Questions tagged [lagrange-multiplier]

This tag is for the questions on Lagrange multipliers. The method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

When are Lagrange multipliers useful?

One of the most common problems in calculus is that of finding maxima or minima (in general, "extrema") of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside conditions or constraints. The method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables.

Put more simply, it's usually not enough to ask, "How do I minimize the aluminum needed to make this can?" (The answer to that is clearly "Make a really, really small can!") You need to ask, "How do I minimize the aluminum while making sure the can will hold $10$ ounces of soup?" Or similarly, "How do I maximize my factory's profit given that I only have $\$15,000$ to invest?" Or to take a more sophisticated example, "How quickly will the roller coaster reach the ground assuming it stays on the track?" In general, Lagrange multipliers are useful when some of the variables in the simplest description of a problem are made redundant by the constraints.

The mathematics of Lagrange multipliers:

To find critical points of a function $f(x, y, z)$ on a level surface $g(x, y, z) = C$ (or subject to the constraint $g(x, y, z) = C$), we must solve the following system of simultaneous equations: $$∇f(x, y, z) = λ∇g(x, y, z)$$ $$g(x, y, z) = C$$ Remembering that $∇f$ and $∇g$ are vectors, we can write this as a collection of four equations in the four unknowns $x, y, z,$ and $λ$ : $$f_x(x, y, z) = λ~g_x(x, y, z)$$ $$f_y(x, y, z) = λ~g_y(x, y, z)$$ $$fz(x, y, z) = λ~g_z(x, y, z)$$ $$g(x, y, z) = C$$ The variable $~λ~$ is a dummy variable called a Lagrange multiplier; we only really care about the values of $x, y,$ and $z$ .

Once we have found all the critical points, we plug them into $~f~$ to see where the maxima and minima. The critical points where $~f~$ is greatest are maxima and the critical points where $~f~$ is smallest are minima.

Note:

Solving the system of equations can be hard! Here are some tricks that may help:

$1.\quad$ Since we don’t actually care what $~λ~$ is, you can first solve for $~λ~$ in terms of $~x,~ y,~$ and $~z~$ to remove $~λ~$ from the equations.

$2.\quad$ Try first solving for one variable in terms of the others.

$3.\quad$ Remember that whenever you take a square root, you must consider both the positive and the negative square roots.

$4.\quad$ Remember that whenever you divide an equation by an expression, you must be sure that the expression is not $~0~$. It may help to split the problem into two cases: first solve the equations assuming that a variable is $~0~$, and then solve the equations assuming that it is not $~0~$.

Reference:

https://en.wikipedia.org/wiki/Lagrange_multiplier

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Find maximum and minimum of a function with constraint

I'm finding maximum and minimum of a function $f(x,y,z)=x^2+y^2+z^2$ subject to $g(x,y,z)=x^3+y^3-z^3=3$. By the method of Lagrange multiplier, $\bigtriangledown f=\lambda \bigtriangledown g$ and $g=3$ give critical points. So I tried to solve these…
user112018
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how to get the dual of that optimization problem

max $1^\top x$ such that $x^\top M x = 0$ and $x_i^2 = x_i$ for all i For the above problem how can I derive the dual form. My main problem is to choose matrix notation or the element-wise notation as I am deriving the dual. This conflict is cause…
erogol
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Help with Lagrangian multiplier method

How do we apply Lagrangian multiplier method for a problem with more than one condition? Here is my work so far, for the below optimization problem:
Niousha
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What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot \int_0^1 \left( 1-x\right)^{n-1} \cdot x^m \cdot…
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Proving an inequality usuing Lagrange multipliers

Cant seem to find the trick, how to proove this one, usuing Lagrange's amazing multipliers. $$ \bigg(\frac{x+y}{2}\bigg)^n \le \frac{x^n + y^n}{2} $$ $ x,y > 0$, and $n\in \Bbb N$. Any tips? Thanks guys!
Simba
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How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function without plugging it into an equation and/or using…
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What is the shortest distance from the origin to the intersection of $xyz=a$ and $y=bx$?

Constraints: $a,b>0$ Here is what I have so far: In order to get the shortest distance from the origin, we set $f(x,y,z)=x^2+y^2+z^2$ subject to the constrains $xyz=a$ and $y=bx$. By Lagrange multipliers, let…
dlaser
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Finding maximum of a function with an ellipse constraint

I'm trying to find the maximum of a function $f = a^T\mu$ where a is a vector when we have a constraint of the form: $$g(\mu) = n\mu^T S^{-1}\mu - C = 0$$ where C is a fixed constant, $S^{-1}$ is a fixed matrix , $n$ is a number and $\mu$ is a…
Sam
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$f$ does not have extrema at the Lagrange multiplier value

I have the function $2x^2+y$ and one constraint $x-y^2=1$ and want to find maximum value by lagrange multiplier. Intuitively, I see the point $(2,1)$ satisfies $c$ and have value of $f(2,1)=9$. Using Lagrange function $f(z)-\lambda c(z)$, I get…
seteropere
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Using Lagrange multipliers to find minimum value

Use Lagrange multipliers to find the minimum value of $$ T = \frac {a}{v \cos \alpha} + \frac {b}{v\cos\beta} $$ subject to the constraint $$ L = a\tan \alpha + b\tan\beta $$ where $a, b, v$ and $L$ are all constants. I have no idea how to do…
Drey1
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Is there any stricy way of eliminating Lagrangian Multipliers or it depends on the specific equations after derivative equations?

I am pretty new for the concept of langragian and might be a naive question. After computing the first derivative equation of lagrange version of the function, it is required to get away multipliers but it is a little hasting for me at the moment to…
erogol
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Problem on Lagrange multipliers

This problem has two parts: $a)$ Let $k>0$, find the minimum of the function $f(x,y)=x+y$ over the set S=$\{(x,y) \in \mathbb R^2_{> 0}:xy=k\}$. $b)$ Prove that for every $(x,y) \in \mathbb R^2_{> 0}$ the inequality $\frac{x+y} {2}\geq \sqrt{xy}$…
user100106
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Finding the shortest distance between a point and a coplex surface

I have a surface which is $z=ax+bx^2+cxy+d$, where $a,b,c$ are coefficients and $d$ is the constant. so an arbitrary point would be $(x, y, ax + bx^2 + cxy+ d)$. there are a set of points that are not in this surface and I want to calculate the…
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Find all extrema for $f(x,y) = 3xy$ subject to the constraint $4x^2 + 2y = 48$

Find all extrema for $f(x,y) = 3xy$ subject to the constraint $4x^2 + 2y =48$. I put it into the form of: $3xy - \lambda(4x^2 +2y -48) = F(x,y,\lambda)$ $3xy - 4x^2\lambda -2y\lambda + 48\lambda$ $F_x = 3y - 8x\lambda = 0$ $F_y = 3x - 2\lambda =…
Ozera
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Lagrange multiplier problem

I am stuck with the following question: Use Lagrange multipliers to determine the shortest distance from a point $x \in R $ to a plane {$y∣b^T y=c$}. Please could someone help me step by step through the problem as I am not even sure where to start…
Natalie
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