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

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1answer
23 views

Distance between two points using Lagrange

Set up the system of equations required according to Lagrange in order to minimize the square of the distance between P1 and P2. $${M_{1}=‎\lbrace(x,y)\in R^2} \mid x^2+\frac{9}{4}(y-1)^2=9‎\rbrace$$ ...
1
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1answer
19 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?
0
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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 ...
2
votes
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 ...
1
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0answers
43 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)\\ ...
0
votes
1answer
39 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 ...
1
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0answers
23 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 ...
0
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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} ...
2
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0answers
22 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 ...
0
votes
2answers
68 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. ...
0
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4answers
36 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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2answers
25 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 ...
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1answer
38 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, ...
1
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0answers
19 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 - ...
0
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0answers
15 views

Lagrangian multiplier vectorial form

When i have this problem: $min f(x)$ $h{_i}(x)=0, i=1,\dots,m$ $g{_j}(x)<=0, j=1,\dots,p$ I can use the Lagrangian multiplier to write function in: $L(x,\lambda,\mu)=f(x)+\sum_{i=1}^{m} ...
1
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1answer
43 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 ...
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0answers
19 views

How to obtain the optimal lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found.?

I am using a blackbox solver to solve the following nonconvex QCQP to global optimality. $$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$ where $Q_0$ is ...
1
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2answers
66 views

Lagrange multiplier optimizing a 3-D ellipse with respect to the origin

I cannot solve this question: The plane $x+y+2z=2$ intersects the paraboloid $z=x^2+y^2$ in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin. My ...
0
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4answers
22 views

extrema with constraints (lagrange?)

I'd like to find the point of $E: 2x+3y+z = 14$ which has the smallest distance to the point of origin (0,0,0). I think I have $ d(x,y,z) = \sqrt{x^2+y^2+z^2}$ with constraint $2x+3y+z = 14$. What ...
2
votes
1answer
17 views

Lagrange Multiplier - equation system

I'm trying to get the extrema of a function $x + y²$ with a constraint $2x² + y² = 1$ using Lagrange multipliers. The Lagrange function is $x + y² + \lambda (2x² + y² - 1)$. I have three partial ...
3
votes
3answers
57 views

find extreme values of $\cos(x)+\cos(y)+\cos(z)$ when $x+y+z=\pi$

How can I find the maximum and minimum of $\cos(x)+\cos(y)+\cos(z)$ if $x,y,z\geq0$ such that they are vertices of a triangle with $x+y+z=\pi$. I don't know how to start, but I feel like the Lagrange ...
0
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1answer
42 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 $x^* = \arg\min_{x} L(x,\lambda)$ ...
2
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0answers
19 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
1
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2answers
35 views

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 ...
0
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0answers
28 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
45 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 ...
1
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1answer
21 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
33 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 ...
0
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0answers
86 views

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. ...
0
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1answer
12 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 ...
0
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1answer
24 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
90 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
42 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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votes
4answers
49 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 ...
-1
votes
1answer
38 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 ...
2
votes
1answer
36 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 $f(x, y)=x^2+y^2-x-y+1$ ...
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 ...
0
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1answer
24 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 ...
1
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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 ...
1
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1answer
76 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
33 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
26 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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0answers
43 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
28 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 $a_i$, $b_i$ ...
0
votes
2answers
41 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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votes
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 ...
0
votes
2answers
56 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 $4x^2 + y^2 + 9z^2=36$. ...
1
vote
1answer
27 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 ...
0
votes
1answer
22 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 ...
1
vote
2answers
43 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 ...