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

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Lagrange Multipliers with Calculus of Variations

We wish to extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to the constraint $$g(\mathbf{y}, t) = 0 \;.$$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha ...
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Lagrange multipliers problem with inequality constraints [on hold]

Let $x, y$ be non-negative real numbers satisfy $x \leq y \leq 5x $. Find the maximum value of $\sqrt{2x+2y} + \frac{5}{8}x^2 - \frac{3}{4}xy - \frac{3}{8}y^2$
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Lagrange Multipliers with $f\left(x,y\right)=x^2-y^2$ and “constraint” $g\left(x,y\right):=2y-x^2=0$

I am working on a problem from my textbook on Lagrange Multipliers. I feel I have these down now, but I am curious about this specific problem. Let \begin{align} f\left(x,y\right)=x^2-y^2\tag{1},\\ ...
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Insight for basic constrained optimization

Consider the basic Constrainted Optimization Problem: Minimize f(x) Subject to: Ax = b The Lagrangian is ...
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Quick Constrained Optimization Huerstistics

I am wondering if there is a way to find very quick optimization heuristics for the form. $$ f(x) = cx^a $$ $$ s.t. $$ $$ L \le Ax \le B$$ $$ 0 \le x \le \infty $$ I know with only a few variables ...
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Finding the maximum and minimum on a constraint.

Let $f(x,y,z) = 2x + y$. Find the absolute maximum and minimum on the constraint $x+y+z=1$ So we know that $\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$ where $g(x,y,z) = x+y+z-1$. Calculating, we ...
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Fixed Length Catenary

Doing a fixed length catenary problem, why is it that adding the constraint $L=\int_A^B ds$ gives us more solutions. A little background: the catenary problem involves minimizing the integral ...
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Determining points on a 3-dimensional intersection closest to the origin

I was presented with this question in a lab: Use the method of Lagrange Multipliers to solve the following. Be sure to let Mathematica do all the heavy lifting for you. Determine the points that ...
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1answer
45 views

Finding the distance between a point and a parabola with different methods

I'm trying to find the shortest distance from point $(3,0)$ to the parabola $y= x^2$ using the method of Lagrange Multipliers (my practice), and by "reducing to unstrained problem in one variable" ...
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Maximum and Minimum Values Subject to Constraints (LaGrange)

Max and min values of $f(x,y,z) = yz+xy$ subject to $y^2 +z^2 = 289$ and $xy =5$. I know this will be a LaGrange problem and two constraints will be utilized so the formula you're looking for is the ...
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1answer
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Undefined case in Lagrangian method

I am trying to finding the minimum distance between the point(1,1,0) and points on the sphere $$x^2+y^2+z^2-2x-4y=4$$ An easy way to do this is to graphical intuition and get the distance, since the ...
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Conditional Extremum, need help finding the extreme points in calculation.

Find the conditonal extremums of the following $$u=xyz$$ if $$(1) x^2+y^2+z^2=1,x+y+z=0.$$ First i made the Lagrange function $\phi= xyz+ \lambda(x^2+y^2+z^2-1) + \mu (x+y+z) $, now making the ...
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Arrow–Debreu Model, Optimization for n-agents to determine security prices

Good day, My question is as to the following Lagrage equations system: $ \scriptsize \left\{-\lambda \left(p_1 \left(2.14865 y_1-0.675676 y_2-3.74324\right)+p_2 \left(0.878378 y_1+0.189189 ...
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SVM Soft Margin Lagrange form

I study the Lagrange multipliers form of SVM. I am particulary interested in values that $\alpha_i$ can get. The following is the Langange multipliers form of hard margin SVM. $min_{w,b} ...
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1answer
24 views

Maximizing the volume of a cylinder with given area

Consider a right circular cylinder of radius $r$ and height $h$. It has volume $V=\pi r^2 h$ and area $A=2\pi r (r+h)$. We are to use Lagrange multipliers to prove the maximum volume with given ...
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38 views

Getting explicit expression of function from the dual function

Considering the following problem : $$ minimize_{x_1,x_2} \ -2x_1+x_2 \\ subject \ to \ x_1+x_2=\frac52 \\ (x_1,x_2) \in X ,\\$$ where $X=\{(0,0),(0,2),(2,0),(2,2),(\frac54,\frac54)\}$ The dual ...
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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 ...
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From utility function (3 products) to demand function (2 products)

I am struggling with this exercise and would appreciate some help. Consider two goods and a representative consumer whose utility is given by: $U(q_{0}, q_{1}, q_{2})= ...
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42 views

Lagrange multipliers

A current of $18$ amperes branches into currents $x$, $y$, and $z$ through resistors with resistances $5$, $7$, and $4$ ohms as shown. It is known that the current splits in such a way that the sum ...
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Optimize monotonic function in calculus of variations

I'm interested in the variational problem $$\min_{y} \int_a^b F(x,y(x),y'(x))dx \qquad \text{subject to} \quad -y'(x)\leq 0 \quad \forall x \tag{1}$$ i.e. $y(x)$ has to be monotonic. I ...
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How to maximize $|(x,\alpha)|^2+\sum_{i=1}^n |x_i|^4$?

I need to calculate the following expression: $$\max_x |(x,\alpha)|^2+\sum_{i=1}^n |x_i|^4,$$ here $x,\alpha$ are unit vectors in $\mathbb{C}^n$. I tried Lagrange multiplier method and used ...
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Optimizing concave function over non-convex set

I have the following problem that I am looking advice on. Let $ \mathcal{F}$ be a convex subset of vector space $X$. The goal it to \begin{align*} \max_{x \in \mathcal{F}} f(x)\\ s.t. \ g(x) \le 0 ...
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1answer
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Lagrangian multipliers in complex optimization

I want to know where is my mistake in solving the problem \begin{equation} \begin{array}{c} minimize \hspace{1cm} z^*z \\ s.t. \hspace{0.5cm} z = z^* + i \\ \end{array} \end{equation} by using ...
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3answers
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Lagrange Multipliers, maximize $f=xy$ restricted to $g=x^2+y^2=r^2$

So I have to solve the system of equations $$\cases{\nabla f = \lambda \nabla g\\x^2+y^2 = r^2}.$$ Then $y=2\lambda x, x=2\lambda y$. Sorry if this is obvious, but how can I get $x$ and $y$ only as ...
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31 views

When is lagrange multiplier the best choice?

Let f(x,y) be a smooth function. If i want to find the min and max of this function in the quarter disk constrained by x^2+y^2=1 in the first quadrant. Can i then use lagrange multiplier to do this ...
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Lagrange's Interpolation

$\log_{10} 2=0.3010300 ,\log_{10} 3=0.4771213 ,\log_{10} 5=0.6989770 ,\log_{10} 7=8450980$ Derive Lagrange's Interpolation formula of degree $5$ for $\log_{10} x$ where $40\le x\le50$ Find ...
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Lagrange Multipliers with two constraints

The problem is to find the maximum value of $f(x,y,z)=z+y+z$ subject to the two constraints $g(x,y,z)=x^2+y^2+z^2=9$ and $h(x,y,z)=(1/4)x^2+(1/4)y^2+4z^2=9$. I got these equations: ...
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Lagrange multipliers on a boundary

Use Lagrange multipliers to find the extrema of $f(x,y,z)=yz+x^2$ on the boundary of $J$ given by $\partial J=\{(x,y,z) \in \mathbb{R}^3 : x^2+2y^2+3z^2=6\}$ . I have started by forming ...
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using lagrange method of undetermined multipliers

A rectangular tank is to have its capacity of 1.0 cubic meter. If the tank is closed and the top is made up of a metal half as thick as its sides and base,use Lagrange method of undermined multipliers ...
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1answer
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solution involving inverse of a rank-1 matrix

I am looking for $\mathbf{y} \in \mathbb{R}^n$ that minimizes the following objective function that involves a real matrix $\mathbf{V} \in \mathbb{R}^{n\times n}$ \begin{equation}\tag{*} ...
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Min/Max with Lagrange Multipliers

Find the max and min values of the function $f(x,y,x)=3x+2y+4z$ with constraint $g(x,y,z) = x^2+2y^2+6z^2=16.$ I set $\nabla f = \lambda \nabla g$. This gives me $3=λ2x$, $2=λ4y$, $4=λ12z$ . ...
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Min and Max with two constraints

I'm asked to find the minimum and maximum values of $f(x, y, z) = x^2+y^2+z^2$ given the constraints $x+2y+z=3$ and $x-y=7$. I'm pretty sure I need to set up Lagrange equations, giving: $$2x = ...
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Global maxima of $f(x,y)=x^2y$ restricted to D

Let $f(x,y) = x^2y$ and $D = \{(x,y): y\geq0 \land 2x^2+y^2 \leq a\}$ with $a>0$. I need to find $a$ such that the global maxima of $f$ restricted to $D$ is $\frac{1}{8}$. I found, using Lagrange ...
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When are there not min/max values of a function subject to a constraint?

How do I know if there are no extreme values of a function subject to a constraint? For example, if $f(x,y,z)=xy+3xz+2yz$ subject to the constraint $5x+9y+z=10$. Why does it not have min/man values?
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Using Lagrange multipliers to find shortest distance between two straight lines

A problem asks me to use the method of Lagrange multipliers to find the shortest distance between the straight lines $x=y = z$ and $x = -y, z=2$ (It also warns me that using this method is a bit ...
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Factoring out the trace of a matrix

This question is related to a derivation step in " A Duality View of Spectral Methods for Dimensionality Reduction" Xiao et al. 2006 When deriving the dual equation for Maximum Variance Unfolding ...
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46 views

Optimization of utility function with Lagrange multiplier

Let u: ${\mathbf R^n_+ \rightarrow \mathbf R}$ be a utility function of n goods which you buy in quantities $x_1,…,x_n$ to the prices $p_1,…,p_n$ under the budget K. So maximize $u(x_1,…,x_n)$ subject ...
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Is there a difference between inequality and equality sign when using Lagrange multiplier?

For example, find the extreme values of z=xy subject to the condition x+y=1 This is quite simple example of finding extreme using Lagrange multiplier When the constrain is changed from x+y=1 to ...
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Minimize $f(X)=trace\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}\right)$

Minimize $$f(X)=trace{\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}X\right)}$$ subject to the condition $g(X)=det(X)=1$. Then for taking $X=\begin{bmatrix} ...
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Variational calculus on integrals and derivatives

I am studying mechanics but am a novice in variational calculus. While reading a book on Lagrangian mechanics, I blocked when the author states that by calculating the variation of the following ...
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Decomposition using Lagrange Multipliers

Using Lagrange multipliers, decompose the number 411 in the sum of three (possibly fractional) nonnegative numbers, making sure that their product is maximized. Determine whether the solution is a ...
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Lagrange multipliers for Bose-Einstein distribution

We have $$ W = \prod_{s=1}^N \frac{(g_s-1+n_s)!}{(g_s-1)!n_s!} $$ For $ g_s,n_s>>1 $. We also have the constraints $$ \sum_{s=1}^Nn_s = n \hspace{10mm} \sum_{s=1}^Nn_sE_s = E $$ We are to ...
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Reformulation through lagrange multipliers

I have the following objective function \begin{equation}\tag{*} \begin{array}{c} \varepsilon = \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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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 $f_1(x_1,\dotsc,x_n)=c_1,\dotsc,f_k(x_1,\dotsc,x_n)=c_k$ ...
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optimization problem resolution

I need to get the value of k that minimizes : $\frac{k}{\sqrt{k^2+\left( \omega-\omega_{c}\right)^2}}$ under the constraints : $k > 0$ and $\omega \ne \omega_{c} $. Thank you.
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Question about Lagrange multiplier and maximum point

Find the maximum of $\log{x}+\log{y}+3\log{z}$ on portion of the sphere $x^2 + y^2 + z^2 =5r^2$ where $x,y,z>o $ I found that maximum is $5\log{r} + 3\log{\sqrt{3}}$ at $(r,r,3\sqrt{3})$ And ...
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48 views

How can I find minimum volume using Lagrange multipliers?

What is the minimum volume bounded by the planes $x=0, y=0, z=0$ and a plane which is tangent to the ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$$ where $x,y,z>0$ I only ...
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How do Lagrange multipliers work with functions of 3 variables?

I've been trying to imagine the workings of Lagrange multipliers for functions $\mathbb{R^3}\rightarrow \mathbb{R}$. So, say we have a function $f(x,y,z)$. If we had one constraint (one level ...
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1answer
16 views

Fiddly optimisation

I'm trying to complete the following problem: $\max_{x_0, x_1, x_2} 2ln(x_0) + \ln(x_1) + \ln(x_2) $ s.t $\ x_0 + p_\alpha\alpha + p_\gamma\gamma = 4, x_1 = 4 + \alpha + \gamma, x_2 = 4 + \gamma $ ...
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solving euler-lagrange equation in constrained functional optimization

The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = ...