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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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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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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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 ...
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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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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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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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48 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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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 ...
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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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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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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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59 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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98 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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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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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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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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139 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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255 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 ...
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134 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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39 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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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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876 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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Confusion over Lagrangian function

When forming the Lagrangian of an optimization problem, why don't we include all constraints in the Lagrangian? For example, the optimization problem $$ \begin{align} \max_{x}&\quad f(x)\\ ...
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Lagrange multipliers, once I use the constraint equation, do I have to worry about it again later?

I am solving $ grad [f(x,y,z)]$ = $\lambda$grad[g(x,y,z)] I have then three equations, one involving x's and lambdas, another involving y's and lambdas and a third involving z's and lambdas. I then ...
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Taking the partial derivative of a Lagrangian with square roots

I have a nasty function that is simplified from an even nastier function: $F(x,y,z,\lambda) = \frac{-0.0129x-0.0051y-0.0066z}{\sqrt{.44^2x^2 + .15^2y^2 + .44^2z^2 + 0.05808xy + .139392xz - ...
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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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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 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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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 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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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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Change of variables for functions with constraints

I want to find critical points of say F = (x1-x2) + (y1^2-y2^3) + (z1^2-z2^3) with constraints x1^1 + y1^2 + z1^2 -1 = 0 and x2^1 + y2^2 + z2^2 -1 = 0 I can do this by finding critical points of ...
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Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...