# 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

f
(
x
)

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

g
(
x
)
=
0

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

L

(
x
,
λ
)
=
f
(
x
)

λ
g
(
x
)

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

L

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

x

{\displaystyle x}
and the Lagrange multiplier

λ

{\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

h
(

x

)

c

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

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1. ### 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...
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15. ### A Lagrange multipliers on Banach spaces (in Dirac notation)

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16. ### Lagrange Multipliers inconsistent system

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18. ### I Minimization using Lagrange multipliers

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19. ### Lagrange Multipliers in Classical Mechanics - exercise 1

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20. ### 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?
21. ### MHB What are the steps for solving a problem using Lagrange Multipliers?

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

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23. ### Minimizing weight of a cylinder using Lagrange multipliers

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26. ### I Lagrange multipliers and critical points

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27. ### A Is this constraint nonholonomic or not?

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28. ### Lagrange Multipliers / Height of a Rocket

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29. ### Ellipsoidal motion (Lagrange multipliers)

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30. ### Lagrange multipliers method?

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31. ### 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...
32. ### Find Relative Extrema of f|s: Explained with Lagrange Multipliers

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33. ### Part Derivs: Minimizing the Weight of a Rocket

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34. ### 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?
35. ### Lagrange multipliers, guidance needed

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36. ### Lagrange Multipliers: Deriving EOM & Conditions for Contact Loss

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37. ### Particle constrained to move on a hemisphere

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38. ### 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...
39. ### Lagrange multipliers open constraint

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40. ### Finding Maxima, Minima, and Saddle Points with Lagrange Multipliers

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41. ### Lagrange multiplier systems of equations -- Help please

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42. ### Lagrange Multipliers and Shadow Prices

This is a homework in mathematical modeling and optimization; we're up to Lagrange multipliers and shadow prices. 1. Homework Statement A manufacturer of PCs currently sells 10,000 units per month of a basic model. The cost of manufacture is 700$/unit, and the wholesale price is$950. The cost...
43. ### Modeling and optimization - HW help

Help please with a problem from my modelling and optimization class. We're doing 2 variable optimization using Lagrange Multipliers. We're also discussing shadow prices. The first part of this problem is to maximize profit using the price and advertising budget assumptions and data. The data...
44. ### Lagrange multipliers: How do I know if its a max of min

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45. ### Lagrange multipliers for multiple constraints of multiple coordinates

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46. ### Lagrange Multipliers with Multiple Constraints?

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47. ### Lagrange multipliers (yes, again)

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48. ### 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...
49. ### Alloy composition from component densities (Lagrange Multipliers)

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50. ### 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.