...
  37. S

    Lagrange Multiplier -> Find the maximum.

    Lagrange Multiplier --> Find the maximum. Homework Statement Find the maximum value, M, of the function f(x,y) = x^4 y^9 (7 - x - y)^4 on the region x >= 0, y >= 0, x + y <= 7. Homework Equations Lagrange multiplier method and the associated equations. The Attempt at a Solution...
  38. A

    Lagrange multiplier problem - function of two variables with one constraint

    Homework Statement Find the maximum and minimum values of f(x,y) = 2x^2+4y^2 - 4xy -4x on the circle defined by x^2+y^2 = 16. Homework Equations Lagrange's method, where f_x = lambda*g_x, f_y= lambda*g_y (where f is the given function and g(x,y) is the circle on which we are looking...
  39. ElijahRockers

    Using Lagrange Multipliers to Solve Constrained Optimization Problems

    Homework Statement f(x,y) = y2-x2, g(x,y) = x2/4 +y2=9 Homework Equations \nabla f = \lambda \nabla g -2x = \lambda \frac{x}{2} 2y = 2\lambda y \frac{1}{4} x^2 + y^2 = 9 The Attempt at a Solution I arrived at the three equations above. So according to the first equation...
  40. I

    Optimizing Elliptical Radius Vectors with Lagrange Multipliers

    Homework Statement The question is : Find the maximum and minimum lengths of the radius vector contained in an ellipse 5x^2 +6xy+5y^2 Homework Equations The Attempt at a Solution Hi I seem to be at a loss here because usually along with an equation a constraint is also given but in this case...
  41. C

    Finding and recognizing infeasible Lagrange multiplier points

    Maximize: 3*v*m subject to: L - m - v >= 0 V - v >= 0 m - 6 >= 0 M - m >= 0 Where L, M, and V are positive integers. Lagrangian (call it U): U = 3vm + K1(L - m - v) + K2(V - v) + K3(m - 6) + K4(M - m) Where K1-K4 are the slack variables/inequality Lagrange...
  42. D

    How to determine maximum and minimum for Lagrange Multiplier?

    Homework Statement Find the minimum and maximum values of the function subject to the given constraint f(x,y) = x^2 + y^2, 2x + 3y = 6 Homework Equations \nablaf, \nablag The Attempt at a Solution After doing all the calculation, x value and y value came out to be...
  43. S

    Lagrange Multiplier problem

    I'm in a bit of a hurry, so this isn't going to be very pretty. Homework Statement Maximize: V(l,d) = pi * (0.5*d)^2 * l Subject to: l + 3.5d = 84 -> C(l, d) = l + 3.5d - 84Homework Equations ∇V(l,d) = λ ∇C(l,d) The Attempt at a Solution ∇V(l,d) = 0.5*pi*d ∇C(l,d) = 0How do I find the...
  44. L

    Lagrange multiplier question

    Homework Statement Find the maximum and minimum values of f(x,y) = x5y3 on the circle defined by x2 + y2 = 10. Do the same for the disc x2 + y2 ≤ 10. The Attempt at a Solution for the first part, if I call the circle g(x,y) defined by x2 + y2 = 10 I need to now define some F(x,y,λ) =...
  45. T

    Lagrange Multiplier question with solid attempt

    Homework Statement Use the method of Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = x + y2 subject to the constraint g(x,y) = 2x2 + y2 - 1 Homework Equations none The Attempt at a Solution We need to find \nablaf = λ\nablag Hence...
  46. F

    Lagrange Multiplier theory question

    Homework Statement I made this up, so I am not even sure if there is a solution Let's say I have to find values for which these two inequality hold x^2 + y^5 + z = 6 and 8xy + z^9 \sin(x) + 2yx \leq 200And by Lagrange Multipliers that \nabla f = \mu \nabla g So can I let f = 8xy + z^9 \sin(x)...
  47. E

    Lagrange multiplier problem

    1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs 2. Function: V(x,y,z) = 2x2y2z = 8xyz Constraint: x^2+Y^2+z^2=9 3. So far I have gone Φ= 8xyz +...
  48. L

    Maximizing the Lagrangian with Constraints: A Guide to Solving Problems

    Homework Statement L = - \Sigma x,y (P(x,y) log P(x,y)) + \lambda \Sigmay (P(x,y) - q(x)) This is the Lagrangian. I need to maximize the first term in the sum with respect to P(x,y), subject to the constraint in the second term. The first term is a sum over all possible values of x,y...
  49. A

    Max/Min of f Using Lagrange Multipliers

    In a exercise says: Find max a min of f=-x^2+y^2 abaut the ellipse x^2+4y^2=4 i tried -2x=\lambda 2x 2y=\lambda 8y x^2+4y^2-4=0 then \lambda =-1 or \lambda =\frac{1}{4} , but, ¿how i find x,y?
  50. BeBattey

    Maximizing f(x,y,z) with Constraint and Lagrange Multipliers

    Homework Statement Maximize f(x,y,z)=x^{2}+y^{2}+z^{2} with constraint x^{4}+y^{4}+z^{4}=1 using Lagrange multipliers The Attempt at a Solution I've got the setup as: \Lambda(x,y,z,\lambda)=x^{2}+y^{2}+z^{2}+\lambdax^{4}+\lambday^{4}+\lambdaz^{4}+\lambda I solve for all partials nice...