For questions on Lagrange multipliers, a strategy to solve constrained optimisation problems.

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6
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2answers
413 views

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: $\ ...
1
vote
0answers
114 views

A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
2
votes
1answer
156 views

Lagrange multiplier constrain critical point

When using Lagrange multipliers in an inequelity, $$ f(x,y) = x^2+y $$ with the constraint $$ x^2+y^2 \leq 1. $$ I have to find the critical points inside the "disk" right? I've done $$ f_x = 2x ...
1
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0answers
20 views

Find the critical curves for the following functional with subsidiary conditions

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1}\left(y'^2+z'^2-xyz'-yz\right)dx$$ with subsidiary conditions : $$\int_{0}^{1}\left(y'^2-xy'-z'^2\right)dx=2$$ ...
1
vote
2answers
180 views

Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
1
vote
1answer
25k views

Deriving demand functions given utility

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$. Derive the ...
0
votes
0answers
28 views

Lagrange identity

I'm aiming to prove that there is a vector $$v$$ equal to ∇f divided by its normal - ∇g divided by its normal. I've being trying to solve it from the point of view of Lagrange multipliers. The ...
0
votes
3answers
396 views

Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
6
votes
1answer
388 views

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 ...
5
votes
1answer
1k views

Calculus of variations: Lagrange multipliers

Given a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$ where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the ...
1
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0answers
90 views

Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
3
votes
2answers
313 views

Minimize $x^TAx$, subject to $||x||=1$. Show that ${x^*}^TAx^*$ is the smallest eigenvalue of $A$ in magnitude.

I'm solving constrained optimization exercises for preparing my final exam. I got stuck at this question. $$ \begin{array}{ll} \text{min} & \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{s.t.} & ...
1
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0answers
36 views

Lagrange multipliers question and my attempt

Question is to minimise the $f(x,y)$ = $3x^{2} + y^{2} - x $$$$$ and constraint is given by $2x^{2} + y^{2} =1 $ Question is simple and ii have got most of points but i seem to miss few points ...
1
vote
2answers
61 views

Question related to Lagrange multipliers

I am stuck with the following problem: A is symmetric $n\times n$ matrix and $f(x)=(Ax)x$ for $x\in {\bf R}^n$. I need to show that the maximum and the minimum values of $f$ on the unit sphere ${x: ...
1
vote
3answers
114 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
4
votes
1answer
190 views

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 ...
2
votes
1answer
118 views

Clarification on optimization problem, continued

Background This is a follow-up to this question. The problem statement is the same: Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ ...
2
votes
2answers
153 views

Using the method of Lagrange multipliers, find the extreme values of a function

Using the method of Lagrange multipliers, find the extreme values of the function $f(x,y)= \frac{2y^3}{3} + 2x^2 +1$ on the ellipse $5x^2 + y^2 = 1/9$ . Identify the (absolute) maximal and minimal ...
1
vote
3answers
107 views

Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
1
vote
2answers
96 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
1
vote
1answer
341 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
1
vote
1answer
396 views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint ...
1
vote
2answers
132 views

Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.

This is not homework, but it is in my book and I find it hard to solve: Determine the points where $f$ is has a local minimum/maximum. Determine if it strong/weak and absolute/relative and ...
0
votes
1answer
41 views

Minimization involving equality constraints

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}} \hspace{4mm} \big(\left( \mathbf{y}^T ...
0
votes
2answers
62 views

Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?

The global maximum and the global minimum of the function $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ can be found using Lagrange multipliers. $\nabla f = \lambda \nabla g$ ...
0
votes
1answer
133 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
0
votes
1answer
2k views

First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
0
votes
1answer
71 views

using the method of LaGrange multipliers find the extreme values of the function

using the method of LaGrange multipliers find the extreme values of the function $f(x,y)=x^3+4y^2$ with constraint $x^2+y^2=1$ currently I have that: $3x^2=2Lx$ which leads to $x=0$ or $L=3/2$ and: ...
0
votes
2answers
72 views

Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
0
votes
3answers
171 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
0
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2answers
343 views

Lagrange Multipliers have no solution

$f(x,y)=2x+y$ subject to constraint $x+y=m$. $(2,1)=\lambda(1,1)$ but this does not satisfy $x+y=m$ So there are no solution?
-1
votes
1answer
55 views

Theorem of Lagrange multipliers - Extremas of $f$ [duplicate]

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are ...