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

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To find the Maximum and minimum value of f over square

Given function $f = (x+y)^2 - (x+y) +1$ .I have to find maximum and value of $f$ over square with unit side in first octant in xy-plane. I calculated $f_x $ and $f_y $ both came out to be ...
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1answer
24 views

Minimize squared distance to origin from a paraboloid

I have to use Lagrange multilpiers to find the minimum distance from the paraboloid with equation $z = \left({x-1/}{\sqrt{2}}\right)^2 + \left({y-1/}{\sqrt{2}}\right)^2$ to the origin, and from this ...
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32 views

Create a fourth order polynomial function f(x,y) with at least two distinct terms

I will be computing the gradient, finding the critical points, and use Lagrange multipliers to either maximize or minimize the function. Any suggestions?
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27 views

How does this optimization problem satisfy Karush-Kuhn-Tucker Conditions?

I am following Andrew Ng's course notes on Support Vector Machines at: http://cs229.stanford.edu/notes/cs229-notes3.pdf There is something in these notes which I do not understand. SVM's basic ...
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30 views

Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
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1answer
15 views

how to setup the Lagrange multiplier for contraints such as $0\leq x \leq 1$

I need to find minimum of objective function $Q=f(x,y,z)$ inside a cube region. Hence the constraints are in the form $0 \leq x \leq 1$ and $0 \leq y \leq 1$ and $0 \leq z \leq 1$. Should the ...
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15 views

Minimization of a weighted least-squares problem by Lagrange multiplier method

Problem: Let $Y = (y_1, y_2, \dots, y_m) \in \mathbb{R}^{m \times n}$ and $k \in \mathbb{R}^{m}$ satisfy $\sum_{i=1}^{m} k_i =1$ and $k \geq 0$. Show that $x=Yk$ is a minimizer for $h(x) = ...
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1answer
23 views

Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
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23 views

Multivariable functions application

We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought. Two squares, both with length $l=1$ intersect ...
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1answer
40 views

Solve the system of equations for x and y

Solve the system of equations for $x$ and $y$: $$ \left(\frac{x}{8-2y}\right)^2 - \left(\frac{y}{-4}\right)^2=4 \\ \frac{x}{8-2y} + \frac{x}{-2}=1 $$ I used Lagrange multipliers with multiple ...
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3answers
67 views

Lagrange Multipliers: Find $\min$ of $f(x,y)=3(x+1) +2(y-1)$ subject to the constraint $x^2+y^2=4$

Find the minumum value of the function $f(x,y)=3(x+1) +2(y-1)$, subject to the constraint that $x^2+y^2=4$. The problem states to use Lagrange Multipliers. In doing so I obtained the point ...
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1answer
19 views

How can I find the minimum distance between the origin $(0,0)$ and the curve $y=1-x^2$ using Lagrange multipliers?

I used the curve as the constraint and the origin as the point but I am not sure if that is correct.
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72 views

Maximum and minimum of $f(x,y)=xy$ when $x^2 + y^2 + xy =1$

It is asked to find the maximum and minimum points of the function $$f(x,y)=xy$$ when $x^2 + y^2 + xy=1$ I've tried Lagrange and obtained $$\lambda = \frac{y}{2x+y}=\frac{x}{2y+x}$$ but what ...
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1answer
35 views

When writing the Lagrange function does it matter if the constraint has minus/plus sign?

My university teaches me like this: However everywhere I look on the internet there is either a 1) minus sign before the lambda 2) or the whole equation is written differently (by placing ...
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44 views

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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1answer
19 views

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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2answers
33 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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60 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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15 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
24 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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19 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
16 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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1answer
33 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
34 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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20 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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51 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
39 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
47 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
55 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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39 views

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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1answer
43 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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32 views

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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23 views

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
32 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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68 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
27 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
60 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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18 views

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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23 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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2answers
47 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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2answers
25 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 ...