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

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Minimum of $x+y+z$ on $\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}$

Find the minimum of $x+y+z$ on $$\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}.$$ My first naive thoughts would be to consider setting up a nasty triple integral and evaluate it or ...
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Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
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30 views

Finding maximum value of a 3-variable function using inequality.

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
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28 views

Could someone explain the Lagrangian Method?

I understand the method, technically, but what is actually going on? We set the gradient of the function equal to the gradient of the constraint (multiplied by a constant), and in doing so, we find ...
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41 views

What is wrong with this parametrization?

I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression: $L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$, ...
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14 views

Lagrange identity

I'm aiming to prove that there is a vector $$v$$ equal to ∇f divided by its normal - ∇g divided by its normal. I've being trying to solve it from the point of view of Lagrange multipliers. The ...
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1answer
23 views

Geometric interpretation of duality and Slater's condition

I am trying to study about optimization problems, Lagrange duality and related topics. I came across some presentation on the net, which claims to show the geometric interpretation of the duality and ...
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18 views

What's wrong with the following trace optimization?

I'm reading a paper that has used the augmented Lagrange function for optimization. I've tried to derive one subproblem but got a different answer from that in the paper. Could you help check it ...
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32 views

Fun Lagrange multiplier problem?

Do any of you have a fun or interesting Lagrange multiplier problem that would be suitable for undergraduate calculus students? I'm planning on working through a standard Lagrange multiplier problem ...
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1answer
14 views

How to tell of extrema lies on the boundary or interior of a function? (Lagrange Multiplier)

For example: Q: Find the extreme values of f(x,y,z) = x + yz on the solid ellipsoid x^2+2y^2+8z^2 <= 32. The solution manual does: " f_x = 1 not equal 0, f has no critical points. -> all ...
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29 views

Lagrange multipliers: More than one constraint

I have more or less understood the underlying theory of the Lagrange multiplier method (by using the Implicit Function Theorem). Now, I try to extend this understanding to the general case, where we ...
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20 views

Lagranges polynomial for jth power of x

Let $l_i(x)$ be the $i$th ’Lagrange’ polynomial corresponding to $x_0, x_1, \dots, x_n$, i.e. $$l_i(x) = \frac{\left[\pi (x - x_j)\right]}{\left[\pi (x_i-x_j)\right]}$$ Show that for $0 \leq j \leq ...
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1answer
31 views

Finite number of critical points on a sphere of a polynomial?

Let $f$ be a polynomial in $x_1,\dots,x_n$. Suppose, I want to find all the critical points of $f$ on the sphere $\left\{ x\in \mathbb{R}^n \colon x_1^2+\dots + x_n^2=1\right\}$. Are there any obvious ...
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22 views

Stuck with an optimization problem with 2 constraints (Lagrangian multiplier method)

I am really stuck with a certain minimization task. I thought I would understand the Lagrangian multiplier method (at least I could solve simple 2-variable optimization problems with 1 constraint). ...
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17 views

Dimensions of rectangular parallelopiped of maximum volume…

i Need to calculate volume of parallelopiped of maximum volume with edges parallel to the coordinate axes that can be incribed in a ellipseoid $(x/a)^{2} + (y/b)^{2} + (z/c)^{2} =1$ . Apparently ...
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37 views

Lagrange Multiplier Question and my attempt

Question is Find the extrema of $xyz$ when $x+y+z=a$ , a>0. Strating with usual Lagrange Multiplier method i get $f_x$ = $yz$ +$\lambda$ =0 $f_y$ = $xz$ +$\lambda$ ...
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1answer
30 views

Finding Triangle with constant perimeter and largest area (Lagrange Multiplier)

Question is to find Finding Triangle with constant perimeter and largest area by method of lagrange multiplier . What i have done is that i have firstly taken $x+y+z=2k$ , where x,y,z are sides of ...
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2answers
39 views

Find Rectangle of Constant Perimeter whose diagonal is maximum (My attempt with Lagrange Multipliers)

Question is to Find Rectangle of Constant Perimeter whose diagonal is maximum (My attempt with Lagrange Multipliers) . I took rectangle with sides $x$ and $y$ . Since Perimeter is constant so i ...
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6 views

Difference between Relative And Absolute Extrema and concept of convex regions

Can anybody explain to me difference between relative and absolute extrema ? Secondly somebody have told me on stackexchange that for convex regions(triangles,rectangles) extrema occurs on corners ...
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2answers
54 views

Can Anyone help me with Lagrange multiplier problem

I need to find absolute maximum and minimum of thi function $$F(x,y) = x^{2} - y^{2} - 2y$$ over $$R = \{ (x,y)\ |\ x^{2} + {y^2} \leq 1\} $$ Thanks for help
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Rectangle surmounted by an isosceles triangle

A window has the shape of a rectangle surmounted by an isosceles triangle. Determine the dimensions of the window, if its if its perimeter is to be at most $M$ and its area is to be maximized. I have ...
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1answer
20 views

Maximum and minimum of function in a curve

Find the points of maximum and minimum of the function $$f(x,y,z) = 2x + y - z^2$$ in the compact space $$C = \{(x,y,z) \in \mathbb{R}^3 : 4x^2 + y^2 -z^2 = -1,z\ge 0, 2z \le 2x + y + 4\}$$ So, I ...
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Lagrange multipliers question and my attempt

Question is to minimise the $f(x,y)$ = $3x^{2} + y^{2} - x $$$$$ and constraint is given by $2x^{2} + y^{2} =1 $ Question is simple and ii have got most of points but i seem to miss few points ...
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39 views

How to use Lagrange Multiplier in this question?

I have to find absolute maximum and minimum values of $f(x,y)$ = $4x^{2} + 9y^{2} -8x - 12y + 4 $ over rectangle in first quadrant bounded by lines $x=2 , y=3$ and coordinate axes I have checked ...
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Proving Lagrange method by using Implicit Function Theorem

I am trying to show the proof of the Lagrange multiplier method. According to this in general, if $f$ and $g$ are $D+1$ dimensional functions such that $f,g : \mathbb{R}^{D+1} \mapsto \mathbb{R}$, and ...
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Showing that the gradient $\nabla f(x)$ is parallel to constraint surface gradient $\nabla g(x)$ at an extreme point on the surface

Let $f(x)$ a function $f:\mathbb{R}^D \mapsto \mathbb{R}$. $g(x)$ is another $D$ dimensional function. Then we have a constraint equation $g(x)=0$. Now, we have a local, constrained extreme point $x'$ ...
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2answers
30 views

Trying to understand Lagrange multipliers

I am trying to understand how the Lagrange multipliers method work for constrained optimization. Let's assume that we have a function $f(x)$ which is $f:\mathbb{R}^D \mapsto \mathbb{R}$. Now we have ...
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45 views

Find the minimum and maximum distances between the ellipse $x^2 + xy + 2y^2 = 1$ and the origin.

I know that I'm trying to maximize/minimize $f(x,y)=x^2+y^2$ with the constraint $g(x,y)=x^2+xy+2y^2-1=0$ Here are the partial derivates: $f_x=2x \qquad$ $f_y=2y \qquad$ $g_x=(2x+y)\lambda \qquad$ ...
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Example of a point (x*) on a hyper surface where tangent plane cannot be expressed in terms of gradients

This question is related to the discussion of regular points in the context of constrained optimization. In the textbook that I am reading, "Linear and Non Linear Programming by David Luenberger", it ...
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1answer
21 views

Solve system of linear equations within a Lagrange multiplier problem

Find points on the ellipse $2x^2-4xy+5y^2=54$ closest to origin using Lagrange multipliers. It's a past paper question. I let $$f=x^2+y^2$$ and the system of equations I got is $$2x=\lambda ...
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1answer
59 views

Doubt on a paragraph regarding Lagrange's multiplier.

I've a topic in my notes "The method of Lagrange's multipliers" which is described as follows: Let $U$ be an open set in $\mathbb R^n$.Let $f\in C^1(U,\mathbb R)$ and let ...
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Constraint Optimization

I have a sequence of iids defined by: $f(x|\theta) = \exp(-(x-\theta))\;\;\;\; \theta<x<\infty$ To find the maximum likelihood estimate, i should maximize the log likelihood with respect to ...
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19 views

Lagrange multiplier and implicit function theorem

I am looking for the proof of the method of lagrange multipliers using the Implicit function theorem. Does anyone know any book that supplies its proof? I have seen the proof in Shifrin's book, but it ...
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Gradient of the function and the contour line

I do not understand, reading the chapter in the book about Lagrange multipliers, why the gradient of the function $f$ is perpendicular to the contour line? There is no sufficient explanation there, ...
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1answer
34 views

How many samples of $y$ and $x$ given variances?

On a homework problem, I am given two variables, $x$ and $y$, with variances $4$ and $16$, respectively. The question is how many observations should I draw of $y$ in order to estimate the difference ...
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44 views

Question related to Lagrange multipliers

I am stuck with the following problem: A is symmetric $n\times n$ matrix and $f(x)=(Ax)x$ for $x\in {\bf R}^n$. I need to show that the maximum and the minimum values of $f$ on the unit sphere ${x: ...
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24 views

Solving Lagrange equation systems?

Given an equation system when using Lagrange multipliers to find maxima and minima, how does one solve it will all these variables that I cannot isolate because I don't know if they are 0 or not, so I ...
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1answer
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lagrange multipliers, find the maximum and minimum values

Can someone can explain it in details to me? I really don't get it. In Q (a), I got the maximum values is 1, and minimum values is -1. but for Q(b), is x & y both equal to +- sqrt(13/2) ? It's ...
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1answer
35 views

maximization using Lagrange

I am maximizing $f(x,y)=-x$ given the constraint $g(x,y)=x^2-y^2=0$ To satisfy the non degenerate constraint qualification I have: $Dg(x,y)= [2x\quad-2y]$ and the set of $(x,y)$ that satisfy it ...
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Non-differentiable variational calculus (Dido's problem)

I wonder what is the alternative to Euler-Lagrange equations when we have non-differentiability issues. I'll give an example: Dido's problem can be stated as: Find the figure bounded by a line ...
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Find maximum hyperplane separating two classes by optimizing the Langrangian function.

I am trying to solve the following problem: I am having difficulty starting. I know this is a constrained optimization problem for support vector machines. I am wondering how the training data is ...
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Bordered Hessian for Kuhn-Tucker

With Lagragian problems, you are often asked to solve for a stationary point and use the bordered Hessian to determine whether it is a maximum or minimum. I have noticed with Karush-Kuhn-Tucker ...
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how to find the maximum and minimum value of the directional derivative using Lagrange Multiplier Method?

I want to prove that the maximum value of $\frac{df}{ds}$ is $\left|\nabla f\right|$. To maximize $\frac{df}{ds}$ given by $\nabla f=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial ...
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3answers
33 views

Lagrange multipliers with inequalities

I have to find the max and min values of $x^2 + y^2 + xy$ bounded by $x^2 + y^2 \le 4$. I know how to do Lagrange multipliers, and have the points 0,0, x=y, -x=y, and -y=x, but I don't know how to ...
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Lagrange Multipliers

The Question: Find the minimum distance between the origin and the surface $x^2y -z^2 +9 = 0$. I've been able to find the critical points when $x =0$ and when x is not equal to zero but lamda is ...
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22 views

Lagrange multiplier, how to show that these two methods gives the same solutions.

I have read about another way of using Lagrange multipliers, but I can not explain why this is the same as I have seen before. I have seen this before: Lets say you want to maximize ...
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1answer
32 views

Lagrange multipliers method with two constraints

Use Lagrange multipliers to find the minimum and maximum values of $y$ when $(x,y,z)$ is constrained to be in the intersection of the plane $x-y+2z=0$ and the ellipsoid $3x^2+2y^2+z^2=4$. I am just ...
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1answer
35 views

Why can we solve eigenvalue problems which are non-convex by Lagrange multiplier methods and get global minima?

while reading the paper "Some Modified Matrix Eigenvalue Problem" by Golub this doubt occurred to me. there he writes that we can minimize $x^TAx$ subject to $x^TBx=1, Cx=0$ As far as I understand ...
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52 views

The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
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Multivariate calculus (Lagrange multiplier)

If we need to use the method of Lagrange multipliers to find extreme values of a function $f(x, y)$ on a triangle-shaped region in $R ^2$ , how many times would we have to run the method? How many ...