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
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
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...
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...
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
$$...
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...
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...
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 ?
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...
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...
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...
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...
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
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...
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 ) -...
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...
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...
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.
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 -...
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...
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...
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...
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 = <...
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...
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 :(
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...
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...
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...
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...
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...
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...
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...
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...
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) =...
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...
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)...
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 +...
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...
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?
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...