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

learn more… | top users | synonyms

0
votes
0answers
9 views

How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
1
vote
0answers
14 views

connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
-1
votes
0answers
11 views

Boundary conditions and Lagrange Constraints in Calculus of Variations

I am trying to learn about Calculus of Variations for some time now. In many problems, there are some boundary conditions defined, for example when we want to maximize a functional ...
0
votes
1answer
40 views

Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
3
votes
1answer
45 views

$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
1
vote
1answer
30 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
1
vote
2answers
60 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
1
vote
2answers
21 views

Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
4
votes
0answers
38 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
0
votes
0answers
39 views

Lagrange Multipliers and optimisation

An open-topped metal water tank with volume $2$ m$^3$ is to be constructed with vertical sides and a right-angled triangle as base. What should be the dimensions of the base to minimise the area of ...
3
votes
3answers
91 views

Why do lagrange multipliers have the form $\nabla G$

I was studying some multivariable Calculus and we were covering the topic of Lagrange multipliers. I didn't understand exactly why the equations take the form: $$ \nabla f = \lambda \nabla G $$ ...
4
votes
1answer
46 views

Lagrange's multiplier not working

Given the function $f(x,y):=xy+x-y$. Let $D:=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq25\wedge x \geq 0\}$. Find the absolute maximum and minimum of $f$ on $D$. My working is as follows: $\begin{array} ...
1
vote
2answers
71 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
1
vote
2answers
43 views

maximizing a coordinate of $x^T A^T A x \leq r^2$

Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by ...
0
votes
1answer
63 views

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? [duplicate]

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? Using Langrange Multipliers, I've set up the standard equation with $$g(x,y) = (x/2)^2 + (y/3)^2 = 1$$ $$f(x,y) = ...
0
votes
1answer
42 views

Can this optimization problem be solved analytically?

Can the following be solved analytically? minimize $$ \ V(x) = |x_1-2| + |x_2-2| \ \ ; \ \ [x_1,x_2] \in R^2$$ subject to: $$ h_1(x) = x_1-x_2^2 \ge 0 $$ $$ h_2(x) = x_1^2+x_2^2-1 = 0 $$ I ...
0
votes
0answers
29 views

Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
1
vote
1answer
20 views

Verifying an inequality with Lagrange multipliers

My question has two parts. Let $p$ and $q$ be numbers with $p > 1$ and $q > 1$ 1) Show $${x^p\over p} + {y^q\over q} \ge {1\over p} + {1\over q}$$ where $x \gt 0$, $y \gt 0$ and $xy = 1$. 2) ...
0
votes
1answer
27 views

lagrange method, linear constraints and unique global maximum

My book in linear programming states two things that I do not understand. We are working with the lagrange method with linear constraints.: From multivariate calculus we have that at a critical ...
1
vote
1answer
34 views

The method of Lagrange's Multipliers

I used the method of Lagrange's multpliers to find the maximum of $f(x,y,z)=\ln x+\ln y+3\ln z$ on the portion of the sphere $g(x,y,z)=x^2+y^2+z^2=5r^2 \ ; r>0$ where $x>0, y>0, z>0$ . I ...
2
votes
2answers
635 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
0
votes
2answers
28 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
1
vote
0answers
7 views

In the envelope theorem, why can I write my inputs $x$ and $y$ as a function of $\xi$?

This is a question about the envelope theorem. Suppose I have a maximization problem $$\max_{x,y} f(x,y,\xi)$$ such that $$g(x,y,\xi) \leq c$$ where $x$ and $y$ are control variables and $\xi$ is a ...
3
votes
0answers
34 views

Lagrange Multipliers Dilemma

In the problem $f(x,y) = xy$ and $g(x, y) = x^2 + 9y^2 = 18$ I get $y = 2λx$, $x = 18λy$ and $x^2 + 9y^2 = 18$ (the constraint). All is fine, but I feel like I'll get two different answers ...
6
votes
1answer
278 views

Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by ...
1
vote
1answer
39 views

Find max/min of $f(x,y,z)$ on closed unit ball $B$ in $\mathbb{R^3}$

$f(x,y,z) = 3x - 2y + z$ Let $B$ be a closed unit ball in $\mathbb{R^3}$, find the max/min of f on $B$. We first need to observe $(a)$ the behavior of $f$ in $B^0$ $(b)$ the behavior of $f$ on ...
0
votes
0answers
47 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?
2
votes
1answer
27 views

for f(x,y,z) find point on surface nearest to origin

$f(x,y,z)=x^2+2y^2-z^2$, $S=\{(x,y,z): f(x,y,z)=1\}$ find point on S nearest to origin. I thought I would use Lagrange multipliers to solve this problem, but when I use $f(x,y,z)=x^2+2y^2-z^2$ and ...
1
vote
1answer
22k views

Deriving demand functions given utility

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$. Derive the ...
1
vote
0answers
23 views

Lagrange Multipliers with Calculus of Variations

We wish to extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to the constraint $$g(\mathbf{y}, t) = 0 \;.$$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha ...
0
votes
1answer
15 views

Lagrange Multipliers with $f\left(x,y\right)=x^2-y^2$ and “constraint” $g\left(x,y\right):=2y-x^2=0$

I am working on a problem from my textbook on Lagrange Multipliers. I feel I have these down now, but I am curious about this specific problem. Let \begin{align} f\left(x,y\right)=x^2-y^2\tag{1},\\ ...
0
votes
0answers
13 views

Insight for basic constrained optimization

Consider the basic Constrainted Optimization Problem: Minimize f(x) Subject to: Ax = b The Lagrangian is ...
0
votes
0answers
24 views

Quick Constrained Optimization Huerstistics

I am wondering if there is a way to find very quick optimization heuristics for the form. $$ f(x) = cx^a $$ $$ s.t. $$ $$ L \le Ax \le B$$ $$ 0 \le x \le \infty $$ I know with only a few variables ...
0
votes
0answers
15 views

Finding the maximum and minimum on a constraint.

Let $f(x,y,z) = 2x + y$. Find the absolute maximum and minimum on the constraint $x+y+z=1$ So we know that $\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$ where $g(x,y,z) = x+y+z-1$. Calculating, we ...
0
votes
1answer
32 views

Optimizing a function with simple equality and inequality constraints

I'm looking to maximize $$f(x_1,..,x_n)=\sum_{i=1}^n\alpha_i \sqrt x_i+\beta $$ subjected to the following constraints $$a_i\le x_i \le b_i \space\forall i \in\{1,...,n\}$$ and ...
0
votes
0answers
20 views

Fixed Length Catenary

Doing a fixed length catenary problem, why is it that adding the constraint $L=\int_A^B ds$ gives us more solutions. A little background: the catenary problem involves minimizing the integral ...
1
vote
2answers
16 views

Determining points on a 3-dimensional intersection closest to the origin

I was presented with this question in a lab: Use the method of Lagrange Multipliers to solve the following. Be sure to let Mathematica do all the heavy lifting for you. Determine the points that ...
1
vote
1answer
52 views

Finding the distance between a point and a parabola with different methods

I'm trying to find the shortest distance from point $(3,0)$ to the parabola $y= x^2$ using the method of Lagrange Multipliers (my practice), and by "reducing to unstrained problem in one variable" ...
0
votes
0answers
21 views

Maximum and Minimum Values Subject to Constraints (LaGrange)

Max and min values of $f(x,y,z) = yz+xy$ subject to $y^2 +z^2 = 289$ and $xy =5$. I know this will be a LaGrange problem and two constraints will be utilized so the formula you're looking for is the ...
0
votes
1answer
20 views

Undefined case in Lagrangian method

I am trying to finding the minimum distance between the point(1,1,0) and points on the sphere $$x^2+y^2+z^2-2x-4y=4$$ An easy way to do this is to graphical intuition and get the distance, since the ...
2
votes
2answers
37 views

Conditional Extremum, need help finding the extreme points in calculation.

Find the conditonal extremums of the following $$u=xyz$$ if $$(1) x^2+y^2+z^2=1,x+y+z=0.$$ First i made the Lagrange function $\phi= xyz+ \lambda(x^2+y^2+z^2-1) + \mu (x+y+z) $, now making the ...
0
votes
0answers
17 views

Arrow–Debreu Model, Optimization for n-agents to determine security prices

Good day, My question is as to the following Lagrage equations system: $ \scriptsize \left\{-\lambda \left(p_1 \left(2.14865 y_1-0.675676 y_2-3.74324\right)+p_2 \left(0.878378 y_1+0.189189 ...
0
votes
0answers
14 views

SVM Soft Margin Lagrange form

I study the Lagrange multipliers form of SVM. I am particulary interested in values that $\alpha_i$ can get. The following is the Langange multipliers form of hard margin SVM. $min_{w,b} ...
0
votes
0answers
48 views

Getting explicit expression of function from the dual function

Considering the following problem : $$ minimize_{x_1,x_2} \ -2x_1+x_2 \\ subject \ to \ x_1+x_2=\frac52 \\ (x_1,x_2) \in X ,\\$$ where $X=\{(0,0),(0,2),(2,0),(2,2),(\frac54,\frac54)\}$ The dual ...
1
vote
1answer
26 views

Maximizing the volume of a cylinder with given area

Consider a right circular cylinder of radius $r$ and height $h$. It has volume $V=\pi r^2 h$ and area $A=2\pi r (r+h)$. We are to use Lagrange multipliers to prove the maximum volume with given ...
5
votes
2answers
53 views

Can anyone tell me if this is correct?

Suppose that the temperature of a metal plate is given by $T(x; y) = x^2 +2x+y^2$, for points $(x, y)$ on the elliptical plate de fined by $x^2 + 4y^2 <= 24$. Find the maximum and minimum ...
1
vote
0answers
26 views

From utility function (3 products) to demand function (2 products)

I am struggling with this exercise and would appreciate some help. Consider two goods and a representative consumer whose utility is given by: $U(q_{0}, q_{1}, q_{2})= ...
0
votes
1answer
56 views

Lagrange multipliers

A current of $18$ amperes branches into currents $x$, $y$, and $z$ through resistors with resistances $5$, $7$, and $4$ ohms as shown. It is known that the current splits in such a way that the sum ...
0
votes
0answers
14 views

Optimize monotonic function in calculus of variations

I'm interested in the variational problem $$\min_{y} \int_a^b F(x,y(x),y'(x))dx \qquad \text{subject to} \quad -y'(x)\leq 0 \quad \forall x \tag{1}$$ i.e. $y(x)$ has to be monotonic. I ...
0
votes
0answers
57 views

Optimizing concave function over non-convex set

I have the following problem that I am looking advice on. Let $ \mathcal{F}$ be a convex subset of vector space $X$. The goal it to \begin{align*} \max_{x \in \mathcal{F}} f(x)\\ s.t. \ g(x) \le 0 ...