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

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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
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
16 views

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
33 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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0answers
12 views

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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14 views

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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23 views

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
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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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9 views

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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20 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
26 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
23 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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2answers
34 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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0answers
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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 ...
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1answer
39 views

Lagrange multipliers subject to a 3-variable, 4th degree constraint function?

I have recently been tackling the following problem: If $a+b+c = 0 $ and $ a^2 + b^2 +c^2 = 1$, work out $a^4 +b^4 +c^4$. Could this problem admit a solution through the method of lagrange ...
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2answers
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Lagrange multipliers and calculus

I'm trying to solve this problema but I don't know what else to do: Let A be a nonzero symmetric 3x3 matrix. Thus, its entries satisfy $a(ij) = a(ji)$. Consider the function $f(x) = (1/2)(Ax)\cdot ...
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1answer
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Solving the euclidian distance squared to kernelize a Lagrangian dual

Homework question, looking for a hint on the following problem: I'm trying to solve this dual lagranging form (which could potentially be wrong already, but let's assume it is right) $\boldsymbol{x}$ ...
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2answers
33 views

Finding minimum of $x_1^p+\cdots+x_n^p$ subject to $x_1+\cdots+x_n=1$

I want to find minimum of $f=x_1^p+\cdots+x_n^p$ ($p>1$) subject to $g=x_1+\cdots+x_n=1$. By Lagrange's multiplier, if it has a local extremum at $P$, it should satisfy $\nabla f(P)=\lambda \nabla ...
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1answer
24 views

Shortest distance from a point to a a Hyperplane

how could I prove the following using Lagrange optimization? Prove that the shortest distance from the hyperplane $$H= \{\vec{x} \in \mathbb{R}^{n} : \vec{a} \cdot\vec{x}=b\} $$ to a point ...
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Maximize and minimize a function using Lagrange multipliers.

I want to maximize and minimize $$h(a,b) = a + b$$ given the constraint $$g(a,b) = a^{\frac{1}{3}}b^{\frac{2}{3}} = l$$ I'm trying to use Lagrange multipliers. Here's what I did: \begin{align} ...
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Explain KKT conditions without reference to duality.

Is it possible to explain (not derive) KKT necessary conditions without reference to the concept of Lagrangian duality?
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25 views

Solving with LaGrange multipliers

With a given point $(p,q)$, I'm trying to find the nearest point on a curve $g$ to the point. So for $g(x,y)=x^2+y^2=1$, easy enough, the point would just be the solution of this system of equation ...
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1answer
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Find the maximum and minimum of $f(x,y)=xy-y+x-1$ on the set $x^2+y^2=2$

So I started the problem by first finding the critical points using the partial derivatives, which turns out that there is only 1 critical point at $(1,-1)$ where $f(1,-1)=0$ Then I know I must look ...
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1answer
32 views

Lagrange Multipliers for Implicit Functions

How can I find the minimum / maximum of a function for one variable defined implicitly (f(x, y, z) = c) with a constraint g(x, y) = c on the domain? For example, say you wanted to minimize for z: ...
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1answer
33 views

Determine the min, max, and saddle points of F(x) on the unit sphere.

$F(x)=x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}$ $x\in \mathbb{R}^{3}$ unit sphere: $x_1^2+x_2^2+x_3^2=1$ I am trying to find the min, max and saddle points of $F(x)$ on the unit sphere. ...
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31 views

A matrix equation with real coefficients

The problem is the following: Find $\lambda$ such that $ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $ where ...
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1answer
30 views

Problems on calculus of variations

I'm reading a paper in which it gives the following Lagrangian $$L[u,\rho,\phi]=L_0[u,\rho]+\phi(x)(\partial_t\rho+\nabla\cdot(\rho u))$$ where $L_0$ is part of Lagrangian and $\phi(x)$ is Lagrange ...
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1answer
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Unit vector that maximizes or minimizes

I know by the Taylor expansion $f(x,y)$ that in order for the origin to be a minimum point, $f_{xx}$ and $f_{yy}$ have to be both positive. Which I know how to prove. I also know other methods like ...
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1answer
44 views

minimum value of $x^2+y^2+z^2$ subject to $ax+by+cz=1$

If $ax+by+cz=1$, what is the minimum value of $x^2+y^2+z^2$ It is obvious that we can do Lagrangian multiplier,$W=x^2+y^2+z^2-\lambda (ax+by+cz-1)$
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1answer
29 views

Optimize $f(x,y,z) = xyz$ restricted to $g(x,y,z)= x^2+2y^2+3z^2= 6$

I'm stuck doing this problem. Optimize $f(x,y,z) = xyz$ restricted to $g(x,y,z) = x^2+2y^2+3z^2 = 6$ First, I found ${\nabla}f$ and ${\lambda}{\nabla}g$, and for Lagrange Multipliers, I got these ...
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Lagrange multipliers problems

I got some question about a problem I was doing. I have to optimize $f(x,y,z) = xy + yz$, restricted to $g(x,y,z) = y^2 + z^2 = 1$, so, for Lagrange multipliers theorem, I have this: $$ {\nabla}f = ...
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How to obtain closed form solution to the constrained optimization problem?

Suppose the following minimization problem: $$ N^*(\lambda)=\min_{X\in\mathbb{R}^8}\left\|D\left(A\cdot X-b\right)\right\|^2_2 \\ s.t. C_\lambda X= r_\lambda, $$ where $X\in \mathbb{R}^{8\times ...
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1answer
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Optimization: KKT conditions statement

I'm currently following this material http://www.math.uh.edu/~rohop/fall_06/Chapter2.pdf And I can't understand why the following statement is true, between the equations (2.9) and (2.10): "The ...
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2answers
27 views

Minimum of a function $f(x,y)=\frac{(1+2y)(1+\frac{x}{2})}{(1+y)(1+x)+x}$

what is the minimum of a function \begin{align} f(x,y)&=\frac{(1+2y)(1+\frac{x}{2})}{(1+y)(1+x)+x}\\ \text {s.t. }& 1 \le y \le x \le y(1+y) \end{align} I asked Wolfram and Alfa and it says ...
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Minimizing sample variance of $n$ functions

$f_n$, $i=1,\dots, n$ are $n$ functions. I would like to minimize the sample variance of these functions subject to a linear constraint: $$\text{minimize}\quad \frac{1}{N}\sum (f_i(x_i) - ...
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1answer
39 views

Lagranges multiplier to minimize function of two variables with two constraints

I have a Cobb Douglas type production function with $K$ and $L$ as inputs; $\alpha$ and $1-\alpha$ as output elasticities and $C$ as efficiency parameter. Now I have to minimize cost $=wL+rK$ w.r.t ...
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1answer
34 views

Do lagrangian multipliers converge to dual variables in LPs?

Can anybody clarify the following to me? Consider an LP, say a maximization problem, with solution x* and optimal value Z*. Its dual will have optimal value W*=Z* (by strong duality) and optimal ...
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2answers
85 views

How do you maximize a polynomial over an integer domain?

I am working on maximizing the polynomial $f:\mathbb{R}^N \rightarrow \mathbb{R}$ $$f(v):=\prod_{i=1}^{N}( v_i+\alpha_i^2)$$ over integer $n$-partitions of $P$, $n\leq N$: $$\left\{v \left|\, v_i \in ...
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Lagrange multipliers for minimax optimization

Does the equally constrained minimax optimization can be solved by using Lagrange multiplier method? Thanks in advance.
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How does the Lagrangian multipliers equation for multiple equality conditions follow?

I understand the intuitive narrative that wikipedia gives. I understand until the part that says: $\triangledown f \in S$, which means $\triangledown f$ is also an "illegal" direction, along with the ...
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2answers
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Lagrange Multipliers to determine min and max

I've got this question in a book of questions I'm doing. Can someone show me step by step how to solve this? Using Lagrange Multipliers for two constraints, determine the maximum and minimum of ...
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2answers
32 views

A Lagrange Multiplier Problem : How to deal with this case when $b< 8$

I was trying to solve the following problem of several variables calculus given in my class.I am stuck in a particular case of the problem.Please help me to solve the problem.Thnx in advance. Find ...
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1answer
18 views

maximization problem with inequalities restriction

I have a function $g(x,y,z)$, and $x+y+z=1, x\geq0,y\geq0,z\geq0$. Now I want to maximize $g$. If I ignore the inequalities, then I can use lagrangian and can solve this thing for maximum. But I am ...
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1answer
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Optimization with probability densities - Lagrange multipliers

This question is concerned with the paper "A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance" by Sigeiti Moriguti appeared in Ann. Math. Statist. ...
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2answers
64 views

Why is alternative approach to constrained optimization incorrect?

I am studying for the math GRE subject test, and my practice exam has a problem that goes as follows: Find the minimum distance from the origin to the curve $3x^2 + 4xy + 3y^2=20$. Apparently I was ...
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1answer
134 views

How to solve this optimization problem with equality constraints?

I want to find $\delta_j$ in the following optimization problem. My variables are $\gamma_i$ and $\delta_j$ (all other symbols are known parameters). Assume $i\in\{1,\ldots,9\}$ and ...
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A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
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2answers
53 views

Augmented Lagrangian

Consider the following equality constraint minimization problem: minimize $\text{ }f(x)$ subject to $Ax=b$ Its Lagrangian is then: $L(x,y) = f(x) + y^T(Ax-b)$ We can use then gradient ascent to ...
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1answer
37 views

Estimator with smallest variance, Lagrange multiplier

I have a question and I think I'm supposed to use the Lagrange multiplier although I haven't been taught it, so I'm not sure if I can use it or not. The question is: Suppose that $X_i$ has mean $\mu$ ...