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

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Find the maximum and minimum of the function $f$

Find the maximum and minimum of $f(x, y)=xy-y+x-1$ at the set $x^2+y^2\leq 2$. I have done the following: Since the region $x^2+y^2\leq 2$ is closed, $f$ has a maximum and a minimum, which is ...
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20 views

Find the extremas of the fuction $f$

I have to find the extremas of $f(x, y)=3x+2y$ subject to $2x^2+3y^2 \leq 3$. Since the region $2x^2+3y^2 \leq 3$ is closed, $f$ has a maximum and a minimum, which is either at the boundary or at ...
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1answer
40 views

Theorem of Lagrange multipliers - Extremas of $f$

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 ...
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1answer
20 views

Help with Lagrange multipliers on an intresting function

Hi guys I am trying to do Lagrange multipliers to figure out $\lambda$ $$F=a \log(x^2-y)+b\log(x^3-z)-\lambda (x^2-y+x^3-z -1)$$ Where a and b are constants and we have the constraint $x^2-y+x^3-z ...
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2answers
22 views

Lagrange Multiplier Method On Linear Equation Set

I am trying to perform a Lagrange constraint problem for a simple set of linear equations (I realize this can be solved by substitution) but I'm curious why/how the Lagrange method is failing and I'm ...
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1answer
33 views

Lagrange primal and duality properties [on hold]

max Lprimal(x,lamb,mu) >= f(x*) + lamb g(x*) + mu h(x*) s.t. lamb >= 0 min Ldual(lamb,mu) = min max Lprimal(x,lamb,mu) What is the purpose and properties of the lagrange dual function, why do I need ...
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34 views
+100

Change of Lagrange Multipliers in terms of constraint

I have the following issue in solving an economics problem. Obviously I am aware that the question could be completely idiotic or on the other hand completely obvious. But I am having a brain block. ...
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1answer
10 views

Minimizing constrained functions on $l^p$

Suppose we have some functionals $H,G:l^p(\mathbb{N}^+)\to\mathbb{R}$, and we want to find some $p \in l^p(\mathbb{N^+})$ which minimize $H$, subject to the constraint that $G(p)=0$ is constant. As ...
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1answer
23 views

What's the detailed derivation from equation 126 to 127 (See Figure)? (problem of Lagrange method with vector transpose)?

I am confused of vector transpose, I do not know how to get equation 127. Thanks.
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1answer
81 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
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1answer
38 views

How can we continue to get the critical points?

A service requires the dimensions of a rectangle box are such that the length plus twice the width plus twice the height do not exceed $274cm$ ($l+2w+2h \leq 274$). What is the maximum volume of the ...
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4answers
46 views

Which function do we want to minimize?

A ray of light travels from the point $A$ to the point $B$ across the border between two materials. At the first material the speed is $v_1$ and at the second it is $v_2$. Show that the journey is ...
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1answer
37 views

Do we have to use the Lagrange multipliers method? [closed]

Draw a cylindrical container (with a lid), so as to contain $1$ liter of water, using a minimal amount of metal. Could you give me some hints how we could do that?? Do we have to use the Lagrange ...
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1answer
27 views

Lagrange multipliers method - absolute maximum and minimum

Using the Lagrange multipliers method I have to find the absolute maximum and minimum value of $f(x, y)=x^2+y^2-x-y+1$ in the unit disc. So, I have to find the extremas of $f(x, y)=x^2+y^2-x-y+1$ ...
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2answers
41 views

prove using Lagrange multipliers that for $x,y>0,\space n\in \mathbb N,\space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2 $

I have been asked to prove using Lagrange multipliers that for \begin{equation*} \space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2,~x,y>0,~n\in \mathbb {N} \end{equation*} I am familiar with the ...
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1answer
23 views

Applying the theorem of Lagrange multipliers

I have to fund the extremas of $f$ subject to the contraints, that are given: $$fx, y)=x-y, x^2-y^2=2$$ I have done the following: We use the theorem of Lagrange multipliers. The constraint is ...
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0answers
23 views

A min-max problem and convex optimization problem.

Let $x^*$ a solution of the convex programming problem $$ \begin{array}{rl} \max & f_0(x)&\\ \mbox{s.t.} & g(x)\leq 0 \end{array} $$ where $f_0:\mathbb{R}^n\to \mathbb{R}$ and the ...
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1answer
35 views

How can I go about solving this group of equations in as simple a way as possible?

They arise from partial derivatives of the Lagrange multiplier function. Here below is the original problem: Goal function: $$f(x,y,z)=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} $$ with ...
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1answer
62 views

Is there any way to make the following function convex?

I need to find optimal lagrangian multiplier vectors for a quadratic programming problem subject to three quadratic equality constraints and several other linear inequality constraints. I would like ...
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1answer
29 views

Potential energy of a hanging string of a prescribed length

Consider a homogeneous, flexible string of a prescribed length hanging in a vertical plane where its ends are fixed at two points P and Q. Determine the equilibrium configuration of the string by ...
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0answers
22 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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26 views

Karush-Kuhn-Tucker conditions for non-linear optimalization

I have the following problem: solve the local conditions (KKT) and find ALL optimal solutions: $$\min f(x,y)$$ subject to $$g(x,y)\le 0$$ $$x\geq0, y\in\mathbb{R}$$ I have some questions to this ...
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2answers
26 views

optimization on two “max” function

Anyone knows how to use lagrange multiplier (or KKT conditions) to minimize an objective function such as $L(\beta,\beta_0)=\sum_{i=1}^n[a_i(1-y_if(x_i))_++b_i(1+y_if(x_i))_+$] where $a_i$, $b_i$ ...
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2answers
37 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
21 views

Lagrange Multiplier with Inequality Constraints

To maximize $f(x, y)$ subject to $g(x, y) \le b$, we define a Lagrangian $$L(x, y, λ) = f(x, y)−λg(x, y).$$ Then the conditions are: $$Lx = 0,\ Ly = 0,\ \lambda(g(x, y) − b) = 0,\ g(x, y) \le b$$ ...
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1answer
13 views

Under what hypotheses is a solution to the Lagrangian multiplier equations automatically a global minimum?

Suppose we are minimizing a function $f(x_1,...,x_n)$ under the conditions $g_1(x_1,...,x_n) = g_2(x_1,...,x_n) = 0$. Under what hypotheses is a solution to the Lagrangian multiplier equations ...
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Find an example of critical point

Find an example have the following property: Let $\Omega $ be open in $\mathbb{R}^{n}$, $f, g : \Omega \rightarrow \mathbb {R}$ be $\mathcal{C}^{1}(\Omega)$ and $S=\begin{Bmatrix} ...
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1answer
33 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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1answer
25 views

How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to ...
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1answer
21 views

Find a critical point satisfied the Lagrange condition is not local extremum

We know that Lagrange Multiplier gives necessary conditions for an extremum.It locates all possible condidates.But not all such points need be extrma. I want to find an example of the point is ...
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2answers
42 views

Simple Lagrange Multiplyers Problem

Can anyone please help me with the following: Find the stationary values of $u=x^2+y^2$ subject to the constraint $t(x,y) = 4x^2 + 5xy + 3y^2 = 9$. The answer is given as $u = 9$ and $x = \pm ...
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1answer
24 views

How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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3answers
110 views

How to create very hard problems on Lagrange Multipliers

This is a rather odd request. I only recently started studying the Lagrange Multipliers, and was given a task to create some challenging (as much as possible) problems on them and also provide ...
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3answers
43 views

ADMM formalization

I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and ...
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1answer
21 views

Prove that the line joining x and y is perpendicular to the surface at x

Let $M_m$ be a $C^1$ surface in $R^n$, and let $y$ be a point in $R^n$ not on $M_m$. If $x$ is a point on $M_m$ that minimizes or maximizes the distance to $y$, how would one prove that the line ...
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1answer
8 views

Restricted Stationary point subject to two constraints

While trying to work on a Lagrange multiplier problem, I encountered a system of linear equations that I'm not really able to solve. I don't know where to start. I've already found that $x=y$ or ...
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1answer
20 views

Question on Lagrange multipliers , don't understand the solutions. An error in the solution maybe?

So I've been trying to solve the question above. The thing I don't get is that why is F(t,u,λ) = h(t,u) + λg(t,u)? Shouldn't it be -λg(t,u) instead?
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0answers
36 views

Lagrange multiplier method

Question 1: Could somebody please refer me to an introduction to Lagrange multipliers which is easy to read but still in full generality? Question 2: I am interested in particular in the following ...
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1answer
18 views

Can one optimize a function with min(x,y) as a constraint?

Let $$h(x,y) = \min\lbrace x,y \rbrace$$ I want to find $$\min ax + by$$ Subject to $$g(x,y) = c - h(x,y)= 0$$ My lagrangian is $$L(x,y,\lambda) = ax+by + \lambda (c - h(x,y)) $$ I have ...
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1answer
28 views

Lagrange Multipliers for farthest distance

I am trying to find the farthest point from the origin to a point on the circle $$(x-2)^2+y^2=1$$ I am not great with the formatting on here but this is what I have so far... $$f(x,y)=x^2+y^2 $$ ...
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1answer
21 views

Lagrange optimization method appears to give the wrong answer

This problem comes from Schaum's Outlines "Calculus for Business, Economics, and the Social Sciences". Problem 9.14 (c) states: Maximize $f(x,y,z)=3x^2yz$ subject to x+y+z=32 So, I set up the ...
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1answer
32 views

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 origin. This is ...
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How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
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connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
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1answer
35 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
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1answer
48 views

Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
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2answers
23 views

Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
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1answer
48 views

$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
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0answers
46 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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0answers
41 views

Lagrange Multipliers and optimisation

An open-topped metal water tank with volume $2$ m$^3$ is to be constructed with vertical sides and a right-angled triangle as base. What should be the dimensions of the base to minimise the area of ...