Questions tagged [lagrange-multiplier]

This tag is for the questions on Lagrange multipliers. The method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

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2answers
29 views

global extrema of $(x,y)\mapsto xye^{-x^2-y^2}$ on $\{(x,y)\in\ \mathbb{R}^2\,:\,x^2+y^2=\frac{1}{2}\}$

Find the global extrema of $(x,y)\mapsto xye^{-x^2-y^2}$ on $\{(x,y)\in\ \mathbb{R}^2\,:\,x^2+y^2=\frac{1}{2}\}$ using Lagrangian multipliers. My approach: Set $L(x,y,\lambda):=xye^{-x^2-y^2}-\lambda(...
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0answers
17 views

How do I solve a Lagrange multiplier with multiple periods

Problem: $\max_{c_{i,t}, x_{i,t},k_{i,t}}$$^\infty_{t=0}$ $E_0$=$\sum$$^{\infty}_{t=0}$$\beta^t$$lnc_{i,t}$ s.t. $c_{i,t}+x_{i,t}=\alpha_{i,t-1}k_{i,t-1}$ $k_{i,t} =x_{i,t}+(1-\delta)k_{i,t-1}$ $c_{i,...
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0answers
71 views

How to solve the following convex optimization problem to get a closed form solution?

Please, can you help to solve the following convex optimization problem to get a closed-form solution? I tried to solve it using Lagrange multiplier but it's not easy. I can't get the closed-form ...
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1answer
25 views

How to find the general form of minimum distance from the point (m,n) to the ellipse by using Lagrange Multiplier?

Excuse me! I have tried to solve this problem for a long time but I have stuck in this step. This is my work picture01. This is my work picture02. I cannot simplify y in term of a b m and n. It’s too ...
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0answers
29 views

KKT and Constraint Qualification

Have studied lagrangian and optimization primarily via Khan academy, as got bogged down with Simon and Blume Mathematics for Economists/Chiang and Wainwright Fundamental Methods of Mathematical ...
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1answer
22 views

Showing properties of a specific maximization problem, as well as finding the maximum.

Let there be $p_1,p_2,..,p_n,q_1,q_2,...,q_n$ with $\sum_{i=1}^n p_i = 1 = \sum_{i=1}^nq_i$ For $M :=\{ x\in (0,\infty)^n: \sum_{i=1}^nq_ix_i = a \}$ With the following Maximization Problem. $$(*) \...
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1answer
28 views

Find extremizers using Lagrange multipliers

Using Lagrange multipliers, $$\begin{array}{ll} \text{extremize} & x + y\\ \text{subject to} & x^{2} + y^{2} \leq 5\\ & x \geq 0\end{array}$$ Graphically, it comes that the maximiser is $(\...
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1answer
20 views

lagrange multiplier determinant

Can someone please explain why in my textbook they write lagrange multiplier like this : $$\begin{vmatrix} \frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}\\ \\ \frac{\partial g}{\...
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3answers
72 views

Find maximum point of $f(x,y,z) = 8x^2 +4yz -16z +600$ with one restriction

I need to find the critical points of $$f(x,y,z) = 8x^2 +4yz -16z +600$$ restricted by $4x^2+y^2+4z^2=16$. I constructed the lagrangian function $$L(x, y, z, \lambda ) = 8x^2 +4yz -16z +600 - \lambda (...
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2answers
62 views

Find the maximum of $f(x)=\sum_{i=1}^{4}x_ia_i^2$ under the constraints

Find the maximum of $f(x)=\sum\limits_{i=1}^{4}x_ia_i^2$ under the constraints $|x|=1$ and $\langle x,a\rangle=0$. My try I set $$L(x,\lambda,\mu)=f(x)-\lambda\left(\sum x_i^2-1\right)-\mu\left(\sum ...
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1answer
50 views

Solution to quadratic program

Let $ (a_{ij}) $ and $ (b_{ij}) $ be two sequences of real numbers. I'm trying to solve the quadratic program $$ \min_{(a_{ij})} \sum_i \Big(\sum_j a_{ij} \Big)^2 \quad \text{s.t.} \quad \sum_i\sum_j ...
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2answers
36 views

With $\vec{x}=(x_1,\ldots,x_n)$, find all the min and max of $\prod_{i=1}^{n} x_{i}^i$ given that $||\vec{x}||=1$

With $\vec{x}=(x_1,\ldots,x_n)$, find all the min and max of $\prod_{i=1}^{n} x_{i}^i$ given that $||\vec{x}||=1$ Now clearly this is Lagrange multiplier. So one might take $\prod_{i=1}^{n} x_{i}^i-\...
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0answers
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Computing the dual of online learning of linear classifier

Given a convex loss function $l(\cdot)$, $T$ training examples with their corresponding labels $\{\textbf{x}_i, y_i\}_{i \in [T]}$, regularization parameter $\lambda$, the primal form for the problem ...
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1answer
37 views

Minimizing sum of functions under constrained domain

Consider this function, I am told to find minimum of it. I considered using lagrange multipliers but I have no constraint curves to work with.
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1answer
33 views

Line that best fits the points $p_1,\ldots,p_n$.

I'm trying to solve this problem: Let $p_j=(x_j,y_j)\in \mathbb{R}^2$ $(j=1,\ldots,n)$ with $p_i\neq p_j$ for $i\neq j$. Find the function of de form $f(x)=ax+b$ such that $$\sum_{j=1}^{n} (f(x_j)-y_j)...
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5answers
224 views

prove $\sum\cos^3{A}+64\prod\cos^3{A}\ge\frac{1}{2}$

In every acute-angled triangle $ABC$,show that $$(\cos{A})^3+(\cos{B})^3+(\cos{C})^3+64(\cos{A})^3(\cos{B})^3(\cos{C})^3\ge\dfrac{1}{2}$$ I want use Schur inequality $$x^3+y^3+z^3+3xyz\ge xy(y+z)+yz(y+...
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0answers
17 views

Intersection of plane curves using Lagrange multipliers

Let f(x) and g(x) be denoting some plane curves in the x-y plane. Now, suppose I want to find intersection of f(x) and g(x). This is precisely the point where distance between curves is minimized. Or ...
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0answers
33 views

Convex optimization lagrange multiplier nash bargaining problem

I have an optimization problem that looks like this: let $\mathbf{X} = \begin{bmatrix} x_{0,0} & x_{0,1} \ ... & x_{0,L}\\ x_{1,0} & x_{1,1} \ ... & x_{1,L}\\ \vdots \\ x_{K,0} & ...
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2answers
37 views

Finding the local extrema for a curve of intersection of two surfaces and Lagrange Multipliers

Suppose that I'm trying to find a local extrema for $f\left(x,y\right)$ subject to the constraint $g\left(x,y\right)=c$, is this equivalent to finding the local extrema for the curve of intesection of ...
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0answers
26 views

Finding the global global extrema of a given function on $\{x^2+\frac{1}{2}y^2+2z^2\leq 1\}$

Let $K=\{(x,y,z)\in\mathbb{R}^3|\,x^2+\frac{1}{2}y^2+2z^2\leq 1\}$. Let $F:\mathbb{R}^3\to\mathbb{R},\,(x,y,z)\mapsto x^2+e^{-y^2}+4z^2$. Find the global extrema of $F$ on K. I've first examined the ...
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1answer
46 views

How to solve the optimization problem of PCA?

I'm learning PCA and I found the following optimization problem in pages 9 and 13 of Afonso Bandeira's lecture notes. $$\begin{array}{ll} \underset{V \in \mathbb{R}^{n \times d}}{\text{maximize}} &...
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0answers
27 views

Question about using Lagrange mutipliers finding the solution of PCA

I'm trying to use Lagrange mutipliers to find the solution of PCA. Assume the data matrix is X $ = [X^{(1)},\dots, X^{(m)}] \in \mathbb{R}^{k \times m}$($k$ features and $m$ observations, already zero-...
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2answers
59 views

Is it the absolute maximum?

Let $x,y$ and $z$ be the interior angles of a triangle. Suppose that $f(x,y,z)=\sin(x)\sin(y)\sin(z)$. Using the Lagrange multipliers, I know $f$ has a local maximum at $x=y=z=\pi/3$. However, I do ...
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1answer
37 views

Lagrange Multipliers to find the maximum and minimum values

Using Lagrange Multipliers I need to find the maximum and minimum values of the function $f(x,y,z)=x^2+y^2+z^2$ subject to the given constraints: $g(x)= x^2/\alpha+y^2/\beta+z^2/\gamma=1$ and also $\...
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2answers
40 views

Help with Lagrange multipliers

I need to find the absolute minima and maxima of the function $f(x,y) = 12 x^2 + 12 y^2 - x^3 y^3 -5$ in the region bounded by the disk $x^2 + y^2 \le 1$. I know that $f(x,y)$ has three critical ...
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2answers
24 views

Finding extremum of a function using Lagrange

$f(x, y) = \ln(x) + \ln(y) $ Restricted to $g(x, y) = x + y/2 - 1 = 0$ I did $\delta f(x, y) = \lambda \delta g(x, y) $ I have the system $1/x = \lambda$ and $1/y = \lambda /2$ But I'm stuck here. How ...
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0answers
20 views

Linear system involving lagrange

Note that: fx is the first-order partial derivative of f(x) and gx the first-order derivative de g(x). I have this system to solve: f x = λ gx f y = λ gy g(x,y) = 2 fx = 2x /(x^2+y^2 +2) fy = 2y /(x^...
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1answer
30 views

Maximum volume using Lagrange multipliers

I need to determine the maximum volume of a rectangular box with these side conditions: its surface has 2m² and the sum of all its edges = 8 m of length. How do I do that ?
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1answer
22 views

Extension of median minimizing the sum of absolute deviations (the $L_1$ norm)

This is an extension of the question asked in The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm) . Except with the extra constraint that $x \in S$. The solutions provided there ...
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0answers
7 views

how can Lagrangian function have saddle points though its linear dependence on multipliers?

I recently read that lagrangian function have saddle point at the optimal point. But as per the definition of saddle points it should be convex along one direction and concave along the other. But how ...
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1answer
38 views

Supremum of $f(x_1,\dots,x_n)=\sum_{i=1}^n a_i x_i-\sum_{i=1}^n x_i \log x_i$

Let $a_1,\dots,a_n\in\mathbb{R}$. Find supremum of: $f(x_1,\dots,x_n)=\sum_{i=1}^n a_i x_i-\sum_{i=1}^n x_i \log x_i$ in the following set: $D=\{x_1,\dots,x_n\ge0 , \sum_{i=1}^n x_i=1\}$. Note: we ...
2
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1answer
31 views

Find the absolute maximum and minimum of $f(x,y,z)=ax+by+cz$ over the function $h(x,y,z)=x^2+y^2+z^2=1$ using Lagrange multiplier

Find the absolute maximum and minimum of $f(x,y,z)=ax+by+cz$ over the function $h(x,y,z)=x^2+y^2+z^2=1$ using Lagrange multiplier. $$\mathcal{L}(x,y,z,\lambda)=f(x,y,z)+\lambda g(x,y,z)$$ $$\;\;\;\;\;...
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1answer
69 views

Meaning of pure Nash equilibrium in the context of Lagrangian game

I'm reading a paper on solving an optimization problem for a non-convex function. This paper is suggesting a method using a game theoretic approach: Optimizing the Lagrangian can be interpreted as ...
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2answers
44 views

Find the minimum and maximum of $f(x,y,z)=3x+2y+4z$ subject to constraint $x^2+2y^2+6z^2=9$

I have $g=g(x,y,z,\lambda)=3x+2y+4z-\lambda(x^2+2y^2+6z^2-9)$ $g_x=3-2x\lambda=0$, so $\lambda=\frac{3}{2x}$ $g_y=2-4y\lambda=0$, so $\lambda=\frac{1}{2y}$ $g_z=4-12z=0$, so $\lambda=\frac{1}{3z}$ ...
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1answer
36 views

Calculus FoC and SoC

I'm not quite sure how to approach this. My thinking is to use the Lagrangian to solve for the FOC and the proceed, however, I'm not sure how to proceed from there with the SOC or if that's missing ...
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1answer
31 views

What is the minimum of this weighted sum, considering all constraints?

I have started to self-study some non-linear programming courses. In one of the references I have encountered a challenging problem which I was unable to solve it: Given a number $q, 0 < 𝑞 < 1$,...
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1answer
139 views

Lagrange multiplier method. Really need help on this question !! :(

A firm produces a good using two raw materials, X and Y , and the corresponding costs per unit of these raw materials are $C_x$ and $C_y$ , respectively. The amount of its good that a firm can produce ...
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1answer
47 views

Find the maximum and minimum of $(1/x-1)(1/y-1)(1/z-1)$ if $x+y+z=1$

I get more confused when I try to solve this. For my first approach, I'm using normal AM-GM inequality: \begin{equation} (\frac{1}{x}-1)(\frac{1}{y}-1)(\frac{1}{z}-1)=\frac{(1-x)(1-y)(1-z)}{xyz} \\=\...
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1answer
33 views

Optimal parameters

Using lagrangian, we could get optimal parameters of linear hard margin SVM which are $w^*$ and $b^*$. How can we approach this problem of proving that these parameters are indeed optimal?
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1answer
55 views

linear hard-margin SVM Proof of Optimal w* and b*

Hello i have this task about hard-margin SVM with only 2 Data points input. Does anyone know how to approach this task? I think it doesnt require to make a numerical example and solving it with the ...
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2answers
57 views

three variable inequality $x+y+z\le xyz+2$ with constraint $x^2+y^2+z^2=2$

Let $x$, $y$ and $z$ be three real numbers such that $x^2+y^2+z^2=2$. it is asked to prove that $$x+y+z \le xyz+2$$ I tried using Lagrange multipliers but I'm stuck with the following system $$\begin{...
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1answer
13 views

Formulating three functions as Lagrangian multipliers :

so I have two functions f(x, y) and g(u, v) to minimize and we know a third function that maps the variables: h(x,y) = h(u,v). How can I define the Lagrange equation for f and g to make an ...
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0answers
33 views

Show that none of the critical points identifies identifies a solution to the maximization problem

Question: a firm produces a single output $y$ using three inputs $x_1,x_2,x_3$ in non-negative quantities through the relationship: $y=g(x_1,x_2,x_3)=x_1(x_2+x_3)$ The unit price of $y$ is $p_y>0$ ...
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1answer
50 views

Can the Weierstraß and/or Kuhn-Tucker theorems be used to obtain and characterize a solution? Why or why not?

Question: An agent who consumes three commodities has a utility function given by: $u(x_1,x_2,x_3)=x^{1/3}_1+\min\{ x_2,x_3\}$ Given an income $I$, and prices of $p_1,p_2,p_3$. Describe the consumer’s ...
2
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2answers
62 views

Solve the following constrained maximization problem [closed]

Question: let $T\geq$ 1 be some finite integer, solve the following maximization problem. Maximize $\sum_{t=1}^T$($\frac{1}{2}$)$^t$$\sqrt{x_t}$ subject to $\sum_{t=1}^{T}$$x_t\leq1$, $x_t\geq0$, t=1,....
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2answers
26 views

Is Lagrange multipliers and (multivariable) extreme value theorem related?

I couldn't find a question answering this concept but they seem to be related. Extreme Value Theorem (two variables) If f is a continuous function defined on a closed and bounded set $A⊂\mathbb{R}^2$,...
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0answers
55 views

Maximizing Marginalized Mutual Information $P(X=x_i)$ using Lagrange Multipliers

First of all, sorry if the question is too big. I researched if there were any questions that could help, but I was unsuccessful. This problem is part of a personal study that I've been doing, a ...
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0answers
13 views

Lagrange multiplier - Budgetary constraints

Price of product X: 65 Price of product Y: 105 Total budget: 500 F(x,y)= x^0.2 * y^0.8 Given this info how can I find the optimal combination of product X and Y using the lagrange multipliers?
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1answer
21 views

Find such points on the ellipsoid $x^2+2y^2+4z^2 = 8$ that are the farthest and nearest to the point $(0;0;3)$

So, I need to compute this optimisation exercise using Lagrange multiplier. I know how to do these sorts of exercises when a constraint function is given, but in this case there's only a point. So, I ...
0
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1answer
24 views

Find the min value of $\min\{x_1x_2x_3:a^2x_1^2+x_2^2+x_3^2\leq1\},a>0$

Find the min value of $\min\{x_1x_2x_3:a^2x_1^2+x_2^2+x_3^2\leq1\},a>0$ using KKT So my try is: if we set $L(x,\lambda)=x_1x_2x_3+\lambda(a^2x_1^2+x_2^2+x_3^2-1)$ and differentiating wrt to all ...

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