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

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17
votes
3answers
653 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be $...
12
votes
2answers
617 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 : $$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$
11
votes
1answer
4k 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 ...
10
votes
2answers
447 views

I found only one critical point using Lagrange multipliers. Must it be a minimizer?

I am trying to minimize $$V(x,y,z) = \frac {a^2b^2c^2}{6xyz}$$ subject to $$\frac {x^2}{a^2} + \frac{y^2}{b^2} + \frac {z^2}{c^2} = 1$$ and for $x,y,z>0$. I found one critical point; ...
10
votes
4answers
250 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
3answers
4k 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 ...
7
votes
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 ...
7
votes
4answers
232 views

Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n ...
7
votes
0answers
134 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 ...
6
votes
3answers
316 views

Lagrange Multipliers Example

Minimize $$f(x,y) = x^2+y^2$$ subject to the constraint $xy=3$. I know the formula for Lagrange multipliers to be $\nabla f = \lambda \nabla g$ so we get a system of equations like this $$\begin{...
6
votes
1answer
1k 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
459 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: $\ (0,0);(3,2);(-3,...
6
votes
3answers
322 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 ...
6
votes
3answers
2k 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
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 ...
6
votes
3answers
381 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
325 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 ...
5
votes
4answers
104 views

Why does taking derivatives of $L$ in Lagrangian multiplier problems let me find solutions to optimizations problems?

Consider the problem Maximize $f(\mathbf{x})$ subject to $g(\mathbf{x})=c$ Using the method of Lagrangian multpliers, I would set up a Lagrangian like $$L = f(\mathbf{x})-\lambda (g(\mathbf{x})...
5
votes
2answers
89 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
1k 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 $f(x_1,x_2,...,x_n)...
5
votes
2answers
83 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} &...
5
votes
1answer
207 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
68 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
198 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 ...
5
votes
2answers
68 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
3k 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
729 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 $f_1(x_1,\dotsc,x_n)=c_1,\dotsc,f_k(x_1,\dotsc,x_n)=c_k$ ...
4
votes
2answers
321 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
2answers
2k 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
699 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 $x^*=4....
4
votes
2answers
133 views

Radius of a largest circle inscribed under $y=\frac{1}{(1+x^2)^n}$, closed form

The curve $y=\frac{1}{1+x^2}$ has an obvious connection to circles, because it's the derivative of the arctangent function. Besides, if we inscribe a circle under it, its radius is exactly $R=\frac{1}...
4
votes
1answer
98 views

Is this a known result?

I heard the following result and I am wondering if anyone can verify its correctness and also provide a source to cite. If the Lagrangian $L(x,\lambda)$ is convex in $x$ at the optimal Lagrange ...
4
votes
1answer
296 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
40 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 $$ h(x,y,z) = x+y+z-2 = 0 $$ ...
4
votes
1answer
68 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 H_0^1(\...
4
votes
1answer
95 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: $$\sup_{(...
4
votes
1answer
88 views

Why is the Lagrange Multipliers Theorem not working?

Consider the function $h: K \to \mathbb{R}$ where $K := \{x \in \mathbb{R}^3:x,y,z \geq 0, x+2y+3z\leq 6\}$. $h$ is defined as: $$ h(x) = xe^{(x+2y+3z)} $$ Find the supremum and the ...
4
votes
2answers
117 views

Finding maximum and minimum with 2 constraints

Let $C$ be the curve of intersection of the plane $x+y-z=0$ and the ellipsoid $$\frac{x^2}4+\frac{y^2}5+\frac{z^2}{25}=1$$ Find the points on $C$ which are farthest and nearest from the origin When ...
4
votes
0answers
153 views

Positive Lagrange multipliers of an equality constraint

Consider the problem \begin{align} \max_{x\in\mathbb{R}^n} f(x)\\ \text{subject to }\quad h(x) = 0\\ x\in X \end{align} where $X$ is a convex and compact subset of $\mathbb{R}^n$. I also know that ...
4
votes
0answers
80 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 ...
4
votes
1answer
230 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
108 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
0answers
227 views

Calculus of variations: Isoperimetric and holonomic constraints.

A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most ...
4
votes
1answer
133 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
160 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 $g(x,y)=(y-1)^2=0$...
3
votes
3answers
62 views

Find minimum and maximum on range

$f(x,y)=x^{4}-x^{2}+y^{2}$ $B={(x,y)\in \mathbb R, x^{2}+y^{2}\leq 1 }$ I should find minimum and maximum of this function on the range B. I tried it with Lagrange Multiplier and I got these points ...
3
votes
4answers
60 views

Finding the Maximum value.

Maximize $xy^2$ on the ellipse $b^2x^2 +a^2y^2= a^2b^2$ The steps I tried to solve: $$\nabla f = (y^2,2yx)\lambda\qquad g = (2xb^2,2y^2a^2)\lambda$$ $$y^2= 2xb^2\lambda$$ $$2yx= 2y^2a^2\lambda$$ ...
3
votes
3answers
88 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
119 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
2answers
58 views

How to maximize the function $f(x,y)= x^2+2y^2$ subjected to constraint $y-x^2+1=0$?

I want to maximize the function $f(x,y)= x^2+2y^2$ subjected to constraint $y-x^2+1=0$ Using Lagrange multipliers $$2x=\lambda(-2x) $$ hence $\lambda=-1$ $$4y=\lambda $$ hence $y=\frac{-1}{4}$ So the ...