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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Use of Lagrange multipliers in pure math problems

I realize that Lagrange multipliers are extremely useful for applied optimization problems. However, I know that the standard analytic proof of the spectral theorem relies on them. I've also seen a few other uses/mentions of them in some pure math…
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I found only one critical point using Lagrange multipliers. Must it be a minimizer?

I am trying to minimize $$V(x,y,z) = \frac {a^2b^2c^2}{6xyz}$$ subject to $$\frac {x^2}{a^2} + \frac{y^2}{b^2} + \frac {z^2}{c^2} = 1$$ and for $x,y,z>0$. I found one critical point; evaluating it gives a function value of $$V= \frac…
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Augmented Lagrangian

Consider the following equality constraint minimization problem: minimize $\text{ }f(x)$ subject to $Ax=b$ Its Lagrangian is then: $L(x,y) = f(x) + y^T(Ax-b)$ We can use then gradient ascent to solve this problem though it is said that it works with…
trembik
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Proving that $f(t)=\frac{n^2}{2}\cdot t^{n-4}(1-t^2)\left(t^2-\frac{n-3}{n}\right)$ is bounded above by $1$, for $n\geq6$ and $t\in[0,1]$

I have a problem that looks like a typical problem of maximizing functions in a compact interval. However, I am not being able to prove the bound I need. Let $n\geq 6$ be an integer number. Consider the function: $$f(t) = \frac{n^2}{2} \cdot…
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Linear optimization for functions

I have the following linear optimization problem. $$ \max \int_0^1 w(t) dt $$ subject to $$ \int_0^1 w(t) \, x_i(t) \, dt \geq 0, \quad i=1,\dots,n $$ and $$ 0 \leq w(t) \leq 1 \quad \text{for all} \quad t\in[0,1]. $$ Here, $x_1(t), \dots, x_n(t)$…
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Maximum and minimum absolute of a function $(x,y)$

I want to know the maximum and minimum absolutes values of this function: $\ f(x,y)= 4x^2 + 9y^2 - x^2y^2 $ $\nabla f(x,y)=(8x-2xy^2,18y-2yx^2) $ I find these critical points: $\ (0,0);(3,2);(-3,2);(3,-2);(-3;-2) $ In$\ (0,0)$ the Hessian…
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Proving $\sum_{\text{cyc}} \frac{a}{b^2+c^2+d^2} \geq \frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$

Prove that $$\frac{a}{b^2+c^2+d^2}+\frac{b}{a^2+c^2+d^2}+\frac{c}{a^2+b^2+d^2}+\frac{d}{a^2+b^2+c^2}≥\frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$$ What I tried, was to say that $a^2+b^2+c^2+d^2=1$ and so…
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Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n x_i =1$ and $x_i \ge0$. I was trying to use KKT…
J.E.M.S
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Lagrange multiplier sign issue

When one has a function of more variables $f(x_1,\dotsc,x_n)$ and wants to find its maxima and minima on a subset of $\mathbb{R}^n$ defined by $f_1(x_1,\dotsc,x_n)=c_1,\dotsc,f_k(x_1,\dotsc,x_n)=c_k$ with $k\leq n$, one can use the Lagrange…
MickG
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Proving Lagrange method by using Implicit Function Theorem

I am trying to show the proof of the Lagrange multiplier method. According to this in general, if $f$ and $g$ are $D+1$ dimensional functions such that $f,g : \mathbb{R}^{D+1} \mapsto \mathbb{R}$, and if the point $p$ with $p=(x',y')$ where $x'$ is…
Ufuk Can Bicici
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Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by $f(x) = \langle Ax,x \rangle$, where $\langle…
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Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought using Lagrange multipliers with the constraint…
ThP
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Calculus of variations: Isoperimetric and holonomic constraints.

A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most textbooks talk about "(1) s.t. (2)" and "(1) s.t. (3)"…
MIMIGA
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Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$

Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that: $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} \ge{\frac{{83}}{10}}\ \ ?$$
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Lagrange multiplier method, find maximum of $e^{-x}\cdot (x^2-3)\cdot (y^2-3)$ on a circle

I attempted to design an exercise for my engineer students and couldn't solve it myself. Maybe here are some experts in calculus who have some better tricks than I do: The exercise would be to find the maxima of $e^{-x}(x^2-3)(y^2-3)$ on the circle…
Julian Kuelshammer
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