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

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3answers
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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 ...
0
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1answer
17 views

Lagrange condition and second-order conditions

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is ...
0
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0answers
24 views

Extreme value Lagrange multiplier (max or min?)

I am to determine the the range of the volume of a tetrahedron enclosed by the coordinate axes and a tangentplane on the ellipsoid $x^2 + 2y^2 + 3z^2 = 1$. The volume of the tetrahedron can be derived ...
1
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0answers
25 views

Inequality constraints in Lagrangian

I was reading Lagrangian multipliers . In the above text I can't understand why $\lambda \ge 0 $ for $g(x)\ge0$ and vice versa . Can anyone give me the explanation to this ?
2
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0answers
21 views

Question about Lagrange multipliers and a basic example

I am trying to understand the Lagrange multipliers from reading the Wikipedia page. Then I tried to apply this to the problem of finding a discrete probability distribution which is the solution of ...
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1answer
26 views

lagrange multiplier---minimize

minimize $f(x,y,z)=x^2 + y^2+z^2 $ constraint is $x^3+1 \leq 0 $ when I did this using slack variable I get $(x,y,z)=(-1,0,0) $ but it is not working out using lagrange multiplier method. Please ...
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0answers
7 views

Optimizing single element of vector wrt. second order cone constraints

Can anybody put me on the right path to solving the following problem analytically: Given a vector $\bf{x}=(x_1,...,x_n)^T$, how do I find the bounds for a single element subject to second order cone ...
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2answers
19 views

Trouble seeing how Lagrange Multipliers are True

So if a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ constrained to the surface $g(x)=c$ for $x\in\mathbb{R}^n$ has a local maximum at $P$, then I'm having trouble seeing how this implies that the ...
2
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3answers
77 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 ...
0
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0answers
29 views

Lagrangian Method Proof

Suppose $f(\mathbf x)$, $g(\mathbf x)$ are smooth functions where $\mathbf x^*$ is a constrained local minimizer of $f(\mathbf x)$ subject to $g(\mathbf x)=0$. If $\nabla g(\mathbf x^*) \neq 0$ and ...
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0answers
17 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 2d blurring matrix and n denotes noise vector and ...
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1answer
41 views

Convex Functions and Subsets

Suppose that $f, g: \mathbb R^n \to \mathbb R $ are $C^1$ convex functions. Show that $C = ${$\mathbf x \mid g(\mathbf x) \leq 0$} is a convex subset of $\mathbb R^n$. Show that if $\nabla f(\mathbf ...
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2answers
31 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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0answers
16 views

Maximizing distance between a region and a point with a Lagrange multiplier

I have a problem from an old midterm exam that I'm trying to complete so I can practice for the midterm tomorrow. Use the Lagrange multiplier method to find the points on the ellipse $x^2$ + ...
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3answers
74 views

Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
0
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1answer
24 views

Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
0
votes
2answers
31 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
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1answer
32 views

Use Lagrange Multipliers to show the distance from a point to a plane

I'm trying to use Lagrange multipliers to show that the distance from the point (2,0,-1) to the plane $3x-2y+8z-1=0$ is $\frac{3}{\sqrt{77}}$. Our professor gave us two hints: We want to minimize a ...
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2answers
58 views

why is the derivative equal to zero here?

In this MIT open course video Denis Auroux said that given a function $f(x, y)$ and a restriction level surface $g(x, y) = c$, for any vector $u$ tangent to this surface $g=c$, we must have: $$ ...
0
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1answer
20 views

Can I use Lagrange Multipliers with inequality constraints?

Suppose I had a problem: Maximize $f(\bf{x}) $ subject to the contraints $g_i(\bf{x})< b$ Can I still use Lagrange multipliers? My text says that the constraints need to be equalities.
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1answer
16 views

Lagrange Polynomial Interpolation - Equation Help

I understand the concept of Lagrange Interpolation but am having issues understanding how to interpret the following general equation (which I will be provided) for n points. For example, how would ...
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0answers
19 views

Convex Minimization Problem with double sum

Given fixed natural number $n$ and two real numbers $A$ and $B$. I'd like to find $c_{12},\dots c_{(n-1)n}$, i.e., ${n\choose2}$ real numbers, such that $\sum_{1\le i<j\le n}^nc_{ij}=1$ which ...
2
votes
2answers
136 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
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1answer
37 views

Lagrange multiplier - Find maximum on surface

I need someone to walk me through a 3 variable lagrange problem, since I haven't been able to find a reliable source to teach me, please. Here it is: Find the maximum of the function $F(x,y,z) = ...
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0answers
24 views

Mixed Integer Non Linear Problem for Relaxation Approach

I have the following problem. I have meat markets$(\mathcal{T}_1)$ and vegetable markets$(\mathcal{T}_2)$. $(\mathcal{T}_1) \cup (\mathcal{T}_2) = T$ and $(\mathcal{M}) \cap (\mathcal{V}) = ...
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2answers
15 views

Estimate the error of interpolating ,,,

Estimate the error of interpolating (${lnx}$) . at ${x=3}$ with an interpolation polynomial with base points ${x=1 , x=2 , x=4 , x=6 }$ .
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votes
4answers
226 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 ...
0
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2answers
38 views

Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
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0answers
15 views

Lagrangian with Nonholonomic Velocity Constraints

I have a Lagrangian $L$ dependent on four generalized coordinates $[\theta(t), \phi(t), l(t), h(t)]$. And I have two differential non-holonomic constraints given by: $$ Eq1: \dot l(t) = -c\dot \phi/ ...
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0answers
39 views

Solve multivariable nonlinear equation with mixed constraints?

How do I solve a multivariable ($x_1$ to $x_m$) nonlinear (in this case a quadratic) objective function with mixed (equality (my Sum) and inequality (bounded variables) in this case linear ...
1
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2answers
45 views

Minimizing $\sum_{i=1}^n \frac{x_i^2}{w_i}$ subject to $\sum_{i=1}^n x_i=1$

Minimize $\displaystyle\sum_{i=1}^n \frac{x_i^2}{w_i}$ subject to $\displaystyle\sum_{i=1}^n x_i=1$. The answer is $x_i=\displaystyle\frac{w_i}{\sum_i w_i}$ but I don't know why apart from ...
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1answer
42 views

Optimization: Finding line connecting non-pareto-optimal allocation in Edgeworth Box to PO allocation

Two people, A and B, with respective utility functions of: $$U_a(X_a,Y_a) = X_a^2 Y_a\\ U_b(X_b,Y_b) = X_b Y_b^2$$ Total $X$ (that is, $X_a+X_b$) is fixed at $36$. Total $Y$ ($Y_a+Y_b$) is fixed ...
0
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2answers
64 views

How do you minimize this summation with the Lagrange multiplier? [duplicate]

How do you minimize $$\sum_{i=1}^n \frac{c_i^2}{i}$$ where $c_i$ are constants such that $\sum c_i=1$?
2
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0answers
54 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
3
votes
1answer
92 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 ...
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votes
3answers
63 views

Lagrange Multiplier with $3$ variables

Here is the question: $f(x,y,z) = x + 2y^2 - 3z$ subject to the constraint $z = 4x^2 + y^2$. I don't understand how to do this because when I take the partial derivative in respect to $z$, I get $-3 ...
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0answers
31 views

Lagrange multiplier uses

I'm currently self studying calculus of variations and stumbled upon something called a Lagrange multiplier, used in solving isoperimetric problems and the like. Not knowing what they are I backtrack ...
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1answer
32 views

How do I setup the lagrangian for this problem?

I have a function $y(x)$, that I would like to maximize, subject to two constraints. It is given by: $$ \max_{x} \ y(x) = a \ cos(x) + b \ sin(x) \\ \text{subject to:} \\ x \geq 0 \\ x \leq ...
0
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1answer
32 views

Optimization using Lagrange Multipliers for conditions with different codomain

I'm trying to maximize the trace of $X^TAX$ subject to the columns of $X$ being orthonormal, where $A$ is a diagonal matrix and X is not necessarily square, but does not have more columns than rows. ...
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votes
3answers
77 views

Using LaGrange multipliers to solve for minimum

I am having troubles with one part of this homework problem. Hopefully somebody can help me out: Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the given constraint ...
3
votes
1answer
61 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 ...
0
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1answer
64 views

How do I maximize the function $W=\min(u_1,u_2)$ subject to the constraint $u_1^2+u_2^2=100$?

How do I maximize the function $W=\min(u_1,u_2)$ subject to the constraint $u_1^2+u_2^2=100$? Usually I would do a Langrangian on the function to be maximized, but here it is difficult to do so. The ...
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1answer
23 views

Rolle and Lagrange

Let $f$ be continuous and differentiable such that $f'+f''=0$. Show that there are constants $a$ and $b$ such that $f(x)=a\sin(x)+b\cos(x)$. Any hints/ideas? Thanks.
1
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1answer
27 views

How to obtain primal problem from Lagrangian?

If you're trying to optimize $\min_x f_0(x)$ subject to $f_i(x) \leq 0$ then the Lagrangian would be $$L(x, \lambda) = f_0(x) + \sum_i \lambda_i f_i(x)$$ The dual problem is $\max_\lambda g(y)$ ...
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1answer
48 views

Confusion related to augmented lagrangian multiplier method

I have this confusion related to the augmented lagrangian multiplier method from this tutorial How come the gradient wrt y is equal to $\rho(Ax^{k+1}-b)$
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0answers
19 views

Lagrange optimimzation with a Diff.eq constraint

sorry in advance if my English isn't perfect here, it is not my first language( or second, for that matter...) im having some issues with understanding some of the details. the question is as ...
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0answers
14 views

How do I find the maximum and minimum values of xy on an off-center ellipse?

What's the maximum and minimum of f(x,y)=xy with the constraint $$\dfrac{(x-x_o)^2 }{A^2} + \dfrac{(y-y_o)^2 }{B^2} = 1$$ Using lagrangian multipliers is simple when the center of the ellipse is ...
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1answer
31 views

Lagrangian Multiplier Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.
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1answer
27 views

Solving lagrange multiplies

The problem is: $$ \begin{align} \operatorname{max} & \quad ax+by \\ \text{subject to} & \quad x+y=m. \end{align} $$ The Lagrangian is: $$L(x,y) = ax+by−λ(x+y−m).$$ And so far I have: $$ ...
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2answers
75 views

Lagrange Multipliers have no solution

$f(x,y)=2x+y$ subject to constraint $x+y=m$. $(2,1)=\lambda(1,1)$ but this does not satisfy $x+y=m$ So there are no solution?