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

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1answer
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How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just ...
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2answers
50 views

Solve this set of Lagrange multiplier equations,

I'm trying to solve $$(yz,xz, xy) = (\lambda\frac{2x}{a^2},\lambda\frac{2y}{b^2},\lambda\frac{2z}{c^2})$$ with the constraint equation $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ ...
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0answers
34 views

Calculus of variations of multiple integrals and multiple constrains

Let $q(x_1,\dots, x_6)$ be a arbitrary multivariate density that we choose beforehand. What I want is to calculate a new multivariate density $p(x_1, \dots, x_6)$, which is obtained by minimizing the ...
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5answers
76 views

Optimize over unit circle to prove $|ax + by| \le \sqrt{a^2 + b^2}$

I have the following problem which, straight off the shelf, seems totally approachable. It's been giving me difficulty however: Let $a,b,x,y \in \mathbb{R}$, and suppose that $x^2 + y^2 =1$. ...
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1answer
12 views

Why is the lagrange dual function concave?

In a book I'm reading it says I'm struggling to understand the last sentence. Why can one conclude concavity from having a pointwise infimum of a family of affine functions?
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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 ...
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2answers
29 views

Using Lagrange's Method in Finding Extreme Values (New to This Method)

Did I do this hw question correctly (at least in theory, I do not expect anyone to check my algebra work)? In particular, did I solve for lambda and plug lambda back into my equations for x,y, and z ...
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1answer
76 views

Lagrange Multipliers with two constraints

The problem is to find the maximum value of $ \ f(x,y,z) \ = \ x+y+z \ $ subject to the two constraints $ \ g(x,y,z) \ = \ x^2+y^2+z^2 \ = \ 9 \ $ and $ \ h(x,y,z) \ = \ ...
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3answers
41 views

Lagrange Question

Alice has only $24$ hours to study for an exam, and without preparation she will get $200$ points out of $1000$ points on the exam. It is estimated that her exam score will improve by $x(50−x)$ points ...
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1answer
26 views

LaGrange question 2

You are building a barn, with no floor, in the shape of a rectangular box with a square base. The roof material costs $\$19/\text{m}^2$, the sides and back material costs $\$13/\text{m}^2$, and the ...
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3answers
43 views

Shortest distance from a point

Find the shortest distance from the point $(0,b)$ to the parabola $y=x^2+8$. Express your answer in terms of $b$. (Comment: If $b \le \frac{33}{4}$ then the answer is just $|b|$, so assume that $b ...
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2answers
28 views

Lagrange Maxima

I am sorry to post this again, but I am still confused. Alice has only $24$ hours to study for an exam, and without preparation she will get $200$ points out of $1000$ points on the exam. It is ...
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0answers
48 views

On lagrange multipliers and constrained optimisation

I have a real-valued function of two real-valued variables $f(x,y)$. The global maxima is sough by solving the system $$ f_x(x,y)=\frac{\partial f(x,y)}{\partial x}=0\\ f_y(x,y)=\frac{\partial ...
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0answers
31 views

How do I solve a under-determined quadratic multi-variate system?

I have the following equation: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} ...
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1answer
17 views

Use lagrange multultipliers to find the indicated extrema

maximize $f(x,y,z)=x+y+z$ subject to $x^2+y^2+z^2=1$ I do not understand this at all or where to go from here would appreciate some insight
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3answers
54 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. $\nabla f = \lambda \nabla g$ ...
4
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1answer
86 views

Is this a known result?

I heard the following result and I am wondering if anyone can verify its correctness and also provide a source to cite. If the Lagrangian $L(x,\lambda)$ is convex in $x$ at the optimal Lagrange ...
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0answers
11 views

How can we constrain lagrange multipliers in svm dual by adding constraints in primal problem?

Consider svm-dual,i.e., \begin{align} &\text{maximize} \sum_{i=1}^n \alpha_i-\frac{1}{2\lambda} \sum_{i,j=1}^n \alpha_i \alpha_j y_i y_j K(x_i,x_j)\cr &\text{subject to, } 0\leq \alpha_i ...
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2answers
35 views

Lagrange multiplier understanding problem

I do have a problem with the lagrange mutiplier method. I understand how it works for something like: maximize $f(x,y)$ subject to $g(x,y)=c$. But how do I handle something like: Maximize f(x,y) ...
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3answers
98 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
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1answer
40 views

An optimization problem, in the form of a word problem,

The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the ticket price is $50$ dollars. A market survey indicates that $10$ additional seats will remain ...
2
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1answer
32 views

Lagrange minimisation

I'm really struggeling with this obviously easy Lagrange question. Been at it for two days and don't really get the hang of where to start.. can someone pointme in the right direction? A consortium, ...
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4answers
87 views

Why does taking derivatives of $L$ in Lagrangian multiplier problems let me find solutions to optimizations problems?

Consider the problem Maximize $f(\mathbf{x})$ subject to $g(\mathbf{x})=c$ Using the method of Lagrangian multpliers, I would set up a Lagrangian like $$L = f(\mathbf{x})-\lambda ...
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0answers
18 views

Existence of lagrange multipliers with polyhedral constraints

I am working with a paper (Exact regularization of polyhedral norms, Schöpfer 2012) which states as a well-known fact that, if $f$ is a polyhedral norm, then for some $\mu^* > 0$ \begin{equation} ...
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2answers
42 views

Lagrange Multipliers-higher dimensional

I'm studying for an exam and am trying to work out this example. Use Lagrange Multipliers to find the maximum value of $(xv-yu)^2$ subject to the constraints $x^2+y^2=a^2$ and $u^2+v^2=b^2$. My ...
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1answer
31 views

A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
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0answers
37 views

Shannon Entropy Maximization with Constraints

I have got a cumulative distribution function $F_X(x)=Pr(X<=x)$. This distribution is described by 2 parameters $\alpha, \beta$. We define $F_k$ as follows: $\forall k<=n_k, ...
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1answer
22 views

Lagrange multipliers method question.

On the wikipedia page and indeed in my own opinion the method of Lagrange multipliers as applied to an equality constraint function is as follows, Extremise $f(x,y)$ subject to $\phi=c$ for some ...
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0answers
30 views

optimization of a function with inequality constraint

I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms) $$L_P = ...
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1answer
51 views

From utility function (3 products) to demand function (2 products)

I am struggling with this exercise and would appreciate some help. Consider two goods and a representative consumer whose utility is given by: $U(q_{0}, q_{1}, q_{2})= ...
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1answer
36 views

Distance between two points using Lagrange [closed]

Set up the system of equations required according to Lagrange in order to minimize the square of the distance between P1 and P2. $${M_{1}=‎\lbrace(x,y)\in R^2} \mid x^2+\frac{9}{4}(y-1)^2=9‎\rbrace$$ ...
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1answer
36 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
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1answer
20 views

Lagrange multiplier derivative condition

For the Lagrangian $\mathcal{L}(\mathbf{x}, \lambda) = f(\mathbf{x}) + \lambda g(\mathbf{x})$, I read that $\partial\mathcal{L}/\partial\lambda$ must equal $0$. Could someone please explain why?
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4answers
86 views

Minimization on compact region

I need to solve the minimization problem $$\begin{matrix} \min & x^2 + 2y^2 + 3z^2 \\ subject\;to & x^2 + y^2 + z^2 =1\\ \; & x+y+z=0 \end{matrix}$$ I was trying to verify the first ...
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0answers
48 views

Confusion over Lagrangian function

When forming the Lagrangian of an optimization problem, why don't we include all constraints in the Lagrangian? For example, the optimization problem $$ \begin{align} \max_{x}&\quad f(x)\\ ...
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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 ...
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2answers
60 views

Calculus,Lagrange Multipliers

Can anybody help me with the following question, Use the method of Lagrange multipliers to find $\max$ and $\min$ values for $f(x,y,z)=x^2+y^2+z^2$ subject to the constraint $4x^2 + y^2 + 9z^2=36$. ...
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0answers
25 views

Lagrange multipliers, once I use the constraint equation, do I have to worry about it again later?

I am solving $ grad [f(x,y,z)]$ = $\lambda$grad[g(x,y,z)] I have then three equations, one involving x's and lambdas, another involving y's and lambdas and a third involving z's and lambdas. I then ...
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1answer
22 views

For given mean $\mu$ of random variable X in [0,1], what is the probability distribution function $p(X)$ that makes $VAR(X)$ maximum?

Given the conditions $\int_{0}^{1} p(x)dx=1$, $\int_{0}^{1} xp(x)dx=\mu$ and $p(x)\ge0$ for $\forall x \in [0,1]$, What probability distribution function $p(x)$ makes $Var(X)$=$\int_{0}^{1} ...
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2answers
61 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
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1answer
82 views

Optimization: KKT conditions statement

I'm currently following this material Optimization Theory: Chapter 2 Theory of Constrained Optimization And I can't understand why the following statement is true, between the equations (2.9) and ...
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1answer
64 views

To find the Maximum and minimum value of f over square

Given function $f = (x+y)^2 - (x+y) +1$ .I have to find maximum and value of $f$ over square with unit side in first octant in xy-plane. I calculated $f_x $ and $f_y $ both came out to be ...
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2answers
28 views

Optimization problem: smallest euclidean distance with positive entries constraints

Suppose there is the simple function: \begin{align} f(x,y,z) &= (x-a)^2 + (y-b)^2 + (z-c)^2 + (x+y-S-z - d)^2 \end{align} where $a,b,c,d$ are nonnegative constants, and $S$ is an integer. I ...
2
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0answers
25 views

How to use find the Lagrange Multipliers in multidimensional Calculus of Variations

Suppose I wish to minimise the integral $$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$ Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie $$F ...
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1answer
45 views

Lagrange multipliers problem with two constraints

Hi guys I am working with the following polynomial and I am trying to find the $\lambda , \mu$. I have a polynomial and I am trying to do Lagrange multipliers. Here is what I have. $f(x,y,z)= a ...
0
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1answer
74 views

Lagrange multipliers problem

I have a two variables function: $f(x,y)=3x+y$ and I wish to find its minimum and maximum values with the constraint $\sqrt{x} +\sqrt{y} =4$. According to the answer, there is a minimum and a maximum. ...
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4answers
38 views

Closest point to $(2,0)$ on with a hyperbola as a constraint

I'm looking to find a point on the hyperbola $y^{2}-x^{2}=4$ which is closest to $(2,0)$. As far as I know I need to find the distance formula and use lagrange multipliers.
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1answer
46 views

Finding min and max under constraints

I have a two variable function: $f(x,y)=4x^2-y^2-xy-2x+6y$. I need to find its absolute minimum and maximum under the constraints: $y=4-2x$, $x \geq 0$, and $y \geq-2$. I am not sure how to do it, ...
0
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1answer
49 views

Gradient of a Lagrange dual function

Consider: $$\min_{x \in \mathbb{R}^n} f(x)$$ $$\ \ \ \ \ \ \ \text{s.t. }\ h(x) \leq 0$$ Lagrangian:$\ \ \ L(x,\lambda) = f(x) + \lambda h(x)$ Suppose $x^* = \arg\min_{x} L(x,\lambda)$ ...
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0answers
20 views

Taking the partial derivative of a Lagrangian with square roots

I have a nasty function that is simplified from an even nastier function: $F(x,y,z,\lambda) = \frac{-0.0129x-0.0051y-0.0066z}{\sqrt{.44^2x^2 + .15^2y^2 + .44^2z^2 + 0.05808xy + .139392xz - ...