What is Lagrange multipliers: Definition and 179 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. L

    Lagrange multipliers understanding

    Here’s my basic understanding of Lagrange multiplier problems: A typical Lagrange multiplier problem might be to maximise f(x,y)=x^2-y^2 with the constraint that x^2+y^2=1 which is a circle of radius 1 that lie on the x-y plane. The points on the circle are the points (x,y) that satisfy the...
  2. VVS2000

    A H-Theorem and Lagrange multipliers

    so I was studying H theorem from Richard Fitzpartic's site. https://farside.ph.utexas.edu/teaching/plasma/Plasma/node35.html Given H, they consider the following equation and set the constants as I want to understand how they got these particular values for a, b &c can we consider the...
  3. SilverSoldier

    B Constrained Optimization with the KKT Approach

    I'm reading the book Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville, and currently reading this chapter on numerical methods--specifically, the section on constrained optimization. The book states the following. Suppose we wish to minimize a function...
  4. benorin

    This is for an Insights article: Bivariate induction proof using Calc3

    Link to my insight Article it's right where I need you to start checking, read the above boxes to, check out the picture to see examples of the kind of sequence of sets we are dealing with. I need you to read the section jusr below the first picture entitled "3.0.2 Lemma 2.1: Nesting Property of...
  5. M

    MHB Optimization - Lagrange multipliers : minimum cost/maximum production

    Hey! :giggle: Business operates on the basis of the production function $Q=25\cdot K^{1/3}\cdot L^{2/3}$ (where $L$ = units of work and $K$ = units of capital). If the prices of inputs $K$ and $L$ are respectively $3$ euros and $6$ euros per unit, then find : a) the optimal combination of...
  6. jk22

    I Can Lagrange multipliers be used to find a function?

    Problem statement : Let ##f\in C^\infty ([-1;1])## with ##f(1)=f(-1)=0## and ##\int_{-1}^1f(x)dx=1## Which curve has the lowest (maximal) absolute slope ? Attempt : Trying to minimize ##f′(x)−\lambda f″(x)## with Lagrange multipliers but to find f not x ? I got...
  7. AndreasC

    Difficulty with Lagrange multipliers in Kardar's Statistical Physics book

    Alright, so I did some progress and then I got stuck. After some time I went to check the solution. Up to some point, it's all well and good: I understand everything that is happening up to the point where he takes the partial derivative of S wrt ρ(Γ). I don't understand how he gets the...
  8. cwill53

    Lagrange Multipliers and Energy Loss Question

    Constraint: ##I=I_{1}+I_{2}## ##P_{diss,R_{1}}=I_{1}^{2}R_{1}##;##P_{diss,R_{2}}=I_{2}^{2}R_{2}## We want to minimize ##P_{diss,TOT}=I_{1}^{2}R_{1}+I_{2}^{2}R_{2}## $$f(I_{1},I_{2})=I_{1}^{2}R_{1}+I_{2}^{2}R_{2};g(I_{1},I_{2})=I_{1}+I_{2}=I(constraint)$$ $$\nabla f= \left \langle \frac{\partial...
  9. 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...
  10. dRic2

    I Question about Lagrange multipliers

    I'm having some trouble understanding the following proof (##a_{ik}## and ##b_{ik}## are constants) Shouldn't it be ##a_{ik}q_iq_k - \frac 1 {\lambda} (b_{ik}q_iq_k-1)## ? (Summation convention is used) Thanks Ric
  11. Morfe

    Lagrange Multipliers Problem

    Hi there! Kindly help me to solve the problem below. A company is using frustum of a cone containers for their products. What are the dimensions of the least expensive container that can hold 300 cubic cm? Use Lagrange Multipliers to solve the problem. Thanks.
  12. dRic2

    Finding the minimum of an integral with Lagrange multipliers

    Using Lagrange multiplier ##\lambda## (only one is needed) the integral to minimize becomes $$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$ Using E-L equations: $$\frac {\partial F}{\partial x} - \frac d {d \tau}...
  13. sams

    A Question about Euler’s Equations when Auxiliary Conditions are Imposed

    In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following: Equation (6.67): $$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...
  14. Rabindranath

    A Lagrange multipliers on Banach spaces (in Dirac notation)

    I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2##, using the Lagrange multiplier method, minimizing ##\left|\left<\psi\middle|\phi\right>\right|^2## subject to the...
  15. benorin

    Lagrange Multipliers inconsistent system

    Homework Statement maximize f(a,d,h,p)= (4a+3d+3h+c1)c2 *(2+0.01*floor(50+0.0001p)) subject to the constraint 1439a+427d+9259+912/5*h=k. This is not a homework problem but it may as well be: it comes from a game, the function f represents damage as a function of 4 stats and the constraint...
  16. R

    Lagrange multipliers: help solving for x, y and lambda

    Homework Statement Find the local extreme values of ƒ(x, y) = x2y on the line x + y = 3 Homework Equations ∇ƒ = λ∇g The Attempt at a Solution 2yxi+x^2j = λi + λj [2yx=λ] [x^2=λ] [x+y=3] [2yx=x^2] & [(2y)+y=3] [2y=x] & [y=1]...
  17. T

    I Minimization using Lagrange multipliers

    Given the following expressions: and that ## \bf{B}_s = \nabla \times \bf{A}_s ## how does one solve for the following expressions given in (12) and (13)? I've attempted doing so and derive the following expressions (where the hat indicates a unit vector): ## bV = \bf{ \hat{V}} \cdot...
  18. mcaay

    Lagrange Multipliers in Classical Mechanics - exercise 1

    Homework Statement The skier is skiing without friction down the mountain, being all the time in a specified plane. The skier's altitude y(x) is described as a certain defined function of parameter x, which stands for the horizontal distance of the skier from the initial position. The skier is...
  19. V

    I Values of Lagrange multipliers when adding new constraints

    Say we have a Lagrange function with one multiplier a times a constrain. I minimize and solve the system to find a. I now add another constrain to the same system multiplied by the constant b. Is the value of a the same or can it change?
  20. harpazo

    MHB What are the steps for solving a problem using Lagrange Multipliers?

    Use Lagrange Multipliers to find the individual extrema, assuming that x and y are positive. Maximize: f (x, y) = e^(xy) Constraint: x^2 + y^2 = 8 My Work: I decided to rewrite the constraint as x^2 + y^2 without the constant 8 as originally given. g (x, y) = x^2 + y^2 I found the gradient...
  21. harpazo

    MHB Finding Extrema with Lagrange Multipliers: A Step-by-Step Guide

    Use Lagrange Multipliers to find the individual extrema, assuming that x and y are positive. Maximize: f (x, y) = sqrt {6 - x^2 - y^2} Constraint: x + y - 2 = 0 My Work: I first decided to rewrite the constraint as g (x, y) = x + y without the constant -2 as originally given. I found the...
  22. M

    Minimizing weight of a cylinder using Lagrange multipliers

    Homework Statement Julia plans to make a cylindrical vase in which the bottom of the vase is 0.3 cm thick and the curved, lateral part of the vase is to be 0.2 cm thick. If the vase needs to have a volume of 1 liter, what should its dimensions be to minimize its weight? Homework Equations...
  23. DuckAmuck

    A Shape of a pinned canvas w/ Lagrange Multipliers

    I'm basically trying to understand the 2-D case of the catenary cable problem. The 1-D case is pretty straightforward, you have a functional of the shape of a cable with a constraint for length and gravity, and you get the explicit function of the shape of a cable. But if you imagine a square...
  24. M

    Deriving the thermodynamic beta from Lagrange Multipliers

    I'm nearly at the end of this derivation but totally stuck so I'd appreciate a nudge in the right direction Consider a set of N identical but distinguishable particles in a system of energy E. These particles are to be placed in energy levels ##E_i## for ##i = 1, 2 .. r##. Assume that we have...
  25. 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...
  26. Q

    A Is this constraint nonholonomic or not?

    I really want to know whether this equation is nonholonomic or not. (As far as I know, Nonholonomic constraint has a term of velocity and do non-integrable. But this formula does not dependent on a path, because it is a total differential form.)
  27. defaultusername

    Lagrange Multipliers / Height of a Rocket

    Homework Statement I am going to paste the problem word for word, so you can have all the exact information that I have: You’re part of a team that’s designing a rocket for a specific mission. The thrust (force) produced by the rocket’s engine will give it an acceleration of a feet per second...
  28. C

    Ellipsoidal motion (Lagrange multipliers)

    Homework Statement Suppose you have an object of mass m that is constrained to move on an ellipsoid with a constraint function f(x,y,z) = x^2+4y^2+4z^2 -1=0. Aside from the force of constraint, the only force acting on the mass is an elastic force \vec{F}=-kx\hat{x}. Find the Lagrangian, the...
  29. M

    Lagrange multipliers method?

    regarding question number 10, we have h = f + λg where g is the constraint (the ellipsoid) and f is the function we need to maximize or minimize (the rectangular parallelpiped volume), now my question : is it right that f is 8xyz ? i mean if we take f to be xyz not 8xyz and solved till we got...
  30. Y

    MHB Solve Lagrange Multipliers Problem with x-4y=1 Constraint

    Hello all I have this problem: Use Lagrange Multipliers to find the min and max of: \[f(x,y)=xy^{2}\] under the constraint: \[x-4y=1\] \[-1\leqslant x\leq 2\] My problem is: I know how to solve if \[-1\leqslant x\leq 2\] wasn't given. I calculate the Lagrangian function, find it's...
  31. RaulTheUCSCSlug

    Find Relative Extrema of f|s: Explained with Lagrange Multipliers

    I am in Calculus 3, and I do not under stand what it means when they ask to find the relative extrema of f|S? The problem is usually something like f:R^n=>R, (x,y,z) |=> (some function) , S= {(x,y) | x e R} What does f|s mean? How does this relate to Lagrange multipliers? The book does not...
  32. kostoglotov

    Part Derivs: Minimizing the Weight of a Rocket

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  33. throneoo

    Constrained Extrema and Lagrange Multipliers

    Suppose I have a function f(x,y) I would like to optimize, subject to constraint g(x,y)=0. Let H=f+λg, The extrema occurs at (x,y) which satisfy Hy=0 Hx=0 g(x,y)=0 Suppose the solutions are (a,b) and (c,d). If f(a,b)=f(c,d) , how do I determine whether they are maxima or minima?
  34. I

    Lagrange multipliers, guidance needed

    Homework Statement f(x,y) is function who's mixed 2nd order PDE's are equal. consider k_f: determine the points on the graph of the parabloid f(x,y) = x^2 + y^2 above the ellipse 3x^2 + 2y^2 = 1 at which k_f is maximised and minimised. The Attempt at a Solution is this the langrange...
  35. C

    Lagrange Multipliers: Deriving EOM & Conditions for Contact Loss

    Homework Statement An object of mass m, and constrained to the x-y plane, travels frictionlessly along a curve f(x), while experiencing a gravitational force, m*g. Starting with the Lagrangian for the system and using the method of Lagrange multipliers, derive the equations of motion for the...
  36. J

    Particle constrained to move on a hemisphere

    Homework Statement A particle slides on the outer surface of an inverted hemisphere. Using Lagrangian multipliers, determine the reaction force on the particle. Where does the particle leave the hemispherical surface? L - Lagrangian qi - Generalized ith coordinate f(r) - Holonomic constraint...
  37. C

    MHB Solving Lagrange Multipliers: Find Extrema of Distance from (1,2,3) to Sphere

    Hello, I am having a bit of trouble with the Lagrange multiplier method. My question is: Use the Lagrange multiplier method to find the extrema points of the distance from the point (1,2,3) to the surface of the sphere {x}^{2}+{y}^{2}+{z}^{2}=4. Find the possible values for of \lambda. This...
  38. P

    Lagrange multipliers open constraint

    Homework Statement Find the maximum and minimum values of the function f(x, y) =49 − x^2 − y^2 subject to the constraint x + 3y = 10. The Attempt at a Solution ∇f = <2x,2y> ∇g = <1,3> ∇f =λ∇g 2x = λ 2y = 3λ 2x = 2y/3 x = y/3 y/3 + 3y = 10 y = 3 x = 1 f(1,3) = 39 Now that is the only...
  39. Y

    Finding Maxima, Minima, and Saddle Points with Lagrange Multipliers

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  40. T

    Lagrange multiplier systems of equations -- Help please

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  41. GFauxPas

    Lagrange Multipliers and Shadow Prices

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  42. GFauxPas

    Modeling and optimization - HW help

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  43. F

    Lagrange multipliers: How do I know if its a max of min

    in the problem f(x,y)=x^2+y^2 and xy=1, I get 2 as a local extrema and it is a min in the problem f(x1,x2...xn) = x1+x2..+xn (x1)^2+...(xn)^2=1 I get sqrt(n) and its a max. How do I know if these are max or min values? If I get more than two extrema, I just compare them and one's a max and the...
  44. E

    Lagrange multipliers for multiple constraints of multiple coordinates

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  45. A

    Lagrange Multipliers with Multiple Constraints?

    Homework Statement Using Lagrange multipliers, find the max and the min values of f: f(x,y,z) = x^2 +2y^2+3x^2 Constraints: x + y + z =1 x - y + 2z = 2Homework Equations ∇f(x) = λ∇g(x) + μ∇h(x)The Attempt at a Solution Using Lagrange multipliers, I obtained the equations: 2x = λ + μ 4y =...
  46. Feodalherren

    Lagrange multipliers (yes, again)

    Homework Statement f=xy^2 C: x^2 + y^2 = 3 Homework Equations The Attempt at a Solution I don't understand how he can say that x=0 is a solution in this one. Looking at the contours, there are no solutions for f if x=0.
  47. Feodalherren

    Lagrange multipliers to find max/min

    Homework Statement Use Lagrange Multipliers to find the minimum value of f(x,y)=x^{2}+(y-1)^{2} that lie on the hyperbola x^{2}-y^{2}=1. Draw a picture to verify your final answer. Homework Equations \nabla f=\lambda \nabla C The Attempt at a Solution So I can find the critical...
  48. J

    Alloy composition from component densities (Lagrange Multipliers)

    Homework Statement An alloy of gold, aluminum, and copper has a density of 10,000 kg/m3. The alloy contains at least 10% aluminum and 5% copper by mass. The densities for the three metals are respectively ρAu = 19320 kg/m3, ρAl = 2712 kg/m3, ρCu = 8940 kg/m3. Find the maximum and minimum...
  49. MarkFL

    MHB Kunal's question at Yahoo Answers regarding Lagrange multipliers

    Here is the question: I have posted a link there to this thread so the OP can view my work.