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

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In Lagrange Multiplier, why level curves of $f$ and $g$ are tangent to each other?

In Lagrange multiplier method, e.g. optimize a function $f(x_1, \dots, x_n)$ under a constraint $g(x_1, \dots, x_n) = 0$. There is a fact that $\nabla f$ is parallel to $\nabla g$ which is given rise ...
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0answers
14 views

Minimizing convex functions without compatible gradients

I've been working on a minimization problem for a while, involving "simple" conditions, but haven't been able to figure it out. I've tried using Lagrange Multipliers and KKT, but the presence of ...
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0answers
29 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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2answers
31 views

Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point (3, 1, -1).

Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point $(3, 1, -1)$. I identified that this is a constrained optimisation problem which I will solve ...
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0answers
38 views

Global extremum when the constraint is not compact?

When the constraint is compact, the function must have both a global maximum and a global minimum somewhere in the constraint. However, if the constraint is not compact, the global extremum may not ...
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1answer
22 views

Minimum and maximum with lagrange multiplier

I have a function with two constraints whose intersection is unitary circumference. $$x^2+y^2+z^2=1$$ and $$x+y+z=0$$ I can't understand why I cannot apply the lagrange multipliers method with only ...
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0answers
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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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3answers
587 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be ...
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0answers
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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0answers
26 views

Dude with Lagrange Multipliers [on hold]

Good morning, i have a problem, i don't understant very good how it work lagrange multipliers. I working in a problem, but i don't know found the $f(x,y)$ equation and the restriction. The problem: ...
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1answer
25 views

show that $\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big)$

Let $\alpha_1,\alpha_2,...,\alpha_n>0.$ How can I show that $$\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big).$$ Please provide me ...
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1answer
31 views

How to prove the following inequality using Lagrangian multipliers?

Find the maximum and minimum values of the function $f(x,y,z)=(xyz)^2$ where $(x,y,z)$ is on the sphere $x^2+y^2+z^2=r^2$. Then show using above that $(abc)^{1/3} \leq (a+b+c)/3$. For non-negative ...
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1answer
382 views

How can I find the minimum distance between the origin $(0,0)$ and the curve $y=1-x^2$ using Lagrange multipliers?

I used the curve as the constraint and the origin as the point but I am not sure if that is correct.
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1answer
43 views

Solving optimization with Lagrange multipliers

I am fairly new to Lagrange multipliers. Can someone please show me how to maximize the following function: \begin{align} f(x,y)=240\sqrt{x}+y \end{align} Subject to: \begin{align} 30x+y=720 ...
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1answer
61 views

How do you solve this system of equations? [closed]

How do you solve this system of equations? Use the method of Lagrange Multipliers to find the minimum and maximum values of the surface area for all closed boxes with volume 800 cubic inches and ...
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0answers
11 views

How to obtain gaussian(normal) distribution

I heard that Gaussian distribution(Normal distribution) is obtained by maximum entropy theorem. Using lagrange mutilplier, Gaussian distribution is easily obtained. However, it's too hard for me. ...
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14 views

SVM: How to normalize |WX| > 0 into |WX| = 1

Question What are the reason/basis/rationale and the actual steps and design/mechanism behind to do the normalization to convert |WX| > 0 into |WX| = 1 in the process of getting the optimal W for the ...
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1answer
932 views

Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
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0answers
5 views

Axis-aligned bound constraints and algebraic optimization

What is the methodology for optimizing a function with a interval bounded constraint? I guess the solution has something to do with KKT conditions and linearizing the constraint, but I'm stuck and I ...
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1answer
12 views

Correct formulation of equality and non-negativity constrained non-linear minimization problem

I am trying to minimize a non-linear function with both equality and non-negativity constraints numerically (not analytically) using gradient based methods and without software packages. ...
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2answers
25 views

Find the point on the ellipse where the cylinder intersects the plane furthest from the origin?

I'm confused about how I should set this problem up. It is a lagrange problem. The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from ...
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2answers
96 views

Finding extrema points with lagrange multipliers

Using lagrange multipliers, find all the extrema points of the function $f(x,y) = x^2 + (y-b)^2$ subject to the constraint $y = x^2$. Using the fact that critical points occur at $\triangledown ...
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1answer
42 views

Finding the maximum and minimum values of the function $u(x,y)=e^x \cos y$.

Let $D$ denote the unit disk centered at the origin. I'm trying to find the maximum and minimum values of the function $$u:D \to \mathbb{R},\,\ (x,y) \mapsto e^x \cos y.$$ I'll try using Lagrange ...
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1answer
12 views

LaGrange Multiplier Problem Maximum Number of Patient Visits

The Cobb-Douglas patient visit function for a clinic is given by f(x,y) = 1000x^(.7)y^(.3) where x represents numbers of doctors; y represents the numbers of nurses. If a doctor gets 42,000 dollars ...
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2answers
38 views

Lagrange Multiplier in 3D

Find the minimum and maximum values of the function $f(x,y,z) = x+2y+3z$ where $(x,y,z)$ is on the sphere $x^2+y^2+z^2=1$ using Lagrange multiplier. So I put them into the Lagrange form and got ...
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1answer
35 views

vector calculus using Lagrange Multipliers

$(1)$ Let $c\in R$ be a constant. Using Lagrange Multipliers, find all the extrema of $$f(x,y) = x^2 + (y-c)^2$$ subject to the constraint $$y = x^2$$ I'm pretty sure I've found the critical ...
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1answer
27 views

Finding extreme values using Lagrange multipliers given constraint

Find the extreme values of the function subject to the given constraint.$$f(x,\, y) = y^2 - x^2,\, x^2 + y^2 = 16$$ I understand how to to compute the extrema using Lagrange multipliers and lambda ...
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1answer
40 views

Lagrange's theorem and convex functions

Let f:UāŠ‚ $\mathbb{R}^n$--->$\mathbb{R}$ a $C^1$ function with U being convex and an open set. Let g: U āŠ‚ $\mathbb{R}^n$ ---> $\mathbb{R}^m$ (with m smaller than n) an affine application. Let M={x $ ...
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1answer
20 views

Why is the lagrange multiplier a constant?

I am trying to understand isoparametric graph partitioning. Specifically, we have a graph defined by a Laplacian Matrix $L=D-W$, where $W_{ij}$ is the weight of the edge from i to j and $D$ a ...
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1answer
66 views

Are there any global extrema in this Lagrange Multiplier problem?

I'm trying to find the max and mins of the equation $f(x,y,z) = xy + 3xz + 2yz$ on the constraint, $g(x,y,z)=5x+9y+z-10$. So according to the Lagrange Multiplier procedure, I take the partial ...
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0answers
7 views

Can an infeasible point be used to initialize an Active Set Method (optimization)

Consider an optimization problem with a quadratic objective function and linear inequality and equality constraints. Consider an Active Set Method for optimization. Say you do not know a feasible ...
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1answer
46 views

How do I classify extrema found using Lagrange multipliers?

Ok so I have found a bunch of local extrema using the method of Lagrange multipliers. Now how do I classify them as minimum or maximum? I cant use the second derivative test because its not a ...
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1answer
404 views

Find maximum of a double integral over a region

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

Find a solution of optimal problem with an inequality constraint

Let $a,b,x$ be vectors in $R^n$, A be a matrix, $c,d \in R, c<d$. Solve the following problem: $$\begin{cases} \text{minimize} \quad (b-Ax)^T(b-Ax)\\ (a^Tx-c).(a^Tx-d) \leq 0 \end{cases}$$ Assume ...
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1answer
23 views

lagrangian multipliers proof

I am not able to understand the theory behind the Lagrangian multipliers . $f(x,y)$ is the function $g(x,y)$ is the constraint , then let $F(x,y)= f(x)- \lambda g(x,y)$ Now can someone please tell ...
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1answer
20 views

Minimization problem with infinite variables and linear constraints

How can this minimization problem be solved? $$ \left\{\begin{matrix} \begin {aligned} &\sum_{i=1}^{\infty}P_i^3 \rightarrow min \\&\sum_{i=1}^{\infty}P_i=1 \\ &P_i\geqslant 0 \:for\: ...
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0answers
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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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1answer
409 views

Using Lagrange multipliers to find shortest distance between two straight lines

A problem asks me to use the method of Lagrange multipliers to find the shortest distance between the straight lines $x=y = z$ and $x = -y, z=2$ (It also warns me that using this method is a bit ...
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0answers
14 views

Solve convex optimization problem with objective function having power \alpha >1

I am not able to figure out how to get the explicit solution of the following minimization problem: $$\min_{\mathbf{w}\in \mathbb{R}^n} = \sum_{i = ...
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0answers
24 views

Use Lagrange multipliers to find the maximum and the minimum of $f$?

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s) $f(x,y)=xyz$ such that $x^2+y^2+z^2=3$. So far we have $$\begin{align} f(x,y)&=xyz\\ ...
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3answers
101 views

Finding the maximum of $f(x,y,z)=x^ay^bz^c$ where $x,y,z\in [0,\infty)$ and $x^k+y^k+z^k=1$

Given $a,b,c,k > 0$, find the maximum of $f(x,y,z)=x^ay^bz^c$ where $x,y,z\in [0,\infty)$ and $x^k+y^k+z^k=1$ The subject is Lagrange multipliers, thus that is what I tried to use, where the ...
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4answers
328 views

How to solve this difficult system of equations?

$$1+4\lambda x^{3}-4\lambda y = 0$$ $$4\lambda y^{3}-4\lambda x = 0$$ $$x^{4}+y^{4}-4xy = 0$$ I can't deal with it. How to solve this?
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How to solve a multiple knapsack problem?

I have the following binary LP max $\sum_{l=1}^{L}\sum_{f=1}^{F}[S_{f} \sum_{k=1}^{K}a_{kl}b_{kf}]x_{lf}$ s.t $\quad 1)\quad \sum_{f=1}^{F}x_{lf}S_{f}\leq C_{l} \quad \forall l$ $\quad 2)\quad ...
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0answers
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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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1answer
21 views

Given $a,b,c,k>0$ and Maximize $f(x)=x^ay^bz^c$ for $x,y,z \in [0, \infty)$ on $A=\{\,(x,y,z)\mid x^k+y^k+z^k=1\,\}$

Given $a,b,c,k>0$ Maximize $f(x)=x^ay^bz^c$ for $x,y,z \in [0, \infty)$ on $A=\{\,(x,y,z)\mid x^k+y^k+z^k=1\,\}$ So I used $g(x,y,z)=x^k+y^k+z^k-1$ with the Lagrange method and obtained ...
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1answer
18 views

Conditional Extreme. Find a point in $\mathbb{R^2}$ that has the smallest sum of squared distances from the lines $x=0,y=0, x-y+1=0.$

I can find the main function, but I do not know the condition, to set up the Lagrange equation. Can anyone see, what condition the point has to satisfy here?(So as to apply the Lagrange multiplier ...
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1answer
20 views

With the given point $M(a,b,c)$ in $\mathbb R^3$, find the tetrahedron with the smallest volume that is formed with a plane that..

With the given point $M(a,b,c)$ in $\mathbb R^3$, find the tetrahedron with the smallest volume that is formed with a plane that contains $M$ and who's points are the intersections of that plane with ...
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1answer
20 views

Trust-region method

The question has to do with the trust-region method for unconstrained optimization. I came across it on p.~392 of Linear and Nonlinear Optimization, by Griva, Nash and Sofer. Let $p(\lambda)$ be ...
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0answers
12 views

Convert Quadratically constrained basis pursuit to LASSO

The Quadratically constrained basis pursuit is to solve \begin{align} \hat{\boldsymbol{x}} &= \arg\min \|\boldsymbol{x} \|_1 \\ s.t. & \| \boldsymbol{Ax} - \boldsymbol{y} \|_2^2 < \eta ...
3
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2answers
54 views

How to maximize the function $f(x,y)= x^2+2y^2$ subjected to constraint $y-x^2+1=0$?

I want to maximize the function $f(x,y)= x^2+2y^2$ subjected to constraint $y-x^2+1=0$ Using Lagrange multipliers $$2x=\lambda(-2x) $$ hence $\lambda=-1$ $$4y=\lambda $$ hence $y=\frac{-1}{4}$ So the ...