Questions tagged [lagrange-multiplier]

This tag is for the questions on Lagrange multipliers. The method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

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variant of Levenberg-Marquardt suitable for Lagrange-Newton method

If a Newton descent is applied to some scalar function of a vector, the Hessian is positive definite in the vicinity of a minimum, but can become indefinite in larger distance from the minimum. This ...
Ralf's user avatar
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A function optimization problem with constraints

Let a, b, c be three real number constants satisfying $a^2 + b^2 + c^2 \leq 1$. Define the function $f(x, y, z) = \frac{x^2 + y^2}{2(1+z)}$ under the constraints $(x-a)^2 + (y-b)^2 + (z-c)^2 \leq \mu^...
L.Roy's user avatar
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Contour plotting for constrained optimization [closed]

I am trying to find the minimum surface area for a bottle that has a maximum volume of $50$. These are the functions: \begin{align} \text{Surface Area} &= 2.5\pi x + \pi(y + 1.25) \left( (y - 1....
Ibrahim Alzir's user avatar
2 votes
2 answers
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Prove $\frac{1}{\sqrt{a+b+7c}}+\frac{1}{\sqrt{c+b+7a}}+\frac{1}{\sqrt{a+c+7b}}\ge 1.$

Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c+abc=4.$ Prove that$$\color{black}{\frac{1}{\sqrt{a+b+7c}}+\frac{1}{\sqrt{c+b+7a}}+\frac{1}{\sqrt{a+c+7b}}\ge 1.}$$ I found the ...
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Finding $\small{\min\limits_{ab+bc+ca=1}\sqrt{a+2}+\sqrt{b+2}+\sqrt{c+2}- \sqrt{2-abc}.}$

Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Find the minimal value of expression $$P=\sqrt{a+2}+\sqrt{b+2}+\sqrt{c+2}- \sqrt{2-abc}.$$ By $a=b=1;c=0$ I get $P=2\sqrt{3}$ so we ...
Dragon boy's user avatar
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Under what circumstances are the arithmetic and geometric mean equal? [closed]

I asked question "a", but I don't understand under which circumstances the two averages will be the same. a) Determine the maximum value of $$\sqrt[n]{x_1x_2 \ldots x_n} \leq \frac{x_1 + x_2 ...
Prince J4's user avatar
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Finding the point which gives the minimum area for a circular cylinder with volume 1

let $f(x,y) = 2\pi x(x+y)$ be the function for a cylinder area with x as radius and y as height let $g(x,y) = \pi x^2 y = 1$ be the constraint (Volume) I solved this by finding the stationary point/...
willaayy's user avatar
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Use the Lagrangian function to find a point of the first octant that lies on $x^3y^2z=6\sqrt{3}$ and is closest to the coordinate origin

From the problem, I can see that $g(x,y,z)=x^3y^2z-6 \sqrt3$ is the subject. How do I find the $f(x,y,z)$, the function that I need to maximize or minimize?
Vile's user avatar
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3 answers
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I can't find the minimum point using Lagrange multipliers in that function

I need to find the maximum and minimum points of the function $f(x,y,z) = x^2 + y^2 + z^2$ restricted to $x^4 + y^4 + z^4 = 1$. I manage to find the maximum point, $\sqrt{3}$. However, the template is ...
Prince J4's user avatar
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Theoretical doubt about lagrange multipliers

while solving some exercises related to finding maximums and minimums of functions with two variables under constraints, I encountered a doubt when using Lagrange multipliers. For instance, ...
Prince J4's user avatar
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Difficult Lagrange multipliers problem

Let us consider the following convex optimization problem: \begin{equation} \begin{aligned} \max_{\mathbf{x}\in \mathbb{R}^{n}} \quad & \mathbf{c}^\top \mathbf{x} \\ \textrm{subject to} \quad &...
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Prove $\color{black}{\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\cdot\sqrt{\frac{a+b+c+5}{3}}, }$ when $ab+bc+ca+abc=4.$

If $a,b,c\ge 0: ab+bc+ca+abc=4.$ Prove that$$\color{black}{\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\cdot\sqrt{\frac{a+b+c+5}{3}}. }$$ I've tried to square both side but it leads to complicated one. ...
Anonymous's user avatar
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How to prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\ge 2+\sqrt{\frac{2}{3a+3b+3c-2}}$?

Question. Let $a,b,c\ge 0: ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\ge 2+\sqrt{\frac{2}{3a+3b+3c-2}}. }$$ I've tried to use Jichen lemma ...
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Lagrange Multipliers Not Identifying All Critical Points

Suppose that $x,y,z \in \mathbb{R}_{\ge 0}$. Consider $f(x,y,z) = xyz$ (the volume of some cube) and $g(x,y,z) = x+y+z-1$. I wish to maximise $f$ subject to $g=0$ with Lagrange multipliers. The zero ...
user9989615's user avatar
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Derive the dual problem $\min_{x,y}\max_{i=1,\dots, m}y_{i}$ such that $\ a_{i}^{\intercal}x+b_{i}\leq y_{i}, i=1,\dots, m.$

Consider the following programming problem: $$\min_{x\in\mathbb{R}^{n},y\in\mathbb{R}^{m}}\max_{i=1,\dots, m}y_{i}$$ $$s.t.\ a_{i}^{\intercal}x+b_{i}\leq y_{i}, i=1,\dots, m.$$ I want to derive the ...
JacobsonRadical's user avatar
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critical points of projection function on low dimensional set

Suppose $S = \{x \in \mathbb{R}^n | P(x) = 0\}$, for some polynomial $P$, and I want to find the critical points of $f: S \rightarrow \mathbb{R}$ which takes $(x_1, \ldots, x_n) \mapsto x_1$. It is ...
Jyothi's user avatar
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How to prove $\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.$

Question. If $a,b,c\ge 0: a+b+c=2,$ prove that $$\color{blue}{\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.}$$Equality holds iff $(a,b,c)=\{(0,1,1);(0,0,2)\}.$ ...
Inequality's user avatar
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What variational problem does the parabolic suspension bridge solve?

The catenary curve $y(x)$ minimizes the gravitational potential energy $$\int \rho g y ds=\int \rho g y \sqrt{1+y'^2}dx,$$ subject to a fixed length, $L=\int \sqrt{1+y'^2}dx.$ It is known that in a ...
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1 answer
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Interpretation of Lagrange Multiplier as Shadow Price

Consider the following problem: \begin{align} &\min_{(x, y)} (c_x \cdot x + c_y \cdot y)\\ \text{s.t.} \quad & K = x + y\\ &(x, y) \in[0, X] \times [0, \infty)\\ & c_x < c_y \\ &...
clueless's user avatar
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Calculating the Maximum Volume of a Rectangular Parallelepiped Inscribable in an Ellipsoid

I'm working on the following optimization problem and I need guidance to proceed: Step 1: Problem Definition I'm tasked with calculating the maximum volume of a rectangular parallelepiped with edges ...
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Minimization of $J(x)=\langle Ax,x\rangle$ on the unit sphere on $\mathbb{R}^{n}$

I would like to show that the minimum of $J(x)=\langle Ax,x\rangle$ (where $A$ is a symmetric $n$ by $n$ matrix) on $K=\{x\in\mathbb{R}^{n} : F(x) = 1 -\lVert x \rVert^{2} = 0 \}$ is the eigenvector ...
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Is this a useful dimensionality reduction on real projective space?

In the following, I will derive a dimensionality reduction for the real projective space. As I will use scalar product as a means of distance, I am unsure if the dimensionality reduction has a useful ...
Niklas Netter's user avatar
1 vote
2 answers
62 views

Finding Extremes of Multivariable Function $f(x,y,z)=xz−yz $

I am seeking assistance in finding the extrema of the function $f(x,y,z)=xz−yz$ when evaluated at points on the curve of intersection of two given surfaces. The surfaces of interest are defined by the ...
Ayesca's user avatar
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1 vote
2 answers
97 views

Find extreme values of $f(x,y) = x^2 + y^2$ subject to $g(x,y) = xy = 1$

Find extreme values of $f(x,y) = x^2 + y^2$ subject to $g(x,y) = xy = 1$ Quite easy calculations, but I get that values of extreme points $f(1,1) = 2$ and $f(-1,-1) = 2$. Still it is easy to determine ...
Nika Kvashali's user avatar
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0 answers
27 views

Reference request: simple Lagrange multiplier theorem/min-max theorem

I am looking for a textbook reference for a result that is something like the following: Theorem: Let $X$, $Y$ and $\Lambda$ be TVSes with $Y,\Lambda$ in strong duality, and denote the associated ...
Oxonon's user avatar
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1 answer
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Complicated Inequality Proof, Variables Subject To Constraint

Let a, b, c be positive real numbers such that $a+b+c=1$. Prove that $\frac{a}{a^2 + b^3 + c^3} + \frac{b}{b^2 + c^3 + a^3} + \frac{c}{c^2 + a^3 + b^3} \le \frac{1}{5abc}$ This is a problem from RMO ...
Yatharth Shrivastava's user avatar
1 vote
1 answer
94 views

optimization problem in the spirit of the Isoperimetric problem

I consider the functional $J(y)= \int_{0}^{a}y(x)dx$ where $a$ is fixed and $v\in H^{1}(0,a)$. We have a set of constraints $$ K = \{ y\in H^{1}(0,a) : F(y) = \int_{0}^{a}\sqrt{1+y’(x)^{2}}dx - l=0\} ...
coboy's user avatar
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2 votes
0 answers
29 views

Help finding extrema of a function using lagrange multipliers

I am trying to find the extreme values of the function $f(x,y,z)=x^2+y^2+z^2$ given the constraints $g(x,y,z)=x-y=1$ and $h(x,y,z)=y^2-z^2=1$. I am rather lost on this question as it feels as though I ...
Abbey Taylor's user avatar
0 votes
2 answers
65 views

Choose the $w_i>0$ such that $\sum_{i=1}^dw_ia_i$ is minimized and $\sum_{i=1}^dw_i=1$

Say $w,a\in\mathbb R^d$. How can we choose the $w_i$ such that they minimize $$\sum_{i=1}^dw_ia_i\tag1$$ and satisfy $$w_i>0\tag2$$ $$\sum_{i=1}^dw_i=1?\tag3$$ Usually I would solve $$\frac{\rm d}{{...
0xbadf00d's user avatar
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Can I use QR decomposition as a last step in a Constrained Orthonormal Matrix Optimization problem?

I have the optimization problem: minimise $f(V)$, where $V$ is $N\times N$, subject to $V$ is orthonormal All entries of the first column of $V$ are $1/\sqrt{N}$ $V \cdot D \cdot V^T \cdot \mathbf{1}...
Piyush Sawarkar's user avatar
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Does the Jacobian matrix of the constraint functions need to be full rank at the points that satisfy the constraints in Lagrange multiplier method?

According to my calculus textbook for example to maximize $f(x,y,z): \mathcal{D} \subset \mathbb{R}^3 \to \mathbb{R}$ with contrains $F(x,y,z)=0$ and $G(x,y,z)=0$ it is required (along with others) ...
user1206899's user avatar
3 votes
1 answer
61 views

Solving a system of equations coming from Lagrange multiplier method

I'm trying to use Lagrange multipliers to find the points on a surface given by $g(x,y) = x^4+y^4-4xy = 6$ in $R^2$ which have the smallest and largest distance to the origin $(0,0)$. So $g(x,y)=6$ is ...
GreenCoffee248's user avatar
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1 answer
78 views

How to minimize $\Phi(c_0):=c_0^2\int_0^\infty\text E\left[g(c_0,X)\left|\int_0^tf(X_s)-\int f\:{\rm d}\mu\:{\rm d}s\right|^2\right]\:{\rm d}t$?

Let $E$ be a $\mathbb R$-Banach space; $\lambda$ be a $\sigma$-finite measure on $E$; $\mu$ be a probability measure on $(E,\mathcal E)$ with density $p$ with respect to $\lambda$; $q$ be a ...
0xbadf00d's user avatar
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How can I minimize this integral functional on a function space?

I want to find a choice for the functions $w_i:E\to[0,\infty)$ satsifying $\sum_{i\in I}w_i=1$ which minimizes the functional $$\Phi(w):=\int_0^\infty\int\mu({\rm d}(i,x))\operatorname E_{(i,\:x)}\...
0xbadf00d's user avatar
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why is it using max for dual form problem for SVM?

Context SVM: From Primal to Dual Primal form: $$\min_{w, b} \frac{1}{2} \|w\|^2 + C \sum_{i=1}^M \xi_i$$ subject to $y^{(i)}(w^T x^{(i)} + b) \geq 1 - \xi_i$, $i = 1, \ldots, M$, and $\xi_i \geq 0$, $...
Kenny Ynnek's user avatar
1 vote
1 answer
44 views

Linearizing SOS1

Guys, I am trying to find ways of linearizing the operator Special Ordered Set Type 1 (SOS1). In order to understand what is my goal, I will, firstly, describe the problem I am facing. Let's consider ...
Matheus Diógenes Andrade's user avatar
-1 votes
1 answer
81 views

Alternative Proof of Lagrange Multipliers?

I think there might be a mistake in this attempted proof of Lagrange Multipliers, but I'm not sure where. Halfway through I set the total derivative of the objective function to $0$. I believe the ...
TurlocTheRed's user avatar
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1 vote
1 answer
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What is the Lagrange Multiplier value?

Is the Lagrange Multiplier typically the magnitude of the gradient of the objective function in the direction of the constraint function? Let $f(x,y,z)$ be an objective function and $g(x,y,z)=0$ be a ...
TurlocTheRed's user avatar
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1 vote
1 answer
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Comparison between surface area (wax used) of an optimized regular hexagonal prism and an optimized rhombic dodecahedron (BEE PRISM)

In my other post I was asking about the equation of volume of a rhombic dodecahedron with hexagonal base (the prism which bees use). The area is 3s(2h+(s√2)/2) and the volume is ((3s^2√3)/2)(h-s/(2√2))...
MathematicsEnjoyer's user avatar
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0 answers
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Closest distance using Lagrange multipliers

I need a help to solve this: Find the points on the curve $x^2 + 4y^2 =4$ that are closest and furthest from the points on the curve $C: x^2 +y^2 +4x+2y=20$. My try: Let $f(x,y)=d^2(x,y)=(x-t)^2 + (y-...
P. M. O.'s user avatar
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1 vote
0 answers
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Using Lagrange multiplier for finding shortest distance to implicit surface constructed by metaballs

I am writing a software in which I need to find the shortest distance from an arbitrary point in 3D space to an implicit surface defined by a set of metaballs. I wanted to achieve this by using the ...
fieldmops's user avatar
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$A= x^3 + y^3 +z^3 + kxyz $

Let $x,y,z$ are three sides of a triangle such that $x+y+z=3$. Find $k \geq 0$ such that: $$A= x^3 + y^3 +z^3 + kxyz $$ have minimum. I tried for this: $k=6,k=15,k=\frac{15}{4}$ and it have all ...
Lục Trường Phát's user avatar
0 votes
1 answer
126 views

Kuhn-Tucker conditions for inequality constraints

So, when we solve the optimization problem using Lagrange Multiplier method, I know that lambda can be positive or negative. Lambda is simply the rate of change in the optimal value when the ...
Confused_intense_thoughts's user avatar
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0 answers
53 views

Lagrange multiplier method in a discretized way

I succeeded in using the Lagrange multiplier method to solve continuous case, but I failed to solve discretized case. Assume $\min f(x,y) = x^2 + y^2$ constraints $s.t. g(x,y) = x + y -1 = 0$ use ...
sdfa's user avatar
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0 answers
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Small changes leave the binding constraints unchanged in an optimization problem

Let $X\subset \mathbb{R}^n$ and $Y\subset \mathbb{R}^m$, and $f:X\times Y\to\mathbb{R}$. $f$ is continuously differentiable, and for each $y\in Y$, $f(\cdot,y)$ is strictly convex. Consider the ...
user_lambda's user avatar
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1 vote
1 answer
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How should I go about solving this lagrange equation containing vectors and matrices?

I have a Lagrange equation which depends on a multiplier $\lambda$. \begin{equation} f(\lambda)=a^HR^{-1}a + \lambda(||a-\bar{a}||^2-\varepsilon) \end{equation} I know $\bar{a}$, $R$ and $\varepsilon$....
Dan Pollard's user avatar
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How to determine total derivative of a multidimensional function with 2 components, when using Lagrange multipliers to determine the critical points?

First of all, $g(x,y,z)= \{{(x^2+y+z),(x-3y+z^5)}\}$ is the constraint given, for which the total derivate has to be determined. Btw the function given was $f(x,y,z)= x^4+y^2+z^3$ which is not a ...
NirvanicUniverse's user avatar
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The dual of this SDP with a simple nonconvex constraint

In this problem we're optimizing over variables $X\in \text{PSD}_n$ and $Y\in\mathbb R^{d\times n}$ for some $d\le n$. \begin{align} &\text{Maximize}&&\langle A_0, Z\rangle\\ &\...
Blake's user avatar
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1 vote
1 answer
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Find $\max_{ x,y \in [b,1],\frac{p}{x} + \frac{1-p}{y} =a, p \in [0,1]} \left( p x^2 +(1-p) y^2 \right)$

I am struggling to solve the following optimization problem: \begin{align} \max_{ x,y \in [b,1],\frac{p}{x} + \frac{1-p}{y} =a, p \in [0,1]} \left( p x^2 +(1-p) y^2 \right) \end{align} where we ...
Lisa's user avatar
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1 vote
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About Implicit Function Theorem and Lagrange Multipliers

I am studying the meaning of the multiplier in Lagrange Multiplier Method and got the following question. I tried it myself, but I am not sure if it is correct. I would really appreciate it if someone ...
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