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

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12
votes
2answers
601 views

the least value for :$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$

For every $a,b,c$ non-negative real number such that:$a+b+c=1$ how to find the least value for : $$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$
9
votes
4answers
239 views

Maximum of $(1-q_1)(1-q_2)\ldots(1-q_n)$

I'm trying to find the maximum of $(1-q_1)(1-q_2)\ldots(1-q_n)$ where $n\ge 2$, on a the set $\{(q_1,\ldots , q_n) :q_1^2+q_2^2+\ldots+q_n^2=1 \ q_i\ge 0 \}$ (With the condition $q_i\ge0$ this is just ...
7
votes
1answer
3k views

Constrained variational problems intuition

Problem: minimise $F(x,y,y')$ over $x$, constrained by $G(x,y,y')=0$. $$J_1(x,y,y')=\large \int_{x_0}^{x_1}F(x,y,y')+ \lambda (x) G(x,y,y')dx$$ I understand the Euler-Lagrange equation and Lagrange ...
6
votes
4answers
2k 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 ...
6
votes
1answer
353 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 ...
6
votes
2answers
407 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: $\ ...
6
votes
3answers
1k views

Lagrange multipliers and KKT conditions - what do we gain?

I'm working through an optimization problem that reformulates the problem in terms of KKT conditions. Can someone please have a go at explaining the following in simple terms? What do we gain by ...
6
votes
3answers
339 views

Lagrange multiplier method, find maximum of $e^{-x}\cdot (x^2-3)\cdot (y^2-3)$ on a circle

I attempted to design an exercise for my engineer students and couldn't solve it myself. Maybe here are some experts in calculus who have some better tricks than I do: The exercise would be to ...
5
votes
2answers
76 views

Lagrange Multiplier Question and my attempt

Question is Find the extrema of $xyz$ when $x+y+z=a$ , a>0. Strating with usual Lagrange Multiplier method i get $f_x$ = $yz$ +$\lambda$ =0 $f_y$ = $xz$ +$\lambda$ ...
5
votes
1answer
127 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
5
votes
1answer
54 views

How to use Lagrange Multiplier in this question?

I have to find absolute maximum and minimum values of $f(x,y)$ = $4x^{2} + 9y^{2} -8x - 12y + 4 $ over rectangle in first quadrant bounded by lines $x=2 , y=3$ and coordinate axes I have checked ...
5
votes
2answers
54 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 ...
4
votes
2answers
2k views

Simple explanation of lagrange multipliers with multiple constraints

I'm studying support vector machines and in the process I've bumped into lagrange multipliers with multiple constraints and Karush–Kuhn–Tucker conditions. I've been trying to study the subject, but ...
4
votes
2answers
235 views

Lagrange multipliers method

I am currently doing some exercise on the Lagrange multipliers methods and have come upon some confusion. following my lectures notes is says: $$L= f(x,y) + \lambda g(x,y) $$ and in some online ...
4
votes
2answers
163 views

Lagrange multiplier - space probe

i am stuck on this question which uses the Lagrange multiplier. I am trying to construct the equations using the partial derivatives but the $x$'s and $y$'s cancel. can anyone help? A space probe in ...
4
votes
1answer
486 views

How to use lagrange multipliers here?

I have a simple QP as below: $\min L(x,y) = (x-5.1)^2+y^2$ such that $(x-3)^2+y^2\geq1$ $(x-5.3)^2+y^2\geq1$ $(x-7)^2+y^2\geq1$ Intuitively, I think the optimal solution of the problem is ...
4
votes
1answer
179 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 ...
4
votes
1answer
1k views

Pointwise infimum of affine functions is concave

So I was just starting on convex optimization and was having a slightly hard time visualizing the lagrangian being always concave because it is the pointwise infimum of a family of affine functions. ...
4
votes
1answer
60 views

The constraint subset of $H_0^1(\Omega)$ is a $C^1$-submanifold.

This problem comes from the constraint problem in CoV. (the lagrange-multiplier case) Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We define the sub-manifold $$ M:=\{u\in ...
4
votes
1answer
542 views

Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
4
votes
1answer
93 views

A question about Lagrange optimization.

This is a statement from a finance textbook - I find it pretty clear everywhere else, but this particular part I am clueless. Hopefully you guys can figure it out. The problem is solving: ...
4
votes
1answer
1k 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 ...
4
votes
0answers
54 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 ...
4
votes
1answer
50 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} ...
4
votes
1answer
99 views

Proving Lagrange method by using Implicit Function Theorem

I am trying to show the proof of the Lagrange multiplier method. According to this in general, if $f$ and $g$ are $D+1$ dimensional functions such that $f,g : \mathbb{R}^{D+1} \mapsto \mathbb{R}$, and ...
4
votes
0answers
81 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
4
votes
1answer
119 views

Use Lagrange's method to find the maximum value

Use Lagrange's method to find the maximum value of $\langle A\mathbf{x},\mathbf{x}\rangle$ subject to condition $\langle \mathbf{x},\mathbf{x}\rangle=1$ and $\langle \mathbf{u}_1,\mathbf{x}\rangle =0$ ...
3
votes
3answers
131 views

Why Lagrange multipliers don't help to find the minimum of $f(x,y)=x^2+y^2$ with the constraint $y=1$?

Please help me understand why the following doesn't work. Say I want to find the minimum of the function $f(x,y)=x^2+y^2$ with the constraint $y=1$. So I declare the helper function ...
3
votes
3answers
57 views

find extreme values of $\cos(x)+\cos(y)+\cos(z)$ when $x+y+z=\pi$

How can I find the maximum and minimum of $\cos(x)+\cos(y)+\cos(z)$ if $x,y,z\geq0$ such that they are vertices of a triangle with $x+y+z=\pi$. I don't know how to start, but I feel like the Lagrange ...
3
votes
2answers
156 views

Lagrange multiplier sign issue

When one has a function of more variables $f(x_1,\dotsc,x_n)$ and wants to find its maxima and minima on a subset of $\mathbb{R}^n$ defined by $f_1(x_1,\dotsc,x_n)=c_1,\dotsc,f_k(x_1,\dotsc,x_n)=c_k$ ...
3
votes
2answers
99 views

Help with Optimization Problem: Matrix Calculus

Can someone please help me with this problem? I am clueless :( $$ \left\{ \begin{array}{rclrcl} \min f(u) &=& u^tAu\\ \text{s.$\,$t.} \sum_{j=1}^n u_j &=& 0,& ...
3
votes
3answers
148 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
3
votes
4answers
80 views

Maximum and minimum of $f(x,y)=xy$ when $x^2 + y^2 + xy =1$

It is asked to find the maximum and minimum points of the function $$f(x,y)=xy$$ when $x^2 + y^2 + xy=1$ I've tried Lagrange and obtained $$\lambda = \frac{y}{2x+y}=\frac{x}{2y+x}$$ but what ...
3
votes
1answer
67 views

Doubt on a paragraph regarding Lagrange's multiplier.

I've a topic in my notes "The method of Lagrange's multipliers" which is described as follows: Let $U$ be an open set in $\mathbb R^n$.Let $f\in C^1(U,\mathbb R)$ and let ...
3
votes
2answers
282 views

Maximizing Area of Triangle in Circle

I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange ...
3
votes
3answers
69 views

Maximize $x^2+y^2+z^2$ on $x^2+y^2+4z^2 = 1$

Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$. I started by the standard ...
3
votes
2answers
293 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.} & ...
3
votes
2answers
202 views

How to use Lagrange Multipliers, when the constraint surface has a boundary?

The method called Lagrange Multipliers is used to find critical points of $f(x_1,x_2,\ldots,x_n)$, when $f$ is constrained to the level set $S = \{ x\in \mathbb{R}^n \, | \, g(x_1,x_2,\ldots,x_n)=0 ...
3
votes
1answer
50 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 ...
3
votes
2answers
58 views

Can Anyone help me with Lagrange multiplier problem

I need to find absolute maximum and minimum of thi function $$F(x,y) = x^{2} - y^{2} - 2y$$ over $$R = \{ (x,y)\ |\ x^{2} + {y^2} \leq 1\} $$ Thanks for help
3
votes
1answer
92 views

Explain Lagrange multipliers?

I am having serious issues with comprehension of this method. In particular, I don't understand the conditions. Thus far, I think it's something like; Given an objective $f: A \to \mathbb{R}^1$ and ...
3
votes
1answer
728 views

Largest box fitting inside an ellipsoid

Find the volume of the largest box with sides parallel to the $xy$, $xz$, and $yz$ planes that can fit inside the ellipsoid $(x/a)^2 + (y/b)^2 + (z/c)^2 = 1$. My answer: We want to maximize $f(x,y,z) ...
3
votes
3answers
110 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: $$ \nabla f = \lambda \nabla G $$ ...
3
votes
3answers
53 views

Finding the distance from ellipsoid to plane

I'm having problems with finding the distance from the ellipsoid $x^2+y^2+4z^2=4$ to the plane $x+y+z=6$. The question hinted that I'm supposed to find the distance from a point to the plane and ...
3
votes
3answers
119 views

Geometric interpretation of duality and Slater's condition

I am trying to study about optimization problems, Lagrange duality and related topics. I came across some presentation on the net, which claims to show the geometric interpretation of the duality and ...
3
votes
1answer
108 views

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. ...
3
votes
3answers
140 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 ...
3
votes
1answer
81 views

Can this system of equations be solved?

I'm doing some Lagrange multiplier examples that I thought up and I was trying to think of an example where the method of 'solving the constraint explicitly' fails. I'm trying to maximize ...
3
votes
1answer
722 views

Lagrange Multipliers for Function Spaces

For some constant $A > 1$ I am trying to solve the constrained minimization problem minimize $F(u)$ in $C$ subject to $H(u) = 0$. Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx - ...
3
votes
0answers
35 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 ...