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Questions tagged [lagrange-multiplier]

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

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Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the condition $ax^2+by^2+cz^2 =1\;\;$ and $\;\;lz+my+ny=0$

Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the condition $ax^2+by^2+cz^2 =1\;\;$ and $\;\;lz+my+ny=0$ and interpret the result geometrically I started with Lagrange's ...
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Hicksian Demand Function

Derive Hicksian demand for - $$u(x,y) = ax+ b\ln(y)$$ Explain in words what they mean? I solved the problem with the Lagrange Multiplier Method and found Hicksian demand for $x$ only. Solution: ...
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A question related Lagrange multipliers in $\Bbb R^4$

I would be happy to get some help with the following problem: let $a=(a_1, a_2,a_4,a_4)\in \Bbb R^4$, and $M= \{x\in\Bbb R^4: |x|=1, \langle x, a \rangle =0\}, $ and $f(x_1 ,x_2, x_3, x_4) = \sum_{...
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extrema(max-min) two variable constrained function $f(x,y)=x^2+y^2$ with “fixed” Hessian matrix

I have this function $f(x,y)=x^2+y^2$ under the constrain : $x^6+3y^2=1$ I use the Lagrange multiplier method : $$ \left\{ \begin{array}{} 2x=6x^5\lambda\\ 2y=6y\lambda\\ x^6+3y^2-1=0 \end{array} \...
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Derivative of expectation expression

I am reading an article on variational inference. I got stuck on the following derivation: $$argmax_{qj}(\int q(z_j)E_{q-j}[log\ p(z_j|z_{-j}, x)] dz_j) - \int q(z_j)log\ q(z_j)dz_j)$$ "To find ...
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Lagrangian dual and matrix constraints

I am starting to work with matrix calculus and I am trying to write the correct dual for the following minimization problem: $$\begin{equation*} \begin{aligned} & \underset{X}{\min} & & ...
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Constrained (and non) extrema of $f(x,y)=2x^2-2xy^3+3y^2$

I need to find the critical points on the boundary,inside $D$,outside $D$, and find the image of this function (constrained on $D$). $f(x,y)=2x^2-2xy^3+3y^2$ $D=\{2x^2+3y^2\le 9\}$ Critical points ...
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Lagrange Multipliers: Absolute minimum of $f(x_1,x_2,…,x_n)$ on the boundary of the region $x^2_1+2x^2_2+3x^2_3+…+nx^2_n≤1$

I'm trying to use Lagrange Multipliers to solve the following problem Let $f(x_1,x_2,...,x_n)=x_1^2+x_2^2+...+x_n^2$ What is the absolute minimum of $f(x_1,x_2,...,x_n)$ on the boundary $x^2_1+2x^...
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Help with solving a support vector regression problem with Lagrange multipliers

Decided to take an example with only 3 data points as to see the steps of the algorithm. The points (x,y) are (10,26) , (11,35) , (12, 40). When I write out the Lagrangian of the dual problem, I end ...
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Forming a constrained optimization from a given problem?

A workshop produces a product out of three infinitely divisible ingredients X, Y , and Z. The ingredients cost p, q, and r pounds per kg, respectively, with p, q, r > 0 all different, and the factory ...
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Shadow price not covering costs

In (environmental) econ class we had to solve the following problem: ...
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Linear inequality constraint - in KKT optimisation

I have a query regarding whether KKT is optimal with some linear inequality constraint and non-linear inequality constraint. For KKT to be optimal the inequality constraints must be convex. We know ...
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What does $\delta L$ mean in variation calculus?

What does $\delta L(x,x')$ mean in variation calculus and Lagrangian mechanics? How is it different from the derivative? What does it mean to take $\delta$ of an action?
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Dual Ascent Method (DAM)

I recently came across the dual ascent method. The method is part of my question so it is written formally below. $$\min_{x} (f(x))\hspace{1cm}subject\hspace{1mm}to\hspace{1mm}Ax = b $$ The Lagrangian ...
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Maximum value of $f(x,y)=x^2+2y^2$ subject to constraint : $y-x^2+1=0$?

$f(y)=2y^2+y+1$ $f'(y)=4y+1=0\Rightarrow y=-\frac14,\; x=\pm\frac{\sqrt3}2$ $f''(y)=4>0$ so I can't obtain a point of maxima. What does this mean? Do I necessarily need to use Lagrange's ...
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Understanding derivation of ADMM update rule for graphical lasso optimization by solving quadratic matrix equation

I'm trying to understand the derivation of an ADMM update rule in some convex optimization lecture notes by Emmanuel Candes [1]. In the course of the solution (on page 25-4 and 25-5), it is required ...
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Bernoulli's lemniscate inside rectangle

Determine the vertices of the rectangle with the smallest field with sides parallel to the axis of $OXY$, which contains the Bernoulli's lemniscate $(x^2+y^2)^2-2(x^2-y^2)=0$ inside. I would be ...
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Equivalency of two optimization problems

Given functions $f$ and $g$, P1 and P2 below are said to be two equivalent formulations of the support vector machine. How does one show the equivalence? P1: \begin{align} \beta^*(\lambda)=\text{...
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Generalizing the Solution to the Lagrangian for Risk Averse Portfolio Construction

I'm interested in generalizing the Lagrangian in Herold (2005) for portfolio construction in finance: which finds the portfolio $h_p$ that maximizes the utility subject to the fully invested $h'_p ...
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Confusing Lagrange multipliers question

Let $a_1,a_2, \dots, a_n$ be reals, we define a function $f: \mathbb R^n \to \mathbb R$ by $f(x) = \sum_{i=1}^{n}a_ix_i-\sum_{i=1}^{n}x_i\ln(x_i)$, in addition, we are also given that $0 \cdot \ln(0)$ ...
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Extremum value of $f(x,y)=ax^2+2hxy+by^2$ subject to constraints $g(x,y)=x^2+y^2-c^2=0$.

Find the extremum value of $f(x,y)=ax^2+2hxy+by^2$ subject to constraints $g(x,y)=x^2+y^2-c^2=0$, where $abc \neq 0$ and $(a+1)^2+4(a^2-b^2) \geq 0$. My attempt: I have use Lagrange Multipliers to ...
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Maximizing $x^4+y^4+z^4 + p(xy+xz+yz)$ on a sphere

What values of $x,y,z$ maximize $f(x,y,z,p) = x^4+y^4+z^4 + p(xy+xz+yz)$ with constant $p \geq 0$ with the constraint $x^2+y^2+z^2=1$? Some preliminary studies in Mathematica showed that the behavior ...
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Calculating the minimum distance to the origin from a curve defined by $\frac{x^2}{4}+y^2+\frac{z^2}{4}=1$ and $x+y+z=1$

I want to calculate the points of the curve given by $$\frac{x^2}{4}+y^2+\frac{z^2}{4}=1,\qquad x+y+z=1$$ which are minimum and maximum distance to the origin. Using Lagrange multipliers, the maximum ...
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Prove Unique Lagrange Multipliers Equality Constraint

I am working through some old test papers in preparation for exams an am trying to scout out potential sneaky questions that might be asked. I've stumbled across this one. Would you please verify or ...
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Simple non linear system - Lagrange multipliers

I'm trying to minimize the function $f(x) = \sum_{i=1}^{n}(\log(|x_i|))^2$ in the closed unit ball $B(0, 1) \subset \mathbb R^n$, where the function is defined to be $\infty$ if $x_i = 0$. What I did ...
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Constrained Optimization problem with Lagrange Multiplier

I came across an example in a text, maximum volume of a rectangular box in $\mathbb{R^3}$, with sides parallel to the coordinate axes, whose vertices are all a distance $R$ from the origin. So I am ...
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Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time. Context I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\...
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find extrema of a multivariable function Lagrange

I need to evaluate the extrema of this function : $f(x,y,z)=\frac{1}{x^2+y^2+z^2}$ restricted on: $R=\{x^2-y^2-z^2+16<=0\}$ \ $\{0,0,0\}$ boundary : $\theta R=\{x^2-y^2-z^2+16=0\}$ \ $\{0,0,0\}$ ...
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find extrema of a $f(x,y,z)$ function using Lagrange multiplier

The function is : $f(x,y,z)=e^y(x^2+z^2)$ restricted on $R=\{x^2-3y^2+z^2+9=0,x^2+y^2+z^2\le 16\}$ $$ \left\{ \begin{aligned} 2xe^y=\lambda 2x+\mu 2x \\ e^y(x^2+z^2)=-\lambda 6y+\mu 2y\\ 2ze^y=\...
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find extrema min,max on a multivariable function

I have to evaluate min,max inside and on the boundary of a domain: $D=\{xy-1\le 0,|y-x|\le1\}$ $f(x,y)=(y-x)e^{xy}$ So That's a ($xy-1$) hyperbola and two lines. I proceeded like so: for $y=1+x $ ...
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Find any local max or min of $x^2+y^2+z^2$ s.t $x+y+z=1$ and $3x+y+z=5$

Find any local max or min of \begin{align} f(x,y,z)=x^2+y^2+z^2 && (1) \end{align} such that \begin{align} x+y+z=1 && (2)\\ 3x+y+z=5 && (3) \end{align} My attempt. Let $...
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Saddle points on Lagrange Multipliers

Suppose a function $f$ of two variables. To find the extremes of a function under certain conditions, I usually apply Lagrange multipliers. For example, Calculate the extremes of function $f(x,y)=-x^...
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Finding extrema of a tri-variable function under a constraint

We need to find extremum of $$f(x,y,z) = yz$$ under the constraint $$g(x,y,z) = 2x^2 + 3y^2 + z^2 - 12xy + 14xz - 35$$ Using the technique of Lagrange Multipliers, leads to four simultaneous ...
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3answers
43 views

Maximize a bivariate function under constraints by Lagrange multipliers

I have been given a function $f(x,y)= x^2 + y^2$ whose maximum and minimum values have been sought (if existent) under the constraint that $3x^2 + 4xy +16y^2=140$. This looks pretty much to be a ...
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Multi-variable extremisation problem when pure substitution fails to deliver all solutions

I have always thought that pure, equality substitution was one of the methods in mathematics which is always guaranteed to work. Yet I appear to have found a case when it fails and if somebody here ...
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Find maximum value under constraints by Lagrange multipliers or another method

Let $f(x,y)=(x + 2y)^2 + (3x + 4y)^2.$ Then for $(x,y)$ on the circle $x^2+y^2=1 $ what is the maximum value of $f$? I tried to use Lagrange multiplier method with the constraint $g(x,y)= x^2 + y^2 ...
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The operation of KKT condition in lagrange function

Does kkt condition do the partial differential to the lagrange multiplier i wanted,and set the equation become zero? i mean, $L=P_E+\alpha [P_T-\sum\limits _{k=1}^{K}p_k]+\gamma [\sum\limits _{j=1}^{...
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For what value of “c” volume of ellipsoid equal to $8\pi$?

The equation of ellipsoid is $$x^2+\bigg(\frac{y}{2}\bigg)^2+\bigg(\frac{z}{c}\bigg)^2=1$$ I have taking the limits of integration $$\int_{0}^{1}\int_{-2}^{2}\int_{0}^{c\sqrt{1-x^2-\frac{y^2}{4}}}...
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Find the maximum and minimum (there are four in total) that the function $f(x,y)=3xy$ achieves when $(x,y)$ travel the ellipse $x^2+y^2+xy=3$

Find the maximum and minimum (there are four in total) that the function $f(x,y)=3xy$ achieves when $(x,y)$ travel the ellipse $x^2+y^2+xy=3$ I have thought about doing the following but I do not ...
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Optimization problem / Derivative of matrix Confusion

I have attached a small snippet from my lecture notes where we are dealing with constrained optimization. $E,C$ are matrices. I am confused with Eq. (4.12) as to how they have differentiated with ...
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Minimum Value of the function of three variables using Lagrange Multipliers

Find the minimum value of the function $f(x, y, z)=4x^2+2y^2+z^2$ with the constraint $g(x, y, z)=xy+yz+zx=16$. I tried and get the following equations $8x=\lambda(y+z)$, $ 4y=\lambda (x+z) $, $...
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1answer
46 views

Solving dynamic optimization with non-binding inequality constraint

I want to solve a problem similar to the following discrete and finite time horizon dynamic optimization problem : \begin{equation} \begin{split} &\max_{\{d_t\}} \sum_{t=0}^{T} - \left [ f(s_t) + ...
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1answer
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Lagrange multiplier method - cannot resolve equations but graphically I have the answer

My problem is similar to this: Intuitive explanation for formula of maximum length of a pipe moving around a corner? However my pipe (20m long) must pass through a specific point (5,5) in the X-Y ...
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1answer
31 views

Finding the minimum and maximum values of a function over a boundary of a compact set.

I want to calculate the minimum and maximum values the function $f:z^2-2x^2-y^2-4xy-2xz-z+x$ takes on the boundary of the compact set $$\begin{cases}6x^2+y^2+z^2+4xy-2zx\leq1 \space\space(1) \\ 4x^2+y^...
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1answer
44 views

Lagrangian method for a silmple linear programming

I'm trying to solve a simple LP using Lagrangian method. But I don't know how to use the soloution of the dual problem to find the solution of the main LP. I consider the following simple problem: $$ ...
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30 views

Is there any relation between Banach duality and Lagrangian duality?

I can't help but notice some similarity between the two dual spaces. For example, the dual space of a Banach space is the space that contains all bounded linear functionals $l:B\rightarrow\mathbb{R}$, ...
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1answer
45 views

Global extrema of $f(x,y)=3x^2-y^3$ on the circle $x^2+y^2 \leq 25$

I wanted to know if my method and my results are correct. There isn't a solution to that question, so I don't know if it's correct. Given $f(x,y)=3x^2-y^3$ and I need to find its global extremas on ...
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1answer
26 views

Lagrange duality compared with Lagrange multiplier method

As we all know, Lagrange multiplier method says: in order to find the extremum of $f(x)$ over $x$, s.t. $g(x)=0$, one instead finds the extremum of $f(x)+\lambda g(x)$ over $x$ and $\lambda$. Note ...
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2answers
74 views

Constrained Optimization Geometry Confusion

In a constrained optimization problem, let's consider the example $$\begin{cases}f(x,\ y) = yx^2\ \Tiny(function\ to\ be\ maximized) \\ g(x,\ y) = x^2 + y^2 = 1\ \Tiny(constraint)\end{cases}$$ why ...
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2answers
70 views

Global extrema of $f(x,y)= e^{-4x^2-9y^2}(2x+3y)$ on the ellipse $4x^2+9y^2 \leq 72$

I'm asked to find the global extremas of the following function :$$f(x,y)= e^{-4x^2-9y^2}(2x+3y)$$ on the ellipse $4x^2+9y^2 \leq 72$ So for the inside, I have $\nabla f=(2e^{-4x^2-9y^2}+(-16x^2-24xy)...