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Questions tagged [lagrange-multiplier]

This tag is for the questions on Lagrange multipliers. The method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

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How to solve second-order objective function with orthogonal constraint

I want to minimize $tr(P^TAP-P^TB)$ with the constraint $P^TP=I$. $P, A, B$ are all matrices, $I$ is the identity matrix, $tr$ means the trace of a matrix. P: size(m,l) A: size(m,m) non-symmetric ...
popura's user avatar
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3 votes
2 answers
129 views

Lagrange multipliers for an optimization problem

Consider the nonlinear program \begin{equation} \underset{\mathbb{R}^2}{\text{min}} f(x_1 , x_2 ) = x_1^2 - x_1 x_2 + x_2^2 - x_1 - x_2 \end{equation} subject to the conditions: \begin{equation} ...
Superunknown's user avatar
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Lagrange multiplier for coupling flux across subdomains

I am solving two pdes: (1) $\nabla\cdot(\kappa \nabla u)=0$ in $\Omega = \Omega_1\bigcup\Omega_2$ where $\Omega$ can be thought of as a unit square vertically divided into two equal halves. (2) $\frac{...
ihsel's user avatar
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0 answers
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How to determine which eigenvectors of the Lagrangian Hessian are constraints?

I am trying to implement a constrained optimisation algorithm where the constraints are not satisfied at the beginning. I am using a Lagrangian multiplier method: $$L(x, \lambda) = F(x) - \sum \lambda ...
S R Maiti's user avatar
  • 267
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1 answer
23 views

Min max problem with a bilinear form

My goal is to solve the problem $\min_y \max_x y^T Ax$ with constraints $x \geq 0$, $y \geq 0$ and $\sum x_i = \sum y_i = 1$ ($x,y$ are probability measures) Is there a general way to solve this? ...
badinmaths's user avatar
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68 views

Relation between values of $ξ_i$ and $\alpha_i$ in SVM?

I have a question in about a property of support vectors of SVM which is stated in subsection "12.2.1 Computing the Support Vector Classifier" of "The Elements of Statistical Learning&...
hasanghaforian's user avatar
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2 answers
52 views

Prove the following inequality where the constraint is an inequality too

For $a,b,c>0$ and $a^2 +b^2 +c^2 \leqslant a+b+c$ prove the following inequality and show when we have equality : $F=\dfrac{1}{a^2 +b^2 +c^2 } +\frac{a^3}{b} +\frac{b^3 }{c} +\frac{c^3 }{a} \...
the_bot_unknown's user avatar
-1 votes
2 answers
48 views

Maximize the strength of a beam [closed]

I have a similar question from here, however, my rectangular beam is cut from a cylindrical torus with elliptical sections such that the major axis is 24cm and the minor axis, 16cm. To find the value ...
mvfs314's user avatar
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1 answer
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Why there not occur a saddle point in Lagrange Min/Max when the point is regular?

We stated the Lagrange Theorem as follows: Let $U\subseteq \mathbb{R}^n,~ h\in C^1(U, \mathbb{R})$ be a function and $\varphi \in C^1(U, \mathbb{R}^k)$ the additional conditions. Let $a$ be a Maximum ...
NameNameName's user avatar
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Computational Difficulty with Solving System of Equations in an Optimization Problem with 3 constraints (2 active & 1 inactive)

I was given the following problem: Minimize: $$f = -2x + 3y^2$$ Subject to: $$g_1 = (x-1)^2 + y^2 > 1 \\g_2=(x-1)^2 + y^2 \leq 4\\g_3 =x \geq 0 $$ Currently, I am trying to find a minimizer ...
Kakakat's user avatar
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2 votes
2 answers
56 views

Identifying maximum and minimum in Lagrange multiplier problems via compactness of constraint

I am currently studying constrained optimization with Lagrange multipliers, and I am confused about an argument that is used in several examples. Say $f$ is the function we are trying to optimize ...
lightweaver's user avatar
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17 views

What is the value of dual/lagrangian variable of an infeasible problem?

I found some materials saying that: if the primal is unbounded, then the dual is infeasible; If the dual is unbounded, then the primal is infeasible; but it's possible for both the dual and the primal ...
Ruihao Wang's user avatar
2 votes
0 answers
33 views

Wrong sign in co-state of optimal control problem

Consider the following deterministic optimisation problem \begin{align} J(t) = \min_{c(t)} \ & \frac{1}{2} \int_0^\infty e^{-\delta u} \left( x(u)^2 + \lambda y(u)^2 \right) du \\ s.t. \ &c(t) ...
NC520's user avatar
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3 votes
1 answer
66 views

Lagrange Multipliers: How does one prove the existence of a global minimum?

Let $a > 0$. Using Lagrange Multipliers, I wish to solve the problem: \begin{align} & \text{find the global minimum of: } f(x) = \sum_{i=1}^{n} x_i^{-1} \\ & \text{subject to } g(x) = \...
V. Elizabeth's user avatar
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1 answer
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Are Lagrange multipliers applicable to Nesbitt's inequality?

See linked Nesbitt's inequality, after normalization, is equivalent with $$\frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c}\ge \frac{3}{2},$$ with $a, b, c$ positive and $a + b + c = 1$. What troubles me is ...
Natrium's user avatar
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Lagrange mutilplier and KKT theorem problem(maybe) with some probabilities

Suppose we want to maximise a expected utility function: $$E_1(u(C_1,C_2,C_3)) $$ subject to following constraints. There are two possible situations each with probability $\frac{1}{2}$. $$C_1 + S_1 = ...
Chang Henry's user avatar
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2 answers
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I don't uderstand why the Lagrange multipliers don't work here $f(x,y,z)=x^2-y-z$ with constraint $x^2-y^2+z=0$

When calculating the above lagrange multipliers you will find that the system has one solution where the point that solves it is $C (0,-1/2,1/4)$ which is $f(C)=1/4$. But if you calculate $f(0,0,0)=0$ ...
A Math Wonderer's user avatar
3 votes
2 answers
101 views

Lagrange Multiplier Problem Leads to Tricky System of Equations

I am a little stumped on the following problem. Problem: Minimize the value of $108a + 27b^2 + 4c^3 + d^4$ subject to the constraint $ab + bc + cd + da = 25$. I have attempted to solve this using the ...
kjamesxyz's user avatar
1 vote
1 answer
37 views

Proposition 3.2. of Bertsekas' paper about Lagrange multipliers

This is a problem about the proposition 3.2 of Bertsekas' paper below. Bertsekas D P, Ozdaglar A E. Pseudonormality and a Lagrange multiplier theory for constrained optimization[J]. Journal of ...
Mike Dai's user avatar
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0 answers
22 views

Lagrange for f(y,z) with constraint g(x,y,z)

I need to classify extrema of $f=(1+z^2)e^{-y^2}$ on constraint: $x^2+4≤8e^{-y^2-z^2}$ Clearly, a st. pt. for $f$ which is also on and within constraint is $(\pm2,0,0)$ Setting up lagrange: $0=λ(2x)$ $...
Nate's user avatar
  • 13
1 vote
2 answers
71 views

Lagrange mutiplier with multivariable function and constraint

As title suggests, I need to optimize a multivariable function with a constraint, specifically; $$f(x,y,z)=z^2e^{xy}, \text{w/ constraint S: } x^2+y^2+z^2≤1$$ clearly one st. pt. will be on the plane $...
Nate's user avatar
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1 vote
1 answer
89 views

Inconsistency with Lagrange Multiplier Method

I recently read a question here and attempted to solve it with the method of Lagrange Multiplier: If $x$ and $y$ are real number such that $x^2+2xy-y^2=6$, then find the minimum value of $(x^2+y^2)^2$...
Yatharth Shrivastava's user avatar
4 votes
1 answer
140 views

Find the extremals of the functional $ F(y) = \int_{1}^{2} \left( (y'(x))^2 x^2 + y^2(x) \right) \ dx $. Made a correction.

Find the extremals of the functional $ F(y) = \int_{1}^{2} \left( (y'(x))^2 x^2 + y^2(x) \right) \ dx $ subject to the conditions $y(1) = 0$, $y(2) = 0$ and the additional condition $ \int_{1}^{2} y^2(...
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Consider optimization problem with an equality constraint :

I am trying to solve a constrained optimization problem, it seems a bit confusing however, I have made an attempt $$\text{minimize}\ xy + yz $$ $$\text{subject to }$$ $$y^2+ z^2 -1 =0 \\ ...
Ninja's user avatar
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0 votes
1 answer
45 views

Optimization of Multivariable Functions Using Lagrange Multipliers [closed]

Q.) Given the function $f(x,y,z) = x^2+y^2+z^2$ find its local minima and maxima to the constrain $x+y+z = 1$ How do I properly set up the equations using the method of Lagrange multipliers for this ...
Krishna's user avatar
  • 33
1 vote
1 answer
39 views

Parameter control problem derived via Dirichlet principle / variational formulation of Poisson equation / Lagrange multipliers

The below is a distilled-down version of a more involved problem I am looking at. Suppose for simplicity that $\Omega = B_R(0) \subset \mathbb{R}^2$ and $R$ is very large. Let us further define $f_s\,\...
Pink and Floyd's user avatar
1 vote
0 answers
28 views

How to derive the dual of an optimization problem when the original variable appears in the constraint related to the Lagrangian multiplier?

Let be $d\geqslant 2$. Let $\rho$ be $d\times d$ positive semidefinite matrix with $\mathrm{Tr}\left[\rho\right]=1$. Let us consider the following problem: $$\begin{align} \max\quad&\sum_{x=1}^d\...
Tristan Nemoz's user avatar
0 votes
2 answers
71 views

Minimizing $f(x,y,z)$ subject to a quadratic constraint

Given that $C(x,y,z) = x^2 + 2 y^2 + 3 z^2 + x y + 2 x z + 3 y z + 5 x + 7 y + 10 z - 100 = 0 $ Find the minimum of $f(x,y,z) = 2 x^2 + 3 y^2 + z^2 + 5 x y + 2 x z + 4 y z + x + 2 y + z $ My Attempt: ...
Quadrics's user avatar
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1 vote
1 answer
54 views

Minimizing $f(x,y,z)$ subject to a linear constraint

If $x + 2 y + 4 z= 10 $, then what is the minimum of $$ f(x,y,z) = x^2 + 2 y^2 + 3 z^2 + x y + 2 x z + 3 y z + 7 z $$ And at what $(x,y,z)$ does this occur ? My attempt: Using Lagrange's multiplier ...
Quadrics's user avatar
  • 24.3k
1 vote
1 answer
32 views

Writing the Lagrange of a Normal Distribution

I am watching this Youtube video on how to derive the Multivariate Normal Distribution using the Principle of Maximum Entropy (https://www.youtube.com/watch?v=7qsB8ElrCC4 @ 3:13). Here, the Lagrange ...
konofoso's user avatar
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1 vote
0 answers
49 views

analytical solution to a toy-sized but interesting optimization problem

My question follows after the initial context. My research includes solving a subproblem many many times - maximizing the Euclidean distance between a pair of convex neighborhoods, e.g. between an ...
William M.'s user avatar
2 votes
0 answers
45 views

Prove that the point on one circle farthest from another circle must lie on the line connecting centers.

This question comes from a section in a text book about Lagrange Multipliers. The context of the question is such however, that it is clearly intended that the solution be based upon reasoning related ...
gnitsuk's user avatar
  • 43
1 vote
2 answers
46 views

Simplifying Minimization on $(x-y)^2-z^2=1$ : Possible?

Let $A=\left\{(x, y, z) \in \mathbb{R}^3 \mid(x-y)^2-z^2=1\right\}$, and I want to find all points in $A$ that are nearest to $(0,0,0)$. I tried to use Lagrange's method: \begin{equation} \mathcal{L}(...
bruno's user avatar
  • 425
3 votes
1 answer
87 views

Closed form for regularized Rayleigh quotient

Let $A \in \mathbb R^{n \times n}$ and $b \in \mathbb R^n$. Assume $A$ is full rank. I am interested in optimizing the following expression over all $x$ on the unit sphere: $$ \max_{x \in \mathbb S^{n-...
Vivek's user avatar
  • 319
1 vote
1 answer
45 views

Prove that $T_a M \subset ker(Df(a))$

I would like to show for an open subset $ U \subset \mathbb{R}^n$ and $f : U \to \mathbb{R}$ continuously differentiable and $M = \{x \in U | g_1(x) = 0, ..., g_r(x) = 0 \}$ with $r \leq n$. Let $a \...
MathMaestro's user avatar
0 votes
0 answers
26 views

Prove that the feasible set is non-empty.

Consider the following optimization problem: $$ \begin{alignat}{3} & \underset{{\bf x}}{minimize} \quad & f({\bf x}) &= x_{0,0} & & \\ & subject~to \quad & h({\bf x}) &...
Wojtek's user avatar
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0 votes
0 answers
29 views

Prove that the inequality constraints are active at the optimal solution.

Consider the following optimization problem: $$ \begin{alignat}{3} & \underset{{\bf x}}{minimize} \quad & f({\bf x}) &= x_{0,0} & & \\ & subject~to \quad & h({\bf x}) &...
Wojtek's user avatar
  • 1
2 votes
2 answers
77 views

linear optimization using lagrange multipliers

there are two sources of energy (say Wind and Solar) and two custumers (say X and Y). I want to calculate the minimum cost distribution of the flow knowing that both costumers must meet their demand (...
Marco Di Gennaro's user avatar
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0 answers
21 views

Lagrange multipliers imply critical point?

Conceptual problem ... In the lagrange multiplier method, let $x$ be a real n-space variable, looking to minimize $F$ subject to $G=0$, we find crictial points of $F + \lambda G$, sometimes called a ...
Ponder Stibbons's user avatar
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0 answers
25 views

Find the points of conditional extremum of the function

Find the points of conditional extremum of the function $u= xy+yz$ subject to the condition $x^2+y^2=2, y+z=2, x>0, y>0$ I expressed $y=2-z$ and combined the two conditions into one $x^2+(2-z)^...
Gleb Cloudy's user avatar
0 votes
4 answers
32 views

Find the minimum and maximum values of a three-variable function

Find the maximum and minimum values of the function $u=x^2+2y^2-3z^2$ within the region $x^2+y^2+z^2 \leq 50$ During an ordinary extremum analysis, I identified the point $(0,0,0),$ which lies inside ...
Gleb Cloudy's user avatar
1 vote
4 answers
73 views

Using Lagrange Multiplier with an inequality Calc 3

Using Lagrange Multiplier when testing for ABS extrema on inequalities. The Stewart textbook doesn’t have much. And following Paul’s Notes, if I read it correctly, the steps are: Check the inequality ...
user41592's user avatar
  • 143
1 vote
2 answers
85 views

A simple constrained optimization problem

Let $v\in \mathbb{R}^n$. Define $E(v) = \sum_{i=1}^n (v_i^2-1)^2$ be the energy to be minimized. Define the constraint $\sum_{i=1}^n v_i = cn$. This means the average value of all $v_i$'s is $c$. ...
900edges's user avatar
  • 2,039
1 vote
1 answer
30 views

Derivation of steepest descent direction in a Riemannian space as in Amari 1998 - Natural gradient works efficiently in learning

I am trying to follow the steps to proof the theorem 1 in the work of Shun-ichi Amari (1998): Natural gradient works efficiently in learning This is the relevant section of the proof: However when I ...
v.tralala's user avatar
  • 299
1 vote
1 answer
40 views

Maximize this function subject to "too general" conditions?

I'm facing a problem I cannot understand how to solve. I have to find the max of $f(x, y) = 2x + y$ over the set $V = \{(x, y) \in\mathbb{R}^2:\ \sqrt{x} + a\sqrt{y} \leq 2, x\geq 0, y \geq 0\}$ with $...
Heidegger's user avatar
  • 3,482
3 votes
2 answers
84 views

Find Maximum and Minimum distance from origin to $f(x,y)$ using the Lagrange method.

"By using Lagrange's method, find the points on the curve $10x^2 + 12xy + 10y^2 = 1$ that are nearest and farthest from the origin." I've used $f(x,y) = x^2+y^2 $from the distance formula $d$...
Oscar's user avatar
  • 33
3 votes
2 answers
68 views

Maximise $f(x,y)=x^2+y^2$ on contraint that looks like infinity sign

I would like to find the maximum of the function $f(x,y)=x^2+y^2$ on the constraint $x^2-y^2=(x^2+y^2)^2$. The level curve $h(x,y)=0$ of the function $h(x,y)=x^2-y^2-(x^2+y^2)^2$ looks like an ...
Apollo13's user avatar
  • 567
0 votes
1 answer
37 views

Proving concavity of the Lagrange dual function

The Lagrange dual function for an optimization problem of form $$\min f_0(\boldsymbol x)\quad\text{subject to}\quad f_i(\boldsymbol x)\le0,h_j(\boldsymbol x)=0\quad i=1,2\dots m,j=1,2,\dots p$$ with ...
reyna's user avatar
  • 2,224
1 vote
0 answers
107 views

Optimisation problem in two variables

I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$. Now, I solved the problem quite easily in a ...
Enrico M.'s user avatar
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2 votes
1 answer
51 views

On the value of $x$ for which a point mass falls off a curve.

Not sure if this an appropriate venue for this question, please close question as opposed to migrating to Physics SE, because it is not appropriate there, thanks. If we have a functional given by: $$I=...
Albertus Magnus's user avatar

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