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

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Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
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0answers
36 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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4answers
25 views

Help with Lagrangian Constrained Optimisation

Question: Maximise f (x, y) = x2y, where (x, y) ∈ R2 given the constraint that all (x, y) are points on a circle with radius √3 around origin (0, 0). Solution: f (±√2, 1) = 2 is the maximal value ...
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0answers
14 views

Determine $P_2 = f(0.7)$ when Neville's method is used to approximate $f(0.5)$

Let $f(x) = \ln(x + 1)$. Neville's method is used to approximate $f(0.5)$, giving the following table. $$x_0 = 0 - P_0 = 0$$ $$x_1 = 0.4 - P_1 = 2/8 - P_{0,1} = 3/5$$ $$x_2 = 0.7 - P_{2=?}- ...
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1answer
38 views

a question about Lagrange multiplier with higher order constrain term

If we want to minimize $f(x, y)$ subject to condition $g(x, y) = c$. Then we treat $\lambda$ as a new variable and consider minimizing $$h(x,y,\lambda)=f(x,y)+\lambda (g(x,y)-c)$$ Question: can ...
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1answer
42 views

Matrix optimization

I'm trying to minimize over $U$ the objective $\|X^{\top}UU^{\top}UU^{\top}X\|_F^2 = \text{trace}(X^{\top}UU^{\top}UU^{\top}XX^{\top}UU^{\top}UU^{\top}X)$ subject to $U^{\top}U = I$, where $X \in ...
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1answer
25 views

How to computer the Lagrange multipliers associated with an optimal solution

Suppose I have a solution $x^*\in\mathbb{R}^n$ to the following problem \begin{align*} \text{minimize}_{x}& \sum_{i=1}^n f_i(x)\\ \text{subject to}\quad &g_i(x) = 0\,\,i=1,\ldots,m\\ ...
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2answers
34 views

using lagrange multipliers to fit a curve through a point

So this is part math/ part statistics. I have a set of data I'm fitting a 2nd order curve through using least squares method (matrix form). However, I've been given the requirement to pass the curve ...
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0answers
22 views

What is the meaning of positive lagrange multiplier?

I'm handling a maximization problem with a constraint and there is a sentence "but, the lagrange multiplier is positive". I can't understand why that sentence is needed. Is there any difference ...
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2answers
145 views

Maximizing Area of Triangle in Circle

I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange ...
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3answers
68 views

Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
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2answers
53 views

Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
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2answers
26 views

Tips on resolving a Lagrange Multipliers equation system

I'm having a very hard time resolving the system of equations after using the Lagrange Multipliers optimization method. For instance: The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 ...
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0answers
24 views

Lagrange multipliers (distance)

Find the closest point of the surface $z=xy-1$ to the origin. How would you do that with Lagrange multipliers?
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0answers
26 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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0answers
26 views

Lagrange multipliers: optimizing with a constant

Hi! I am currently working on some Calc3 online homework questions in preparation for an upcoming test. I really wish I could provide work that I have done in an attempt to answer this problem, but ...
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1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
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0answers
31 views

Explain notation?

What is the meaning of this notation $$S = g^{\mathrm{pre}(c)}$$ for some constant $c$ and a set $S$? (Context: This $g$ is the constraint in a Lagrange multipliers problem).
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0answers
33 views

Help getting a closed-form solution to a maximisation problem

I'm working through a maximisation problem that I can't seem to get a closed-form solution to. It may be the case that there is no closed-form solution, but I would like a second opinion, since I've ...
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1answer
22 views

Unable to solve system of equations in Lagrange multiplier problem.

The problem: Find the right triangular prism of given volume and least area if the base is required to be a right triangle. As for parameters of the right triangular prism, $V$ is volume, $A$ is ...
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0answers
12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
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3answers
40 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
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2answers
49 views

Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
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1answer
39 views

extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
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2answers
26 views

In regards to lagrange multipliers, Confusion about derivation.

In my calculus III textbook, the following sentence is causing trouble for me and preventing me from understanding the theory behind Lagrange multipliers. "Since the gradient vector for a given ...
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1answer
60 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
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1answer
42 views

Explain Lagrange multipliers?

I am having serious issues with comprehension of this method. In particular, I don't understand the conditions. Thus far, I think it's something like; Given an objective $f: A \to \mathbb{R}^1$ and ...
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1answer
43 views

Lagrange's Multiplier Method

I need to find the distance between the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ and the line $y = 10 - 2x$ using Lagranges' Multiplier Method. So far I know how to find the minimum distance ...
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1answer
37 views

Use Lagrange Multipliers to determine max and min

Using Lagrange Multipliers, determine the maximum and minimum of the function $f(x,y,z) = x + 2y$ subject to the constraints $x + y + z = 1$ and $y^2 + z^2 = 4$: Justify that the points you have found ...
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0answers
16 views

Lagrangian method for Numerical Optimization

I know of a technique, but I don't know of its name and I don't have any real literature on the technique. On the wikipedia page for Lagrange multipliers a method is provided to convert a Lagrangian ...
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1answer
32 views

Optimization of a convex target function with inequality constraints

I want to solve the following optimization problem: \begin{equation} \begin{split} \text{maximize} &\;\;\; \ln x_1+\ln x_2+\ln x_3+\ln x_4 \\ \text{s.t} &\;\;\; x_4\le4 \\ ...
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1answer
46 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
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0answers
44 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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1answer
41 views

How to interpret Lagrangian function (specifically not Lagrangian multiplier)

I am reading the following tutorial on Lagrangian multipliers (http://www.cs.berkeley.edu/~klein/papers/lagrange-multipliers.pdf). My goal is to gain an intuitive understanding of why the Lagrangian ...
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1answer
90 views

Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by ...
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1answer
76 views

The shortest path connecting three points

I have 3 points X,Y,Z, lets call them buildings. I need to find the shortest amount of path that connects the 3 buildings, these buildings can be in any sort of shape and any distance from each ...
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1answer
49 views

Maximum distance from the origin to the surface

I am having trouble getting the maximum distance from the origin to the surface $$ \frac{x^4}{16} +\frac{y^4}{81} + z^4 = 1 $$ Knowing I have to maximize $x^2 +y^2+ z^2$ and that the constrain ...
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1answer
59 views

Finding the shortest distance between two planes using Lagrange multipliers

A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
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1answer
54 views

First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
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1answer
22 views

Question about vector optimization using Lagrange Multiplier

I try to find the vector $x = (x_1, \cdots, x_n)$ to maximize function $f(x)=f(x_1, \cdots, x_n)$ subject to the constraint $x_1^2 + \cdots +x_n^2 = a$, where $a$ is a positive constant. I use ...
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1answer
44 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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2answers
45 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
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3answers
57 views

Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers

Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers. So I set it up: $$ 1 = 2x\lambda_1 + 2\lambda_2 \\ 1 = -2y\lambda_1 \\ 1 = \lambda_2 $$ Plug ...
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0answers
12 views

Trouble with Largrange Multipliers and expectation.

I am reading the following argument: Maximize $E[Log(X(T))]$ subject to $E[Z(T) X(T)]=x$, where $X(T),Z(T)$ are random variables, $x$ is a constant, and E is expected value (You can read this as ...
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0answers
38 views

KKT Conditions and Convexity

min $x^2 -xy +y^2 -5x+6y$ subject to $1 \leq y$, $y^3 \leq 2x$, and $x \leq 8$ Write out the KKT conditions for this problem. Show that $(x,y) = (4,2)$ is a KKT point, and is therefore a global ...
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2answers
51 views

Lagrange Multiplier inconsistent

max $f(x,y,z)$ subject to $x+y+z=1$ $$f(x,y,z)= - (x+0.5y)\log(x+0.5y) -2(0.25y+0.5z)\log(0.25y+0.5z)-\log(2)(1.5y+z)$$ $$\frac{\partial F}{\partial x} = - (1+\log(x+0.5y))$$ $$\frac{\partial ...
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1answer
18 views

Substitute for constraints in constrained optimisation problem

I have to solve the following optimisation problem $$\max_{c,\ell, L} \{\ln(c) + \ln(\ell)\}$$ subject to the constraints: $c=Lw$ and $1=\ell +L$. Is it okay to solve the problem ...
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1answer
34 views

Finding critical points using Lagrange multipliers

So I got a bit stuck with the technicalities of this exercise. Let $f(x,y,z)=\sqrt {x^2+y^2+z^2}$. Find $f$ minimal points under the constraints $x^2+y^2=1$ and $x^2-xy+y^2-z^2=1$ using Lagrange ...
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2answers
30 views

Does Lagrange multiplier have solution if functions doesn't intersect

I am trying to get intuition behind Lagrange multiplier and question that bothers me is: Does Lagrange multiplier have solution if two functions(main function and constraint) doesn't intersect. Thank ...
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1answer
30 views

Lagrange multipliers…what is my constraint?

How would I use Lagrange multipliers to determine which point on the surface $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ with $x,y,z>0$ is closest to the origin? I'm not sure what the constraint would ...