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

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Lagrange multiplier method

Question 1: Could somebody please refer me to an introduction to Lagrange multipliers which is easy to read but still in full generality? Question 2: I am interested in particular in the following ...
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Can one optimize a function with min(x,y) as a constraint?

Let $$h(x,y) = \min\lbrace x,y \rbrace$$ I want to find $$\min ax + by$$ Subject to $$g(x,y) = c - h(x,y)= 0$$ My lagrangian is $$L(x,y,\lambda) = ax+by + \lambda (c - h(x,y)) $$ I have ...
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Lagrange Multipliers for farthest distance

I am trying to find the farthest point from the origin to a point on the circle $$(x-2)^2+y^2=1$$ I am not great with the formatting on here but this is what I have so far... $$f(x,y)=x^2+y^2 $$ ...
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Lagrange optimization method appears to give the wrong answer

This problem comes from Schaum's Outlines "Calculus for Business, Economics, and the Social Sciences". Problem 9.14 (c) states: Maximize $f(x,y,z)=3x^2yz$ subject to x+y+z=32 So, I set up the ...
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Smooth manifold

$A=M\cap N$, $M={(x,y,z\in\Bbb R^3)| x^2+y^2=1}$, $N=(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1$. $1$. Is $A$ is smooth manifold? $2$. Find the points of $A$ that are farthest from the origin. This is ...
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How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
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connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
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Boundary conditions and Lagrange Constraints in Calculus of Variations

I am trying to learn about Calculus of Variations for some time now. In many problems, there are some boundary conditions defined, for example when we want to maximize a functional ...
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A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
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Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
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Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
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$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
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Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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Lagrange Multipliers and optimisation

An open-topped metal water tank with volume $2$ m$^3$ is to be constructed with vertical sides and a right-angled triangle as base. What should be the dimensions of the base to minimise the area of ...
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Why do lagrange multipliers have the form $\nabla G$

I was studying some multivariable Calculus and we were covering the topic of Lagrange multipliers. I didn't understand exactly why the equations take the form: $$ \nabla f = \lambda \nabla G $$ ...
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Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
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Lagrange's multiplier not working

Given the function $f(x,y):=xy+x-y$. Let $D:=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq25\wedge x \geq 0\}$. Find the absolute maximum and minimum of $f$ on $D$. My working is as follows: $\begin{array} ...
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What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? [duplicate]

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? Using Langrange Multipliers, I've set up the standard equation with $$g(x,y) = (x/2)^2 + (y/3)^2 = 1$$ $$f(x,y) = ...
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maximizing a coordinate of $x^T A^T A x \leq r^2$

Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by ...
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Can this optimization problem be solved analytically?

Can the following be solved analytically? minimize $$ \ V(x) = |x_1-2| + |x_2-2| \ \ ; \ \ [x_1,x_2] \in R^2$$ subject to: $$ h_1(x) = x_1-x_2^2 \ge 0 $$ $$ h_2(x) = x_1^2+x_2^2-1 = 0 $$ I ...
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Verifying an inequality with Lagrange multipliers

My question has two parts. Let $p$ and $q$ be numbers with $p > 1$ and $q > 1$ 1) Show $${x^p\over p} + {y^q\over q} \ge {1\over p} + {1\over q}$$ where $x \gt 0$, $y \gt 0$ and $xy = 1$. 2) ...
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The method of Lagrange's Multipliers

I used the method of Lagrange's multpliers to find the maximum of $f(x,y,z)=\ln x+\ln y+3\ln z$ on the portion of the sphere $g(x,y,z)=x^2+y^2+z^2=5r^2 \ ; r>0$ where $x>0, y>0, z>0$ . I ...
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Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
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Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
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In the envelope theorem, why can I write my inputs $x$ and $y$ as a function of $\xi$?

This is a question about the envelope theorem. Suppose I have a maximization problem $$\max_{x,y} f(x,y,\xi)$$ such that $$g(x,y,\xi) \leq c$$ where $x$ and $y$ are control variables and $\xi$ is a ...
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Lagrange Multipliers Dilemma

In the problem $f(x,y) = xy$ and $g(x, y) = x^2 + 9y^2 = 18$ I get $y = 2λx$, $x = 18λy$ and $x^2 + 9y^2 = 18$ (the constraint). All is fine, but I feel like I'll get two different answers ...
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lagrange method, linear constraints and unique global maximum

My book in linear programming states two things that I do not understand. We are working with the lagrange method with linear constraints.: From multivariate calculus we have that at a critical ...
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Find max/min of $f(x,y,z)$ on closed unit ball $B$ in $\mathbb{R^3}$

$f(x,y,z) = 3x - 2y + z$ Let $B$ be a closed unit ball in $\mathbb{R^3}$, find the max/min of f on $B$. We first need to observe $(a)$ the behavior of $f$ in $B^0$ $(b)$ the behavior of $f$ on ...
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for f(x,y,z) find point on surface nearest to origin

$f(x,y,z)=x^2+2y^2-z^2$, $S=\{(x,y,z): f(x,y,z)=1\}$ find point on S nearest to origin. I thought I would use Lagrange multipliers to solve this problem, but when I use $f(x,y,z)=x^2+2y^2-z^2$ and ...
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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 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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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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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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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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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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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 ...