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
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...
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...
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...
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...
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...
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...
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...
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'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
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.
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:
$$\frac {\partial F}{\partial x} - \frac d {d \tau}...
In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following:
Equation (6.67):
$$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...
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...
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...
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]...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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.)
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...
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...
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...
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...
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...
Homework Statement
This is actually an Applied Project in the text, and overall is quite a large problem, so I won't post the entire thing, as there are lots of equations and steps where the text guides me by saying "show that...this thing...then...show that this other thing..."
What I need...
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?
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...
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...
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...
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...
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...
I'm currently having some trouble, after the procedure of finding the actual values for the multipliers and the points, but how come can I figure out whether which points that I've collected are maxima, minima or just saddle ones. I've taken a look on lots of books, but I can't seem to find...
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...
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...
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...
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...
Homework Statement
Sorry for the long derivation below. I want to check if what I derived is correct, I can't find it anywhere else, feel free to skip to the end. Thanks!
I am confused by how to write the EL equations if I have multiple constraints of multiple coordinates. For example, let's...
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 =...
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.
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...
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...