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

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1answer
9 views

Monotonic optimal value function

Are there any theorems/sufficient conditions about when the optimal value function of a parametrized optimization problem is monotonic in the parameter? Specifically, are there simple conditions ...
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12 views

Derivation of SVM algorithm (Lagrangian)

I have a question about the derivation of the SVM algorithm (for example, page 3 here ). The question is about the math, so that's why I'm asking this here. Suppose I have the following optimization ...
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1answer
21 views

Lagrange multipliers/dual problem: A simple worked example for max(2-x^2) s.t. x=1

I'm trying to derive the dual problem of a very simple example of a Lagrange multiplier (note: please correct my terminology if it's off). This isn't homework, I've just picked the example off some ...
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1answer
27 views

Find orthonormal basis of quadratic form

Q: Let $$A = \begin{pmatrix} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{pmatrix}$$ Find the quadratic form of $q: \mathbb{R}^3 \to \mathbb{R}^3$ represented by A. and find ...
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2answers
41 views

Method of Lagrange multipliers determining nature?

Let us say I have a function, $g$ that I want to find the extrema of subject to the condition $h=0$ via the method of Lagrange multipliers. i.e. so that I find the extrema of the function: ...
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On a max-min problem from an exam.

I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems ...
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20 views

Minimize $\|\mathbf{x-y}\|^2 $ subject to $x \in $ set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$

We are given the set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$ and a point $\mathbf{y} \in \mathbb{R}^n$. Our goal is to find point $\mathbf{\hat{x}}$ ...
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Find the minimum of $f(x,y,z)=3x+2y+z+5$ subject to the constraint $g(x,y,z)=9x^2+4y-z=0$

Let $ \displaystyle L(x,y,z,\lambda) =(3x+2y+z+5)+\lambda(9x^2+4y-z)$ By using Lagrange Multiplier $L_x=3+18\lambda x=0$ $L_y=2+4\lambda=0$ $L_z=1-\lambda=0$ It implies that $\lambda=1=-2$? How ...
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1answer
27 views

Duality and the Positive Lagrange Multiplier

Suppose I have the following optimization problem: \begin{align} \min &f(x) \\ & f_1(x) \leq 0 \\ & \vdots \\ & f_k(x) \leq 0 \\ & g_(x) = 0 \\ ...
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17 views

Connection between method of Lagrange multipliers and KKT conditions?

I understand that in general, the KKT conditions are not sufficient for optimality. However, if the primal problem is a convex optimization problem, then the KKT conditions are sufficient for ...
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1answer
40 views

Using Lagrange multipliers to identify the Extremes of function $f(x, y)=x-y$, under condition $g(x,y)=x^2 + y^2 - 4=0$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 3 of 4, part $b$ and graded ...
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1answer
30 views

Maximise volume given inequality constraint on its dimensions without using Lagrange, KKT or Linear Programming

The problem (from Calculus for Business, Economics, Life Sciences and Social Sciences 12e): I found this and that, but they use Lagrange/KKT. What I tried: Girth $= 2w + 2h$ Maximise ...
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0answers
18 views

Minimizing mutual information using Lagrange multipliers

Im trying to follow a minimization of mutual information using Lagrange multipliers in a highly cited paper called The Information Bottleneck Method (1999), page 4: $$R(D) = \min_{p(\tilde{x}|x): ...
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1answer
16 views

Can lagrange multiplier(Kuhn tucker multipliers?) change in corner solution?

If we want to maximize $f(x)$ subject to two constraints, one which says that $x< c$ $c>0$, and another that says that $x\geq 0 $. Assume there are no problems with either $x=0, x>0$ or $\mu ...
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2answers
173 views

Book on applied mathematics/analysis

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
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1answer
18 views

Can I use Lagrange multipliers with redundant constrains?

Can I use Lagrangian multipliers with redundant constrains? For example, suppose I have the following problem: Find the maximum of $F(x,y,z)$, subject to $f(x,y,z)=0$ and $g(x,y,z)=0$. But you also ...
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25 views

Pattern of collision of bouncy balls in a sphere?

Suppose that you have two infinitely bouncy golf balls that exist inside a perfect sphere in weightless suspension, and both golf balls start bouncing at a random angle and are 10 or 100 times ...
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17 views

Find the maximum of this function involving pseudo inverse.

I need to find \begin{equation} \mathbf{d}_i=\max \;( \sum_i \Vert \mathbf{P}x_i \Vert_2^2 ) \;\;s.t.\;\; \Vert \mathbf{d}_i\Vert=1 \end{equation} where $\mathbf{P}$ is the projection operator ...
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29 views

Maximum area of a rectangle whose vertices lie on ellipse $x^2+4y^2=1$

Maximum area of a rectangle whose vertices lie on ellipse $x^2+4y^2=1$. I try to do it by lagrange multiplier as $F(x,y,t)= xy + t(x^2+4y^2-1=0)=0$. Differentiating w.r.t to x,y and solving i get ...
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15 views

HMM optimization: Lagrange multiplier problem

In David Barber's textbook "Bayesian Reasoning and Machine Learning" he hints at the derivation of the Baum-Welch algorithm for HMM parameter learning: Textbook excerpt, (cannot include images yet, ...
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1answer
35 views

What does the Lagrange multiplier of an equality constraint mean, intuitively?

Consider a nonlinear optimization problem of the form \begin{align} \min_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,I\\ \nonumber \quad&g_j(x) \le ...
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1answer
50 views

Maximum and minimum of the function $xy+z^2$

Find the maximum and minimum values of the function $f(x,y,z)=xy+z^2$ in the circumference obtained by intersections between the sphere $x^2+y^2+z^2=4$ and the plane $y-x=0$. I did Lagrange and found ...
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28 views

Minimization/maximization of system of nonlinear equations

Consider a system of nonlinear equations of the following form: $$F_1(x_2, x_3, x_4...x_n)$$ $$F_2(x_1, x_3, x_4...x_n)$$ $$...$$ $$F_n(x_1, x_2, x_3...x_{n-1})$$ And we wish to simultaneously ...
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28 views

KKT conditions (equations) for Generalized Assignment Problem or Binary integer programming problem

I have this formulated Generalized Assignment Problem (GAP) or it can also be considered as Binary integer programming problem. Solving this problem can be achieved through Branch and Bound Technique. ...
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1answer
26 views

Explain why max 2x+y s.t. x+y=m does not have a single solution?

I dont know how to answer the following question. Explain why max 2x+y s.t. x+y=m does not have a single solution? Hope someone can explain.
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0answers
14 views

optimal derivative position through optimization

So I have the following optimization problem: min. $-E^Q[u(h(x))]$ s.t $\int h(x)q(x)dx \leq \frac{V_0}{B_0}$ Where $Q$ is the subjective probability which then gives: $E^Q[u(h(x))]=\int ...
2
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1answer
17 views

If $x_0$ is extreme point of $f(x)$, why $\nabla f(x_0)$ is the normal vector of $z=f(x)$?

If $x_0$ is extreme point of $f(x)$, why $\nabla f(x_0)$ is the normal vector of $z=f(x)$ ? I am not sure whether it right. But when I try to understand why $\nabla F=\lambda \nabla G$ in lagrangian ...
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maximum entropy principle: CDF of its PDF

In goodness-of-fit (gof) tests (COD, R2, X2) to discriminate PDFs, we need their CDFs. With wind speed, another PDF is by Maximum Entropy Principle or Method, of the form: $$f(v)=\exp\left\{-a_0 - ...
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1answer
24 views

help with $\nabla$ and Lagrangian in optimization / portfolio theory?

So $\nabla$ as I know it from calculus means gradient. We have $\min \ \ \frac{1}{2}w^T\Sigma w$ $s.t. \ \ \ \ \ w^T1 = V_0, \ \ V_0 = 100$ where $w$ is weights in vector, $\Sigma$ is the ...
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2answers
41 views

Understanding the solution to a problem on Lagrange Multipliers

This is a problem from Riley-Hobson "Mathematical Methods For Physics And Engineering". QUESTION: Two horizontal corridors, $0\le x \le a$ with $y\ge 0$ and $0\le y \le b$ with $x\ge 0$, meet at ...
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2answers
29 views

Basic questions about finding a volume formula for a box inside of an ellipsoid,

I want to maximize the volume of a box, with sides parallel to the $xy$, $xz$ and $yz$-planes, with the box inside of an ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}=1$$ So, my ...
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1answer
21 views

Optimization problem: Find shortest distance between two vectors

$$\min (u-v)^T(u-v)$$ $$s.t. \space Ru=p, \space Sv=q$$ where $u$ and $v$ are in $R^4$ and $R$ and $S$ are $3x4$ matrices. When I expanded the expression I got this: $$u^Tu - 2u^Tv +v^Tv$$ Is this ...
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1answer
35 views

Partial derivative involving ln()

I'm looking at a paper involving a constrained optimisation problem using Lagrange multipliers, in which the following Lagrangian function appears: $\gamma = \max ...
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0answers
16 views

Lagrange/KKT multipliers and additional constraints

Any ideas on the following would be much appreciated. I’m interested in how the shadow price of non-renewable resource changes when new constraints are introduced to the problem. So, assume that there ...
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33 views

How to use the Karush–Kuhn–Tucker conditions?

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method. For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to ...
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1answer
60 views

shortest distance from point to hyperplane lagrange method

I need to find the shortest distance, in D-dimensional Euclidean space ($\mathbb{R}^D$) from a point $\textbf{x}_0$ to a hyperplane $H: \textbf{w}^T \textbf{x} + b = 0$, using the method of Lagrange ...
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0answers
26 views

How to solve this convex optimization problem with inequalities constraint?

I am trying to understand how to solve the SVM optimization problem. It is usally written : $$\text{Minimize} $$ $$\|\textbf{w}\|$$ $$\text{Subject to}$$ $$y_i(\mathbf{w}\cdot\mathbf{x_i} - b) \geq ...
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1answer
12 views

Langrange multiplier, confusion at setting up the problem

Question Find the rectangular box with the largest volume that fits inside the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, given that it sides are parallel to the axes Solution Clearly the box will have ...
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Hamiltonian Systems

I come from Physics where I am used to writing down Hamiltonians and Lagrangians for physical systems. I recently read "A Theoretical Framework for Back-Propagation" where they use Lagrangian ...
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1answer
38 views

Optimization on manifold via Lagrange multipliers

Let the manifold $S$ in $\mathbb R^n$ be defined by $g(x)=0$. If $p$ is a point not on $S$, and $q$ is the point of $S$ which is closest to $p$, show that the line from $p$ to $q$ is perpendicular to ...
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1answer
38 views

How can I find the two critical points of this system of equations?

I'm currently trying to use Lagrange Multipliers to find the 2 critical points of the function $$ f(x,y,z) = \frac{1}{2}x^{2}+yz+\frac{1}{3} y^{3} - z^{2} $$ subject to $$ h(x,y,z) = x+y+z-2 = 0 $$ ...
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Lagrange multipliers to find maximum and minimum

Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x, y, z) = x - y +z$ on the sphere $x^2 + y^2 +z^2$. Since gradient vectors have have $x$ to the ...
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0answers
47 views

Distance between point and a parabola using Lagrange multiples

I am trying to find the distance between the point $(p, 4p)$ and the parabola $y^{2} = 2px$, where $p$ is a fixed positive parameter. So far, I have got ...
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1answer
28 views

Finding extreme values using Lagrange Multipliers

So the question asked to find the extreme values of the function $$f(x,y,z)=e^{xyz} $$ under the constraint $$ 2x^2+y^2+z^2=24$$ My attempt : I'm a bit lost on what to do now, because I tried ...
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1answer
24 views

Lagrange Multipliers and Lambda

We know that: The upshot of all this is the following: at a local maximum, the gradient of $f$ and the gradient of $g$ are pointing in the same direction. In other words, they are proportional. ...
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2answers
25 views

Lagrange Multipliers: find points farthest/closest to a point

Find the points of the ellipse: $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ which are closest to and farthest from the point $(1,1)$. I use the method of the Lagrange Multipliers by setting: ...
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1answer
10 views

Lagrange multiplication, find (x,y) of the plane where the sum of the squares of the distances to these coordinates is minimal?

Lagrange multiplication, find the point (x,y) of the plane in which the sum of the squares of the distances to the points (0,1), (0,0), (2,0) is minimal? I don't understand if I'm supposed to use ...
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2answers
35 views

Finding extrema with Lagrange multipliers

I'm trying to find the extrema of $f(x,y)= \cos(x^2-y^2)$ constrained to $x^2+y^2=1.$ Using Lagrange Multipliers I get this far: $-x(\sin(2x^2-1)=\lambda x$ $-y(\sin(-2y^2+1)=\lambda y$ But I don't ...
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2answers
31 views

Applicability of Lagrange Method

The textbook has the following theorem: Suppose $f(x,y)$ and $g(x,y)$ have continuous partial derivatives in a domain $D\subset \mathbb R^2$, and that $(x^*,y^*)$ is both an interior point of ...
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1answer
23 views

Hyperplane problem by Lagrange multiplier method

Solve the problem using Lagrange multiplier method. Find the point that belongs to both hyperplanes xT c = β and xT d = γ which is closest to the origin.