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

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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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151 views

Positive Lagrange multipliers of an equality constraint

Consider the problem \begin{align} \max_{x\in\mathbb{R}^n} f(x)\\ \text{subject to }\quad h(x) = 0\\ x\in X \end{align} where $X$ is a convex and compact subset of $\mathbb{R}^n$. I also know that ...
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70 views

How to use the Karush–Kuhn–Tucker conditions?

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method. For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to ...
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106 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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221 views

Calculus of variations: Isoperimetric and holonomic constraints.

A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most ...
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38 views

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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183 views

Lagrange multipliers in the context of the calculus of variations

Suppose we wanted to extremise the function (of a finite number of variables) $f$ subject to the constraint $g = 0$. The Lagrange multiplier approach is to extremise without constraint the function ...
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2k views

Optimization with symmetric matrix constraint

Consider the following optimization problem: ''Minimize some objective $f(A)$ over all matrices $A$ s.t. $A \mathbf{1} = \mathbf{1}$ and $A = A^T$.'' I wonder in which ways one can handle the ...
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27 views

Pattern of collision of bouncy balls in a sphere?

Suppose that you have two infinitely bouncy golf balls that exist inside a perfect sphere in weightless suspension, and both golf balls start bouncing at a random angle and are 10 or 100 times ...
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24 views

optimal derivative position through optimization

So I have the following optimization problem: min. $-E^Q[u(h(x))]$ s.t $\int h(x)q(x)dx \leq \frac{V_0}{B_0}$ Where $Q$ is the subjective probability which then gives: $E^Q[u(h(x))]=\int ...
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52 views

Distance between point and a parabola using Lagrange multiples

I am trying to find the distance between the point $(p, 4p)$ and the parabola $y^{2} = 2px$, where $p$ is a fixed positive parameter. So far, I have got ...
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56 views

Optimizing over an infinite set of variables

This may be a very basic question, but it's been a while since I did any optimization. Suppose I have a sequence $(x_i)$, $i=1,2,\ldots$ in the $\ell^2$ space and the following optimization problem: ...
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67 views

How to use find the Lagrange Multipliers in multidimensional Calculus of Variations

Suppose I wish to minimise the integral $$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$ Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie $$F ...
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47 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
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68 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 ...
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83 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 $$\max~f(x)\quad \mbox{s.t.}~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can ...
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93 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 ...
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141 views

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 : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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64 views

What is wrong with this parametrization?

I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression: $L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$, ...
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91 views

Non-differentiable variational calculus (Dido's problem)

I wonder what is the alternative to Euler-Lagrange equations when we have non-differentiability issues. I'll give an example: Dido's problem can be stated as: Find the figure bounded by a line ...
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158 views

Maximize and minimize a function using Lagrange multipliers.

I want to maximize and minimize $$h(a,b) = a + b$$ given the constraint $$g(a,b) = a^{\frac{1}{3}}b^{\frac{2}{3}} = l$$ I'm trying to use Lagrange multipliers. Here's what I did: \begin{align} ...
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52 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 ?
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50 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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59 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 ...
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174 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 ...
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44 views

Position-dependent Lagrange multipliers for functionals

I'm trying to extremise the functional $$ S = \int L(Q_i) \, dx\,dy $$ Where the $Q_i$ are functions of $x$ and $y$, subject to the constraint $$ \vec{\nabla} \cdot \vec{Q} = A(x,y) \,.$$ My ...
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197 views

Lagrange multipliers for matrix calculus problem

I have what is probably an elementary matrix calculus question. Let $P$ be a linear operator and $||\cdot||$ be the ordinary unsquared vector norm. Suppose one wants to minimize $||Px||$ subject to ...
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14 views

Is convexity of the objective function sufficient for a local maxima to be a global maximum?

In my problem, I have to maximize a convex function $f(x_1,x_2,\cdots,x_n)$ subject to two equality constraints $g_1=0$ and $g_2=0$. As usual, I constructed the Lagrangian ...
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52 views

Find the maximum distance from origin to the surface?

I am having trouble with this problem... I need to find the maximum distance from the origin to the surface $$f:=\frac{x^4}{2^4}+\frac{y^4}{3^4}+\frac{z^4}{(\sqrt2)^4}-1$$ I think I have to use ...
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19 views

Value of Lagrangian Multiplier

I have a two dimensional optimization problem of the form $$ v = \max_{x,y} f(x,y)+g(x,y) $$ Both $g,f$ are concave and continuously differentiable. Assume the solution can be reached by first order ...
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31 views

An optimisation problem

I have an optimisation probem given below $$argmax_{x_i \ \ \forall x_i=1,2...n} \sum_{i} S_ie^{-\alpha x_i}$$ subject to $$\sum x_i = 1$$ $$\sum C_i x_i \leq B $$ $$\forall i \ \ x_i \geq 0 $$ ...
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Problem about find the extreme of a function (Multipliers of Lagrange)

Good morning, i have a problem with this: Find the maximum and minimum distances from the origin to the curve $g\left(x,y\right)=5x^{2}+6xy+5y^{2}$ I make this: Function to optimize: ...
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14 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...
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16 views

Time independent vs. time dependent lagrange multiplier

What are the differences between these two in applications? For example: $$max\sum_{t=0}^{\infty} \beta^t u(c_t)$$$$s.t.f(c_t,c_{t+1},x_t,x_{t+1})=0$$ What are the differences between: ...
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Lagrange multipliers with angular diameters question

Let $a, b \in \mathbb{R}^n$ be linearly independent, |a| = 5, |b| = 10. Functions $f_a, f_b$ on the sphere $S_1(0) = ${$x : |x| = 1$}$ \subset \mathbb{R}^n $ are defined as follows: $f_a(x)$ is the ...
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24 views

Find optimal points using Lagrangian multipliers

Problem: Consider the problem of maximizing and minimizing the function $f(x, y) = 9x^2y$ under the constraint $g(x, y) = x^2 + 2y^2 = 1$. (1) Find all points which satisfy necessary conditions for ...
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On a max-min problem from an exam.

I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems ...
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25 views

Find the minimum of $f(x,y,z)=3x+2y+z+5$ subject to the constraint $g(x,y,z)=9x^2+4y-z=0$

Let $ \displaystyle L(x,y,z,\lambda) =(3x+2y+z+5)+\lambda(9x^2+4y-z)$ By using Lagrange Multiplier $L_x=3+18\lambda x=0$ $L_y=2+4\lambda=0$ $L_z=1-\lambda=0$ It implies that $\lambda=1=-2$? How ...
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24 views

Minimizing mutual information using Lagrange multipliers

Im trying to follow a minimization of mutual information using Lagrange multipliers in a highly cited paper called The Information Bottleneck Method (1999), page 4: $$R(D) = \min_{p(\tilde{x}|x): ...
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29 views

maximum entropy principle: CDF of its PDF

In goodness-of-fit (gof) tests (COD, R2, X2) to discriminate PDFs, we need their CDFs. With wind speed, another PDF is by Maximum Entropy Principle or Method, of the form: $$f(v)=\exp\left\{-a_0 - ...
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Lagrange/KKT multipliers and additional constraints

Any ideas on the following would be much appreciated. I’m interested in how the shadow price of non-renewable resource changes when new constraints are introduced to the problem. So, assume that there ...
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Lagrange multipliers to find maximum and minimum

Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x, y, z) = x - y +z$ on the sphere $x^2 + y^2 +z^2$. Since gradient vectors have have $x$ to the ...
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31 views

Is this correct use of Lagrange multipliers?

Given the standard isoperimetric problem: Minimize the functional $$ A[y]=\int_a^bF(y,y',x)dx $$ subject to the functional-constraint $$ B[y]=\int_a^bG(y,y',x)dx = c=\text{constant}$$ (where $ ...
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46 views

how to define the lagrange problem on the boundaries

Im reading a paper where they solve a problem of the form: minimize $$f(a,b,c) = \sum_{i=1}^n(a+by_i+cy_i^2-u_i)^2$$ Where $y_i,n$ and $u_i$ are given over the compact domain $K$ $$ ...
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28 views

Lagrange Multiplier FEM for Navier-Stokes

I'm trying to derive the weak formulation for the Navier-Stokes equations with boundaries imposed via Lagrange Multipliers. This technique is used by Urquiza 2014 for the Stokes equations. It's done ...
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67 views

Minimize $5\sqrt{36+x^2}+4(20-x)$ using Lagrange Multipliers

I am not familiar with Lagrange multipliers. I am trying to minimize the following equation $$\text{minimize}~ 5\sqrt{36+x^2}+4(20-x)$$ $$\text{subject to}~ 0 \leq x \leq 20$$ Can I use the Lagrange ...
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If I am asked to minimize f, subject to several constraints g_i, and I find only one critical point, must this point be where f attains a minimum?

I am guessing that it must be, but I feel that I have only found a unique critical point, so that this point could be either a unique absolute max or a unique absolute min -- I say uniqueness because ...
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26 views

SVM why to switch to dual Lagrange?

I am newbie reading Support Vector Machines Explained. Translation to dual Quadratic program (p.3 - p.4) in SVM explaining article I mentioned confused me because I understand basic Lagrange ...
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37 views

Multiple equations Lagrange multipliers.

I understand Lagrange multiplier for one equation constraint task. But what about several equations formal proof of Lagrange multipliers method? My own non-formal proof for $R3$ case: Suppose we ...
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45 views

Solving linear constrained optimization problem

So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on: Minimize the following expression (power dissipation): $$I_{B1}(V - C_1) ...