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

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1answer
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Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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2answers
458 views

Maximum and minimum absolute of a function $(x,y)$

I want to know the maximum and minimum absolutes values of this function: $\ f(x,y)= 4x^2 + 9y^2 - x^2y^2 $ $\nabla f(x,y)=(8x-2xy^2,18y-2yx^2) $ I find these critical points: $\ ...
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2answers
152 views

Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?

The global maximum and the global minimum of the function $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ can be found using Lagrange multipliers. $\nabla f = \lambda \nabla g$ ...
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0answers
117 views

A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
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3answers
119 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 ...
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1answer
36k 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 ...
2
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1answer
163 views

Lagrange multiplier constrain critical point

When using Lagrange multipliers in an inequelity, $$ f(x,y) = x^2+y $$ with the constraint $$ x^2+y^2 \leq 1. $$ I have to find the critical points inside the "disk" right? I've done $$ f_x = 2x ...
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2answers
282 views

Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
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3answers
653 views

Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
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1answer
2k views

Lasso - constraint form equivalent to penalty form

We know that there are two definitions to describe lasso. Regression with constraint definition: $$\min\limits_{\beta} \|y-X\beta\|^2, \sum\limits_{p}|\beta_p|\leq t, \exists t $$ Regression with ...
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4answers
3k views

Is it possible for the Lagrange multiplier to be equal to zero?

I would like to find the extrema of the function $f(x,y)=x^2+4xy+4y^2$ subject to $x^2+2y^2=4$ using Lagrange Multipliers. Is it possible to get for the Lagrange multipliers the value zero? I don't ...
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1answer
971 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 ...
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1answer
2k views

Calculus of variations: Lagrange multipliers

Given a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$ where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the ...
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2answers
561 views

Minimize $x^TAx$, subject to $||x||=1$. Show that ${x^*}^TAx^*$ is the smallest eigenvalue of $A$ in magnitude.

I'm solving constrained optimization exercises for preparing my final exam. I got stuck at this question. $$ \begin{array}{ll} \text{min} & \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{s.t.} & ...
2
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1answer
56 views

Maximizing and minimizing dot products

Given 2 vectors $u,v \in \mathbb{R^n}$ such that $\|u\| = 1$ and $\sum_{i=1}^n v_i= c$ where $c<1$, I would like to maximize $$\sum_{i=1}^n u_i v_i \log (v_i)$$ and minimize $$\sum_{i=1}^n u_i v_i ...
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0answers
160 views

Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
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2answers
611 views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint ...
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3answers
101 views

Minimize : $\sqrt{(1+{1\over a})(1+{1\over b})}$ subject to $a+b=\lambda$.

Given positive real variables $a$ and $b$, find the minimum of $$f(a,b)=\sqrt{\left(1+{1\over a}\right)\left(1+{1\over b}\right)}$$ subject to $a+b=\lambda$ where $\lambda$ is a constant . [ISI ...
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0answers
34 views

Finding the Maximum and Minimum values w/constraint [duplicate]

I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is. Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ ...
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0answers
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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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2answers
80 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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3answers
135 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 ...
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2answers
555 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
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2answers
187 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 ...
4
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1answer
287 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
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4answers
49 views

In Lagrange Multiplier, why level curves of $f$ and $g$ are tangent to each other?

In Lagrange multiplier method, e.g. optimize a function $f(x_1, \dots, x_n)$ under a constraint $g(x_1, \dots, x_n) = 0$. There is a fact that $\nabla f$ is parallel to $\nabla g$ which is given rise ...
2
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1answer
2k 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" ...
2
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1answer
120 views

Clarification on optimization problem, continued

Background This is a follow-up to this question. The problem statement is the same: Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ ...
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2answers
251 views

Using the method of Lagrange multipliers, find the extreme values of a function

Using the method of Lagrange multipliers, find the extreme values of the function $f(x,y)= \frac{2y^3}{3} + 2x^2 +1$ on the ellipse $5x^2 + y^2 = 1/9$ . Identify the (absolute) maximal and minimal ...
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1answer
68 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
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3answers
155 views

Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
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1answer
4k views

First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
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2answers
189 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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3answers
260 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
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2answers
133 views

Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.

This is not homework, but it is in my book and I find it hard to solve: Determine the points where $f$ is has a local minimum/maximum. Determine if it strong/weak and absolute/relative and ...
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1answer
33 views

Find the extrema of $\sum_{i=1}^n u_i v_i \log \left| \frac{v_i}{u_i} \right|$

This question is similar to the following one: Maximizing and minimizing dot products. However there are significant differences, hence I opened a new question. Maximize and minimize $$\sum_{i=1}^n ...
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2answers
36 views

How to design a closed rectangular box of minimum cost using Lagrange Multipliers

Suppose the box is to be of volume $V_0$ cubic cm; and the cost of material for the front and back sides is $b$ dollars per square cm, $c$ dollars per square cm for the left and right two sides, ...
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2answers
31 views

By using slack(free) variable, solutions of both min and max problems

$$F(x,y)= 2x^2-3y^2-2x$$. $$\text{s.t.} \ \ \ \ \ \ x^2+y^2\le 1$$ By using slack variable (free variable,), I want to solve minimum and maximum problems. I want to solve this problem for maximum ...
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1answer
53 views

Minimization involving equality constraints

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}} \hspace{4mm} \big(\left( \mathbf{y}^T ...
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1answer
215 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
0
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1answer
101 views

using the method of LaGrange multipliers find the extreme values of the function

using the method of LaGrange multipliers find the extreme values of the function $f(x,y)=x^3+4y^2$ with constraint $x^2+y^2=1$ currently I have that: $3x^2=2Lx$ which leads to $x=0$ or $L=3/2$ and: ...
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2answers
95 views

Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
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1answer
63 views

Can I use Lagrange Multipliers with inequality constraints?

Suppose I had a problem: Maximize $f(\bf{x}) $ subject to the contraints $g_i(\bf{x})< b$ Can I still use Lagrange multipliers? My text says that the constraints need to be equalities.
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544 views

Lagrange Multipliers have no solution

$f(x,y)=2x+y$ subject to constraint $x+y=m$. $(2,1)=\lambda(1,1)$ but this does not satisfy $x+y=m$ So there are no solution?
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1answer
62 views

Theorem of Lagrange multipliers - Extremas of $f$ [duplicate]

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are ...