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

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1answer
35 views

Hottest and coldest points on a heated circular plate (use Lagrange multipliers)

A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using ...
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2answers
393 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
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0answers
14 views

Is convexity of the objective function sufficient for a local maxima to be a global maximum?

In my problem, I have to maximize a convex function $f(x_1,x_2,\cdots,x_n)$ subject to two equality constraints $g_1=0$ and $g_2=0$. As usual, I constructed the Lagrangian ...
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1answer
30 views

Word Problem Lagrange Method

I am studying for my exams and got very very stuck at a word problem on the Lagrange Methods, my biggest difficulty is to properly identify the function to be maximized (in this case) and so its ...
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0answers
29 views

Find the maximum distance from origin to the surface?

I am having trouble with this problem... I need to find the maximum distance from the origin to the surface $$f:=\frac{x^4}{2^4}+\frac{y^4}{3^4}+\frac{z^4}{(\sqrt2)^4}-1$$ I think I have to use ...
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2answers
35 views

How is f(4,4,4)=48 a local minimum? Can it be inferred that it is either a maxima or minima & only one extreme value within the constraint?

Disclaimer: In the definition (Stewart Calculus, 7E): "Method of Lagrange Multipliers" part (b)- Evaluate $f$ at all extreme points $(x,y,z)$ from step a. The largest of these values is the maximum ...
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1answer
47 views

Maximized surface area of box with fixed length

Assuming we have a box given that the sum of all intervals is $a$. What is the maximal surface area of the box? I know I need to use Lagrange multiplier but when I find the hessian matrix I get that ...
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1answer
29 views

Solving a quadratic convex optimization problem

There's this convex optimization problem which I got stuck after writing the Lagrange equation. I simply couldn't find a way to eliminate the Lagrange multiplier. $$\begin{array}{ll} \text{minimize} ...
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1answer
79 views

Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
1
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1answer
70 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
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2answers
153 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. $\nabla f = \lambda \nabla g$ ...
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0answers
35 views

Finding the Maximum and Minimum values w/constraint [duplicate]

I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is. Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ ...
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1answer
18 views

Lagrange method with vectors?

How does one apply the Lagrange method to vectors? The problem I have (it's financial) is $$\max_w w^T r - w^T\Sigma ^{-1} w $$ under the condition that $w_1 + w_2 = 1$. $r$ is a known vector with 2 ...
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0answers
19 views

Value of Lagrangian Multiplier

I have a two dimensional optimization problem of the form $$ v = \max_{x,y} f(x,y)+g(x,y) $$ Both $g,f$ are concave and continuously differentiable. Assume the solution can be reached by first order ...
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2answers
18 views

KKT conditions with two inequality constrains

I need to minimize $f(x,y,z)=x^{2}+2y^{2}+3z^{2}$ subject to \begin{align*} &x-y-2z\leq 12\\ &x+2y-3z\leq 8. \end{align*} So I wrote the lagrangian of $f$. \begin{align*} ...
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2answers
33 views

Solving for $\lambda$ in a Lagrange multiplier problem

A question from a past examination paper reads as follows: Use the method of Lagrange multipliers to find the extreme points of the function $$f(x,y,z) = x^{2}+y^{2}+z^{2}$$ subject to the ...
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2answers
22 views

utility function question from my textbook

Suppose there are two goods with prices $ p₁ = 2, p₂ = 5, $ the income is $ M = 40 $ and the utility function is $ U (x₁, x₂) = (x₁)^⅓ . (x₂)^ ½, $ Find the optimum consumption plan. Attempt: I do ...
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3answers
94 views
2
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4answers
49 views

In Lagrange Multiplier, why level curves of $f$ and $g$ are tangent to each other?

In Lagrange multiplier method, e.g. optimize a function $f(x_1, \dots, x_n)$ under a constraint $g(x_1, \dots, x_n) = 0$. There is a fact that $\nabla f$ is parallel to $\nabla g$ which is given rise ...
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1answer
29 views

Where does lambda come from in Lagrange multipliers? Specifically related to find maximum and minimum values on a constraint.

The book states that the $\nabla f(x,y,z) = \lambda\nabla g(x,y,z)$ It talks about the slope of the tangents being parallel, but wouldn't they technically just be same line? It also brings up norms ...
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0answers
32 views

Help finding the maximum and minimum values using lagranges method [duplicate]

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ after solving Equation 1 $=$ $\nabla$$f_x$ $=$ $\lambda$ $\nabla$$g_x$ $=$ [ $yz$ $=$ $\lambda$2$x$] for x and obtain: $x$ ...
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2answers
16 views

extremum under constraint

I had to find the extremum of $z=x^2+y^2$ under the constraint of $x+y=3$; I used Lagrange multipliers to reach the conclusion that $(1.5,1.5)$ is an extermum point, but had no way of determining ...
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1answer
88 views

Is the Intersection of these Two Sets a Smooth Manifold?

$A=M\cap N$, $$M=\{(x,y,z)\in\Bbb R^3| x^2+y^2=1\},$$ $$N=\{(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1\}.$$ 1. Is $A$ is smooth manifold? 2. Find the points of $A$ that are farthest from the ...
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2answers
21 views

Find the extreme values of $f(x,y)=xy$ on $D=\{(x,y)|1 \leq x^2+y^2 \leq 4\}$

This would have to be done using conditional extremes(Lagrange method), and maybe some topological properties.I do not know how to do this, I have only done cases where the $D$ would be defined with ...
2
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1answer
32 views

Relationship between two-equation constrained optimization and one-equation version

I am learning about the Lagrange multiplier. Here's what I understand so far. Suppose a point $P$ is a minimizer of $f(x)$ subject to $g(x)=0$. Then any movement along that level-curve of $g$ must ...
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0answers
11 views

Discrete Approximation to Dynamic Lagrangians

Suppose I have the following dynamic optimization problem, where I want to maximize the function $u(c,h)$ over time that's differentiable in both $c$ and $h$. I'm going to assume that the function $u$ ...
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1answer
63 views

Elimination of λ and μ in Lagrange method of multipliers both constraints are nonzero

QUESTION: Determine maximum value of OP, O being origin of coordinates where P describes the curve $x^2 + y^2 +2z^2=5, x+2y+z=5$? Here using lagrange method of multipliers we have two constants λ and ...
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1answer
431 views

Using Lagrange multipliers to find shortest distance between two straight lines

A problem asks me to use the method of Lagrange multipliers to find the shortest distance between the straight lines $x=y = z$ and $x = -y, z=2$ (It also warns me that using this method is a bit ...
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1answer
46 views

Cannot Find Mistake In Lagrange Multipliers Problem

I am working on a problem asking me to use the method of constrained extrema to find the global maximum and minimum for the following functions: $$f(x,y) = xy$$ constrained to $$g(x,y) = 4x^2 + 2xy ...
2
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2answers
39 views

Conditional Extremes/Lagrange multipliers: proving: $\frac{1}{x_1}+…+\frac{1}{x_n} \geq \frac{n^2}{x_1+…+x_n}$

$$\frac{1}{x_1}+...+\frac{1}{x_n} \geq \frac{n^2}{x_1+...+x_n};x_i>0.$$ This is supposed to be proven using conditional extremes. I tried the main function being ...
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2answers
34 views

How to find the shortest distance between $z^2 -xy = 1$ and the origin using Lagrange multiplier?

My task is this: Find the points on the surface $z^2 -xy = 1$ with the shortest distance to the origin. My work so far: Let $f(x,y,z) = x^2 + y^2 + z^2$ and $g(x,y,z)=z^2 -xy -1$ then we have to ...
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0answers
39 views

How to solve these simultaneous equations?

I'm doing questions from this page: http://tartarus.org/gareth/maths/tripos/IB/Variational_Principles.pdf and I'm doing Question 2013 1/I/6A The question asks to find the cylindrical cup of least ...
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1answer
41 views

Use Lagrange multiplier to find the distance between the point $(3,4,0)$ and the surface of the cone $z^2=x^2+y^2$

Use Lagrange multiplier to find the distance between the point $(3,4,0)$ and the surface of the cone $$ z^2=x^2+y^2 $$ I wrote the equation of the distance: $$f(x,y,z)=(x-3)^2+(y-4)^2+z^2$$ and ...
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0answers
18 views

Minimizing convex functions without compatible gradients

I've been working on a minimization problem for a while, involving "simple" conditions, but haven't been able to figure it out. I've tried using Lagrange Multipliers and KKT, but the presence of ...
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0answers
31 views

An optimisation problem

I have an optimisation probem given below $$argmax_{x_i \ \ \forall x_i=1,2...n} \sum_{i} S_ie^{-\alpha x_i}$$ subject to $$\sum x_i = 1$$ $$\sum C_i x_i \leq B $$ $$\forall i \ \ x_i \geq 0 $$ ...
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2answers
34 views

Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point (3, 1, -1).

Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point $(3, 1, -1)$. I identified that this is a constrained optimisation problem which I will solve ...
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0answers
48 views

Global extremum when the constraint is not compact?

When the constraint is compact, the function must have both a global maximum and a global minimum somewhere in the constraint. However, if the constraint is not compact, the global extremum may not ...
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1answer
24 views

Minimum and maximum with lagrange multiplier

I have a function with two constraints whose intersection is unitary circumference. $$x^2+y^2+z^2=1$$ and $$x+y+z=0$$ I can't understand why I cannot apply the lagrange multipliers method with only ...
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0answers
12 views

Problem about find the extreme of a function (Multipliers of Lagrange)

Good morning, i have a problem with this: Find the maximum and minimum distances from the origin to the curve $g\left(x,y\right)=5x^{2}+6xy+5y^{2}$ I make this: Function to optimize: ...
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3answers
614 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be ...
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0answers
14 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...
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1answer
25 views

show that $\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big)$

Let $\alpha_1,\alpha_2,...,\alpha_n>0.$ How can I show that $$\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big).$$ Please provide me ...
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1answer
31 views

How to prove the following inequality using Lagrangian multipliers?

Find the maximum and minimum values of the function $f(x,y,z)=(xyz)^2$ where $(x,y,z)$ is on the sphere $x^2+y^2+z^2=r^2$. Then show using above that $(abc)^{1/3} \leq (a+b+c)/3$. For non-negative ...
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1answer
396 views

How can I find the minimum distance between the origin $(0,0)$ and the curve $y=1-x^2$ using Lagrange multipliers?

I used the curve as the constraint and the origin as the point but I am not sure if that is correct.
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1answer
44 views

Solving optimization with Lagrange multipliers

I am fairly new to Lagrange multipliers. Can someone please show me how to maximize the following function: \begin{align} f(x,y)=240\sqrt{x}+y \end{align} Subject to: \begin{align} 30x+y=720 ...
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0answers
15 views

How to obtain gaussian(normal) distribution

I heard that Gaussian distribution(Normal distribution) is obtained by maximum entropy theorem. Using lagrange mutilplier, Gaussian distribution is easily obtained. However, it's too hard for me. ...
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0answers
14 views

SVM: How to normalize |WX| > 0 into |WX| = 1

Question What are the reason/basis/rationale and the actual steps and design/mechanism behind to do the normalization to convert |WX| > 0 into |WX| = 1 in the process of getting the optimal W for the ...
5
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1answer
966 views

Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
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0answers
5 views

Axis-aligned bound constraints and algebraic optimization

What is the methodology for optimizing a function with a interval bounded constraint? I guess the solution has something to do with KKT conditions and linearizing the constraint, but I'm stuck and I ...
0
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1answer
16 views

Correct formulation of equality and non-negativity constrained non-linear minimization problem

I am trying to minimize a non-linear function with both equality and non-negativity constraints numerically (not analytically) using gradient based methods and without software packages. ...