For questions on Lagrange multipliers, a strategy to solve constrained optimisation problems.
12
votes
2answers
527 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 :
$$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$
6
votes
2answers
72 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:
$\ ...
4
votes
2answers
56 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
0answers
69 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
1answer
149 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 ...
3
votes
1answer
108 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 ...
2
votes
4answers
123 views
Minimize $\sum a_i^2 \sigma^2$ subject to $\sum a_i = 1$
$$\min_{a_i} \sum_{i=1}^{n} {a_i}^2 \sigma^2\text{ such that }\sum_{i=1}^{n}a_i=1$$ and $\sigma^2$ is a scalar.
The answer is $a_i=\frac{1}{n}$.
I tried Lagrangian method. How can I get that ...
2
votes
1answer
47 views
Finding the critical point of $\int_0^1(f(t))^2dt$ subject to $\int_0^1f(t)dt=k$
I wish to find the critical point of the functional $F[X]=\int_0^1(f(t))^2dt$ subject to $\int_0^1f(t)dt=k$ for a constant $k$. I read something about using a Lagrange multiplier to convert it to a ...
2
votes
1answer
33 views
Minimizing Mean Squared Error for Exponential Function
I have a function that I'm trying to model using an exponential function and I'm trying to determine the constants for the exponential. I know I could optimize it using trial-and-error in R or another ...
2
votes
1answer
149 views
Lagrange multipliers - maximum and minimum values given constraint
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.)
$ \ \ f(x, y, z) = xyz \ ; \ \ x^2 + 2y^2 ...
2
votes
1answer
84 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 - ...
2
votes
0answers
58 views
Optimization with symmetric matrix constraint
Consider the following optimization problem:
''Minimize some objective $f(A)$ over all matrices $A$ s.t. $A \mathbf{1} = \mathbf{1}$ and $A = A^T$.''
I wonder in which ways one can handle the ...
1
vote
2answers
62 views
Finding the maximum of a function using Lagrange multipliers
Consider the function $x^a e^{-x} y^b e^{-y}$, over the triangle defined by $x \ge 0, y \ge 0, x + y = 1$.
How can one find the maximum of such a function in terms of $a,b \ge 0$? This deeply ...
1
vote
2answers
34 views
Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.
This is no 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 ...
1
vote
3answers
62 views
Constrained optimization max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$
max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$
I need to solve this by the Kuhn Tucker conditions without using concavity of the Lagrangian.
1
vote
4answers
351 views
Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. *help needed please*
Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant.
This is what i've done so far....
Let $a,b,c$ be some points that the ...
1
vote
2answers
2k views
Find Max/Min volume of rectangular box using Lagrange Multipliers
Can anyone help me solve the problem below? This is question number 14.8.42 in the seventh edition of Stewart Calculus.
Here is the problem definition:
"Find the maximum and minimum volumes of a ...
1
vote
1answer
48 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 ...
1
vote
1answer
55 views
Points nearest and farthest from origin, Lagrange Multipliers
The plane $x + y + 2z = 30$ intersects the paraboloid $z = x^2 + y^2$ in an ellipse.
Find the points on the ellipse that are nearest to and farthest from the origin.
I know you have to find the ...
1
vote
1answer
108 views
What is the intuition behind the Lagrange multiplier?
I know that the minimum or maximum point is achieved when the gradient in the constraint function is parallel to the gradient on the $f$ function. But why the Lambda is called the Lagrange ...
1
vote
1answer
44 views
Which $n$-gons, inscribed in a circle, have the largest area?
Let $n$ be an integer, $nā„3$, and $H$ be a circle. Which (convex) $n$-gons, inscribed in $H$, have the largest area?
How can I use Lagrange multipliers to solve this?
1
vote
4answers
267 views
Use Lagrange multiplier to find absolute maximum and minimum
Use Lagrange multiplier to find absolute maximum and minimum of $f(x,y) =x^2+xy+y^2, x^2+y^2 =8$.
What i've done so far..
$f_x = \lambda g_x \Rightarrow 2x+y =\lambda2x, \\f_y = \lambda g_y ...
1
vote
0answers
80 views
K.K.T. conditions, Lagrangian gradient not defined for zero.
When I write the K.K.T. conditions for the problem I have, I get the following expression for the gradient of the Lagrangian:
$$\frac{\partial
\mathcal{L}}{\partial x} = -
\frac{\sqrt{x} +
...
1
vote
0answers
106 views
Lagrange multipliers for finding geodesics on a sphere
The problem statement, all variables and given/known data
Find the geodesics on a sphere $g(x,y,z)=x^{2}+y^{2}+z^{2}ā1=0$
arc-length element $ds=\sqrt{dx^{2}+dy^{2}+dz^{2}}$
Relevant equations
...
1
vote
0answers
47 views
Closed form for Lagrange dual
Can Lagrange dual always be computed in closed form? Can you give me a simple example where the dual is not analytically computable?
0
votes
1answer
65 views
lagrange multiplier with interval constraint
Given a function $g(x,y,z)$ we need to maximize it given constraints $a<x<b, a<y<b$.
If the constraints were given as a function $f(x,y,z)$ the following equation could be used.
...
0
votes
1answer
1k views
Use LaGrange multipliers to find maximum and minimum values
I am having trouble understanding how to solve the problem below. Can anyone show me how to solve this? Here is the problem definition:
"Use LaGrange multipliers to find the maximum and minimum ...
0
votes
2answers
49 views
Solution to a system of symmetric equations
After applying the Lagrange multiplier method, I got the following system of equations, which is quite symmetric:
$(x+y)^2 + (x+z)^2 = \frac{2}{3} \lambda x$
$(y+x)^2 + (y+z)^2 = \frac{2}{3} \lambda ...
0
votes
1answer
57 views
HELP please with Optimization with constrain using lagrangian
I am reading this book on optimization and they present the following problem: Lisa wants to maximize her utility U(q1,q2) subject to a budget constrain, budget constrain is $p1*q1+p2*q2=I$.
Ok , I ...
0
votes
2answers
44 views
Finding constrained optima of $f(x,y) = x^3 - y^3 -x$
For a function $$f(x,y) = x^3 - y^3 -x$$, the minimum and maximum under the constraint $$x^2 + y^2 =1 $$
is searched.
So as usual, my approach is to set up the Lagrangian and the FOC:
$$ L(x,y, ...
0
votes
3answers
55 views
Finding the max/min distance from an ellipse to a line (Lagrange Multiplier Method)
An ellipse is specified $ x^2 + 4y^2 = 4$, and a line is specified $x + y = 4$. I need to find the max/min distances from the ellipse to the line.
My idea is to find two points $(x_1, y_1)$ and ...
0
votes
1answer
187 views
Find the point on the line of intersection of the two planes using Lagrange's method.
Use Lagrange's method to find the point on the line of intersection of the two planes $a_1x+a_2y+a_3z+a_0 = 0, b_1x+b_2y+b_3z+b_0 = 0$ which is nearest to the origin. you may also assume that the tow ...
0
votes
1answer
52 views
What is the constraint in this LaGrange Multipliers ??
$x$ and $y$ are real numbers where satisfied the equation $x^2+y^2+xy-3x-3y-9=0$
Find the max. and min. values of $x^2+y^2$
I don't know how to find the constraint
0
votes
0answers
36 views
Right Triangles and Lagrange Multipliers
Suppose that you have a right triangle $a^2+b^2=c^2$ with integral sides. Given a perimeter $p=a+b+c$, how can you use Lagrange multipliers to determine the maximum length of $a$?
0
votes
1answer
22 views
Need help with Lagrange Multipliers in Support Vector Machine
In linear support vector machine with binary classification if we know all the support vectors, geometrically we can create hyperplanes on both sides of the main hyperplane using these support vectors ...
0
votes
0answers
36 views
Minimize a three variable function using Euler-Lagrange theorem
I have to minimize the function $g(x,y,z)=x^2+y^2+2z^2-x-yz$ in two cases. First, with the restriction $x+y+z=35$, and after with $x+y+z\geq35$
I know how to do this using Lagrange multipliers ...
0
votes
0answers
21 views
Trouble with geometrical application of Lagrange multiplier
Let $A\in\mathbb R^{n\times n}$ be positive-definite and $\langle Ax,x\rangle=1$ be the equation of an ellipsoid $M\subset\mathbb R^n$. Use Lagrange multipliers to prove that the greatest distance of ...
0
votes
0answers
32 views
Lagrange Multipliers and max area
You have a straight line of length b. You want to connect the ends of this fence so as to enclose a maximum area. You have a cost constraint. In the area between x=0 and x=b/2 costs 1 dollar per ft ...
0
votes
0answers
40 views
What do I need to know to understand Lagrange multipliers?
I've seen Lagrange multipliers used as a powerful method for tackling inequalities and some IMO problems, and I'm aware that it's a part of calculus. I'm currently taking BC Calculus in high school, ...
0
votes
0answers
199 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 ...
0
votes
1answer
42 views
Lagrange multipliers for x,y,z
I have this question, I have run completely blind into.
Find by Lagrange multipliers the volume V=xyz of where the largest box with sides adding up to x+y+z = k.
I have found the gradient of V:
...
0
votes
0answers
37 views
Infinite dimensional optimization--Specific example
I would appreciate some help with the following functional optimization, arising in an economic model.
Objective function for maximization:
$\int_0^1\,K\alpha_t\phi_t-(\frac{1}{Z}-1)\alpha_t\,dt$.
...
0
votes
1answer
25 views
Multiple or scalar Lagrange multipler, which is a simpler method?
Without loss of generality, consider minimizing
$F(x,y,z)$
subject to constraints
$G(x,y,z)=0$
$H(x,y,z)=0$
Multiple constraints can be reduced to a single one:
$G^2+H^2=0$
Then, Lagrange ...
0
votes
1answer
32 views
non convex optimisation
\begin{eqnarray}
{\textbf{maximise}} \hspace{2mm} Ar^{-(a+b)} + Br^{-(a+b+c)}-C \nonumber
\end{eqnarray}
such that,
\begin{eqnarray}
c= l(h-m_{0}) \nonumber\\
m_{1} \leq h \leq m_{2} \nonumber\\
...
0
votes
1answer
290 views
Constrained Optimization - Lagrange Multipliers (Example)
Let $f(x, y, z) = xyz$
$h1(x, y, z) = x + y + z ā 4$5
and $h2(x, y, z) = 2x ā y$.
Goal:
Minimize $f(x, y, z)$ subject to $h1(x, y, z) = 0$ and $h2(x, y, z) = 0$.
First part: Show that every ...
0
votes
1answer
78 views
Lagrange method - non-linear system of equations
I have to compute optimal parametres of truncated cone so that its Volume is fixed (lets say it is 1) and its surface is minimal using Lagrange method
These are equations desribing my object:
...
0
votes
0answers
84 views
Optimization problem for a probability density function
I have an minimization problem about a discrete probability density function $p=[p_1,p_2,\cdots,p_n]$
$$E(p) = \left(\sum_{i=1}^np_i(x_i-y)\right)^2+2\sum_{i=1}^np_i\omega_i$$
with constraints that
...
