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

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69 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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51 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
2
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30 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 ...
2
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45 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 $$max~f(x)\quad s.t.~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can show ...
2
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21 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 ...
2
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96 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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63 views

What is wrong with this parametrization?

I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression: $L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$, ...
2
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34 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 ...
2
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45 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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40 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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35 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 ...
2
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38 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 ...
2
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104 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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205 views

Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
2
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108 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 ...
2
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33 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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103 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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145 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 ...
2
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89 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
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676 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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31 views

Sign of the Lagrange multiplier associated with an equality constraint

I am trying to determine conditions under which the Lagrange multiplier(s) associated with an equality constraint is(are) positive. In general, the multiplier of an equality constraint is not ...
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39 views

Continuity of Lagrange multipliers

Suppose I have an optimization problem of the from $\min f(x)$, s.t. $g(x)=a$, where $f,g$ are real polynomials and $a\in\mathbb{R}$. Then using the Lagrange multiplier rule, I have to find the ...
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109 views

A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
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32 views

Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
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23 views

Dimensions of rectangular parallelopiped of maximum volume…

i Need to calculate volume of parallelopiped of maximum volume with edges parallel to the coordinate axes that can be incribed in a ellipseoid $(x/a)^{2} + (y/b)^{2} + (z/c)^{2} =1$ . Apparently ...
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42 views

Rectangle surmounted by an isosceles triangle

A window has the shape of a rectangle surmounted by an isosceles triangle. Determine the dimensions of the window, if its if its perimeter is to be at most $M$ and its area is to be maximized. I have ...
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32 views

Lagrange multipliers question and my attempt

Question is to minimise the $f(x,y)$ = $3x^{2} + y^{2} - x $$$$$ and constraint is given by $2x^{2} + y^{2} =1 $ Question is simple and ii have got most of points but i seem to miss few points ...
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26 views

Gradient of the function and the contour line

I do not understand, reading the chapter in the book about Lagrange multipliers, why the gradient of the function $f$ is perpendicular to the contour line? There is no sufficient explanation there, ...
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15 views

Find maximum hyperplane separating two classes by optimizing the Langrangian function.

I am trying to solve the following problem: I am having difficulty starting. I know this is a constrained optimization problem for support vector machines. I am wondering how the training data is ...
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40 views

Multivariate calculus (Lagrange multiplier)

If we need to use the method of Lagrange multipliers to find extreme values of a function $f(x, y)$ on a triangle-shaped region in $R ^2$ , how many times would we have to run the method? How many ...
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29 views

Explain KKT conditions without reference to duality.

Is it possible to explain (not derive) KKT necessary conditions without reference to the concept of Lagrangian duality?
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17 views

Minimizing sample variance of $n$ functions

$f_n$, $i=1,\dots, n$ are $n$ functions. I would like to minimize the sample variance of these functions subject to a linear constraint: $$\text{minimize}\quad \frac{1}{N}\sum (f_i(x_i) - ...
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32 views

Lagrange multipliers for minimax optimization

Does the equally constrained minimax optimization can be solved by using Lagrange multiplier method? Thanks in advance.
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21 views

A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
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38 views

What is the meaning of positive lagrange multiplier?

I'm handling a maximization problem with a constraint and there is a sentence "but, the lagrange multiplier is positive". I can't understand why that sentence is needed. Is there any difference ...
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45 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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31 views

Lagrangian method for Numerical Optimization

I know of a technique, but I don't know of its name and I don't have any real literature on the technique. On the wikipedia page for Lagrange multipliers a method is provided to convert a Lagrangian ...
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26 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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24 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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17 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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30 views

How can I find the Min and max of this question?

I have been trying for the past 2 hours on this question and cannot seem to figure out the answer. So far I have gotten the 'green' bits correct. Someone Help please
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69 views

a question about Lagrange multiplier?

Q)Given $x_1+x_2+...+x_n=a$ where $a>0$, find the extremum value of $f(x_1,x_2,...,x_n)=x_1^k+x_2^k+...+x_n^k$ Also, find the range of $k$ in which the extremum value of $f$ is a maximum ...
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Strong Duality and Duals of linear programming problem

I have the following problem: $ max_{x,y} \ x + y $ subject to $ 2x + y \leq 1 $ $ x + 3y \leq 3 $ $ x,y \geq 0 $ How to find the dual of this problem using the Lagrangian? I have done the ...
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78 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
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52 views

Lagrange multiplier on inequalities

Let's say we have an inequality, then we change it into a function so for example we have: $f(x,y) = x^2-xy-2$ Also we have the constraint $g(x,y) = x+y - 1 = 0$ Using Lagrange multiplier, I get ...
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123 views

What do I need to know to understand Lagrange multipliers?

I've seen Lagrange multipliers used as a powerful method for tackling inequalities and some IMO problems, and I'm aware that it's a part of calculus. I'm currently taking BC Calculus in high school, ...
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163 views

K.K.T. conditions, Lagrangian gradient not defined for zero.

When I write the K.K.T. conditions for the problem I have, I get the following expression for the gradient of the Lagrangian: $$\frac{\partial \mathcal{L}}{\partial x} = - \frac{\sqrt{x} + ...
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62 views

Closed form for Lagrange dual

Can Lagrange dual always be computed in closed form? Can you give me a simple example where the dual is not analytically computable?
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9 views

Assessing the “Quality ” of a solution sobtained by using lagrangian multipliers

I have an ill-defined question. I work in machine learning and am trying to learn the parameters of a model, such that my problem amounts to constrained optimization. That is, I have some training ...
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33 views

Point of intersection closest to the origin

How do I find the point of intersection of $𝑥 + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = 𝑥^2 + 𝑦^2$ that is closest to the origin? I know I have to use the LaGrange multiplier in order to minimize the ...