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

learn more… | top users | synonyms

0
votes
3answers
20 views

How to solve this system of equations (Lagrange Multipliers)

I was doing a question on Lagrange multipliers and stucked when trying to evaluate the point. The system of equations that I can't solve is this: $$y^2-x^2+3x-3y=0$$ $$-y^2-yx+3y-xy=0$$ I just ...
1
vote
0answers
18 views

Lagrange multiplier over two constraints

I'm having two constraints $g_{1}$=x+y-z+2=0 and $g_{2}$=$z^{2}$-$x^{2}$-$y^{2}$=0 and I want to determine the point on the intersection which is closest to the origin. The question asks us to use ...
2
votes
2answers
30 views

Lagrange Multipliers Calculus II Question

It is given me that $f(x,y)=x^2+-x+2y^2$ subject to $g(x,y)=x^2+y^2=1$ and asks for maximum and/or minimum. What I did... Equalize the partial derivatives and add the function $g$ to the system: ...
2
votes
0answers
68 views
+50

Find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
0
votes
0answers
33 views

Point of intersection closest to the origin

How do I find the point of intersection of $𝑥 + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = 𝑥^2 + 𝑦^2$ that is closest to the origin? I know I have to use the LaGrange multiplier in order to minimize the ...
0
votes
2answers
19 views

Lagrange method over two constraints

plane $x+y-z=-2$ intersects $z^2=x^2+y^2$ I need to use Lagrange multipliers to determine the point of intersection which is the closest to the origin. As far as I understand, to use Lagrange I need ...
0
votes
1answer
25 views

Maximized surface area of box with fixed length

Assuming we have a box given that the sum of all intervals is "a". What is the maximal surface area of the box? I know I need to use Lagrange multiplier but when I find the hessian matrix I get that ...
0
votes
0answers
5 views

Proving minimisation constraints using fundamental suspaces

I have the following question: Let $R^H=R$ be a positive semi-definite (non-negative) m$\times$m matrix, $L$ an n$\times$m matrix, $b$ a fixed m-vector, $y_c$ a fixed n-vector and $c$ a real ...
4
votes
1answer
109 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 $\langle \mathbf{u}_1,\mathbf{x}\rangle =0$ ...
2
votes
3answers
37 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 ...
4
votes
1answer
55 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 ...
2
votes
1answer
17 views

How to set up Lagrangian optimization with matrix constrains

Suppose we have a function $f: \mathbb{R} \to \mathbb{R} $ which we want to optimize subject to some constraint $g(x) \le c$ where $g:\mathbb{R} \to \mathbb{R} $ What we do is that we can set up a ...
2
votes
0answers
38 views

Regularized least squares

In Image Restoration, a true image $f$ (in vector form) can be related to degraded data $y$ through a linear model of the form $$y = Hf + n$$ where $H$ is a 2D blurring matrix and $n$ is a noise ...
1
vote
5answers
54 views

Optimize over unit circle to prove $|ax + by| \le \sqrt{a^2 + b^2}$

I have the following problem which, straight off the shelf, seems totally approachable. It's been giving me difficulty however: Let $a,b,x,y \in \mathbb{R}$, and suppose that $x^2 + y^2 =1$. ...
2
votes
0answers
45 views

Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$max~f(x)\quad s.t.~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can show ...
0
votes
2answers
27 views

Finding minimum using Lagrange multipliers

I'm having difficulty finding the minimum of this equation, $f(x,y,z) = xy + 2xz + 3yz$, subject to the constraint $xyz = 6$ and $x \ge 0$, $y \ge 0$, $z \ge 0$. I tried using Lagrange multipliers but ...
2
votes
0answers
21 views

Dual Decomposition with multiple coupling constraints

This is probably a a simple question, but have been stuck on this for a while and unable to figure out my issue from the standard Boyd/Vandhenbergen decomposition references. I am interested in dual ...
1
vote
1answer
30 views

If Lagrangian multipliers aren't a sufficient condition, what is sufficient for optimality in constrained problems?

According to Wikipedia, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. In particular, a typical problem says ...
0
votes
1answer
26 views

When using the method of Lagrangian multipliers, does it matter whether I subtract or add the lambda term?

As I understand it, the method of Lagrangian multipliers follows the form Minimize $f(x,y)$ subject to constraint $g(x,y)= c$ and involves an equation of the form $L(x,y,\lambda) = f(x,y) + ...
1
vote
0answers
29 views

Sign of the Lagrange multiplier associated with an equality constraint

I am trying to determine conditions under which the Lagrange multiplier(s) associated with an equality constraint is(are) positive. In general, the multiplier of an equality constraint is not ...
0
votes
2answers
51 views

How to solve the function $\max \sum_{i=1}^n \log(x_i \cdot \mu)$ with $\sum _{j=1}^b \mu_j = 1$

$$ \max_{\mu} \sum_{i=1}^n \log(x_i \cdot \mu)\qquad\text{with}\qquad \sum _{j=1}^b \mu_j = 1,\qquad \mu_i \ge 0,\qquad x_{ij} \ge 0 $$ The function is shown as above, where $x_i$ and $\mu$ are ...
1
vote
0answers
36 views

Continuity of Lagrange multipliers

Suppose I have an optimization problem of the from $\min f(x)$, s.t. $g(x)=a$, where $f,g$ are real polynomials and $a\in\mathbb{R}$. Then using the Lagrange multiplier rule, I have to find the ...
2
votes
3answers
74 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 ...
1
vote
4answers
84 views

Finding minimum of $x_1^p+\cdots+x_n^p$ subject to $x_1+\cdots+x_n=1$

I want to find minimum of $f=x_1^p+\cdots+x_n^p$ ($p>1$) subject to $g=x_1+\cdots+x_n=1$. By Lagrange's multiplier, if it has a local extremum at $P$, it should satisfy $\nabla f(P)=\lambda \nabla ...
1
vote
0answers
29 views

Maximization question [duplicate]

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 ...
1
vote
0answers
108 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 ...
4
votes
1answer
51 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 ...
3
votes
2answers
79 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 ...
0
votes
0answers
15 views

Global minimization of the product $\prod_{i=1}^K (1-x_i)$ subject to $\sum_{i=1}^K x_i=1$

I am interested in finding the $x_i$ which minimize $$\prod_{i=1}^K \frac{1}{x_i}$$ where $x_i\in (0,1]$ and $\sum_{i=1}^K x_i=1$. In particular I see that if $K=1$ we get $x_1=1$, and if $K=2$ we ...
1
vote
2answers
26 views

Nash Bargaining Equilibrium with exponential utilities

I'm trying to derive the answer to the following question: Two players play the classic divide-the-dollar game, which is an imperfect information version of the ultimatum class of games. Utility ...
-1
votes
1answer
59 views

Langrage multiplier [closed]

A rectangular box having no top and having a prescribed volume V m3 is to be constructed using two different materials. The material used for the bottom and front of the box is five times as costly ...
2
votes
1answer
32 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...
0
votes
2answers
37 views

How do I find this distance?

Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2 = 1$ and the origin. This is what I've attempted so far: Maximize $x^2+y^2+z^2$ with respect to $x^2+xy+2y^2 = 1$. Using ...
0
votes
0answers
40 views

To find the Maximum and minimum value of f over square

Given function $f = (x+y)^2 - (x+y) +1$ .I have to find maximum and value of $f$ over square with unit side in first octant in xy-plane. I calculated $f_x $ and $f_y $ both came out to be ...
0
votes
1answer
28 views

Minimize squared distance to origin from a paraboloid

I have to use Lagrange multilpiers to find the minimum distance from the paraboloid with equation $z = \left({x-1/}{\sqrt{2}}\right)^2 + \left({y-1/}{\sqrt{2}}\right)^2$ to the origin, and from this ...
0
votes
0answers
36 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?
0
votes
3answers
215 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 ...
0
votes
0answers
39 views

How does this optimization problem satisfy Karush-Kuhn-Tucker Conditions?

I am following Andrew Ng's course notes on Support Vector Machines at: http://cs229.stanford.edu/notes/cs229-notes3.pdf There is something in these notes which I do not understand. SVM's basic ...
1
vote
0answers
32 views

Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
1
vote
1answer
20 views

how to setup the Lagrange multiplier for contraints such as $0\leq x \leq 1$

I need to find minimum of objective function $Q=f(x,y,z)$ inside a cube region. Hence the constraints are in the form $0 \leq x \leq 1$ and $0 \leq y \leq 1$ and $0 \leq z \leq 1$. Should the ...
0
votes
0answers
24 views

Minimization of a weighted least-squares problem by Lagrange multiplier method

Problem: Let $Y = (y_1, y_2, \dots, y_m) \in \mathbb{R}^{m \times n}$ and $k \in \mathbb{R}^{m}$ satisfy $\sum_{i=1}^{m} k_i =1$ and $k \geq 0$. Show that $x=Yk$ is a minimizer for $h(x) = ...
0
votes
2answers
26 views

Multivariable functions application

We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought. Two squares, both with length $l=1$ intersect ...
0
votes
1answer
25 views

Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
0
votes
1answer
42 views

Solve the system of equations for x and y

Solve the system of equations for $x$ and $y$: $$ \left(\frac{x}{8-2y}\right)^2 - \left(\frac{y}{-4}\right)^2=4 \\ \frac{x}{8-2y} + \frac{x}{-2}=1 $$ I used Lagrange multipliers with multiple ...
1
vote
3answers
70 views

Lagrange Multipliers: Find $\min$ of $f(x,y)=3(x+1) +2(y-1)$ subject to the constraint $x^2+y^2=4$

Find the minumum value of the function $f(x,y)=3(x+1) +2(y-1)$, subject to the constraint that $x^2+y^2=4$. The problem states to use Lagrange Multipliers. In doing so I obtained the point ...
0
votes
1answer
25 views

How can I find the minimum distance between the origin $(0,0)$ and the curve $y=1-x^2$ using Lagrange multipliers?

I used the curve as the constraint and the origin as the point but I am not sure if that is correct.
3
votes
4answers
77 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 ...
0
votes
1answer
41 views

When writing the Lagrange function does it matter if the constraint has minus/plus sign?

My university teaches me like this: However everywhere I look on the internet there is either a 1) minus sign before the lambda 2) or the whole equation is written differently (by placing ...
1
vote
1answer
23 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
0
votes
0answers
49 views

Minimum of $x+y+z$ on $\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}$

Find the minimum of $x+y+z$ on $$\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}.$$ My first naive thoughts would be to consider setting up a nasty triple integral and evaluate it or ...