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

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simplify/solve nonlinear equations for constrained least squares problem

I am trying to find a simple, ideally closed form formula for the (not necessarily unique) unit vector $\vec{x}$ minimizing total squared cosine distance from a collection of unit vectors $\vec{v_i}$. ...
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0answers
7 views

How to optimize a system of equality an $\geqslant $ constraints?

In many cases, for example when we work with probably mass functions We may need to solve a system of this form: $$ max f(\vec{p_1})+g(\vec{p_2}) $$ when there are the obvious constraints of : $$ ...
3
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2answers
102 views

Maximize $xy^2$ on the ellipse $x^2+4y^2=4$

I was using Lagrange multiplier, any steps gone wrong? $$f(x,y)=xy^2$$ $$c(x,y)=x^2+4y^2$$ Partial Derivatives $$\frac {\partial f}{\partial x} = y^2 $$ $$\frac {\partial f}{\partial y} = 2xy $$ ...
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0answers
18 views

Help for solving this optimization problem

Are given $2$ square matrices $M_1$ and $M_2$ of dimension $d \times d$ and two points in a $d$-dimensional space $p_1$ and $p_2$ ($d \times 1$). Now I need to find two other square matrices $X$ and ...
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1answer
25 views

Lagrange multiplier problem with two constraints [closed]

Can someone please verify an answer for me for the following Lagrange multiplier problem? Consider the function in three variables defined by $$ f(x,y,z) = xy + yz$$ and subject to the ...
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2answers
28 views

Lagrange's multiplier method find the highest and the lowest point

Plane $x+y+z=12$ intersects the paraboloid $z=x^2+y^2$ find the highest and the lowest point of this cross-section. What should i do here? I need help solely when it comes to transforming this ...
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2answers
29 views

Use the lagrange's multipliers method to find a points on an ellipse

Question: Using the Lagrange's Multipliers method, find the points on the ellipse $x^2+2y^2=1$, that are situated in the longest and shortest distance from the line $x+y=2$. I know how to use ...
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1answer
41 views

How can I find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$
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0answers
47 views

Optimizing over an infinite set of variables

This may be a very basic question, but it's been a while since I did any optimization. Suppose I have a sequence $(x_i)$, $i=1,2,\ldots$ in the $\ell^2$ space and the following optimization problem: ...
2
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3answers
37 views

Finding the distance from the origin to the surface $xy^2 z^4 = 32$ using the method of Lagrange Multipliers

Problem: Find the distance from the origin to the surface $xy^2z^4 = 32$. Attempt: The Lagrange equation for this problem is $L(x,y,z, \lambda) = x^2 + y^2 + z^2 + \lambda (xy^2 z^4 - 32)$. Setting ...
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1answer
70 views

Show that inf $f(x)$ is achieved. Find $\inf f(x)$.

Let $$\Sigma = \{x\in R^3: x_1x_2 +x_1x_3 +x_2x_3=1 \}$$ and $$f(x) = x_1^2 + x_2^2 + \frac{9}{2} x_3^2$$ a) Show that $\Sigma$ is a smooth surface in $R^3$. b) Show that $\inf_{x\in\Sigma}$ f(x) ...
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1answer
28 views

Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...
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3answers
163 views

Lagrange Multipliers Example

Minimize $$f(x,y) = x^2+y^2$$ subject to the constraint $xy=3$. I know the formula for Lagrange multipliers to be $\nabla f = \lambda \nabla g$ so we get a system of equations like this ...
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0answers
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Solving linear constrained optimization problem

So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on: Minimize the following expression (power dissipation): $$I_{B1}(V - C_1) ...
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0answers
19 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 : $$\int_{0}^{1}\left(y'^2-xy'-z'^2\right)dx=2$$ ...
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1answer
35 views

Using Lagrange Multipliers Better?

My question is that "Can we use lagrange multipliers to solve any problem where we need to find local/global minima/maxima?" Also, "Is it much easier to use lagrange multipliers, and if so what cases ...
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1answer
58 views

Lagrange Multiplier Method: Why is the Langragian function defined as $f(x,y)+\lambda \cdot g(x,y)$?

Edit: As AlexR points out in this comment, there is no mathematical reason behind defining the Lagrangian, except because it makes the Lagrange Multiplier Method easier to memorize. I find this ...
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0answers
41 views

May be a trivial question regarding constrained optimization

Optimization problem is to find $x$>0 which $min \ \ L=\frac{A\left ( B(\frac{C}{Cx-B}+\frac{1}{x})+2C\log(\frac{B}{x}-C) \right )}{B^3}$ $s.t \ \ x\leq K $ Rewriting the objective ...
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1answer
44 views

How to check if three dimensional surface has endpoints

A big part of my calculus class is using Lagrange multipliers to find max/min values of a given function subject to some constraint. One thing I'm struggling with however, is that the endpoints of ...
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2answers
41 views

Proving inequality using Lagrange multipliers, somehow?

While going over assignments preparing for an upcoming exam, I noticed the question Prove that $x^{4} + y^{4} - 4b^{2}xy \geq -2b^{4} \text{ }\forall\text{ } x,y \in \mathbb{R}$ I had used ...
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2answers
28 views

Lagrange Multipliers: Rectangular Box

Problem Glass costs twice as much as plywood, per square meter. Use Lagrange multipliers to answer: What is the shape of the cheapest rectangular box, with 5 rectangular plywood sides and 1 ...
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0answers
23 views

How to find out if point is local Maximizer or local Minimizer ? Lagrangian is given

The Lagrangian is: $L(x,\lambda) = x_1x_2-2x_1-\lambda (x_1^2-x_2^2)$ Taking the derivatives and setting it equal to zero gives: $x_2-2\lambda x_1-2=0$ $x_1+2\lambda x_2=0$ $x_1^2-x_2^2=0$ The ...
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1answer
16 views

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
37 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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1answer
19 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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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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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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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
42 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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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
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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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3answers
46 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 ...
0
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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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1answer
92 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
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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
36 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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1answer
44 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
34 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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0answers
19 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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0answers
50 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
24 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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4answers
90 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
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
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
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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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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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
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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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2answers
41 views

Lagrange min max

Find the max/min of $ \ f(x,y) \ = \ x^2+y^2-12x+16y \ $ . Is my proof completely right? Solution: By Lagrange for $x^2+y^2=25$ we have $$ f_x \ = \ 2x-12 \ = \ \lambda \ \cdot \ 2x \ \ , \ \ f_y \ ...