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
18 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
31 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 $$\sum x_i = 1$$ $$\sum C_i x_i \leq B $$ $$\forall i \ \ x_i \geq 0 $$ ...
2
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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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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
10 views

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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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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3answers
594 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
26 views

Dude with Lagrange Multipliers [closed]

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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0answers
40 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
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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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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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
13 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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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 ...
0
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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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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
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 ...
0
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1answer
36 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
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
41 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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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 ...
2
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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
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
21 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
16 views

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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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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0answers
13 views

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
14 views

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 ...
2
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1answer
67 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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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 ...
0
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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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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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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 ...
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0answers
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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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 ...
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1answer
32 views

Find the extrema of $\sum_{i=1}^n u_i v_i \log \left| \frac{v_i}{u_i} \right|$

This question is similar to the following one: Maximizing and minimizing dot products. However there are significant differences, hence I opened a new question. Maximize and minimize $$\sum_{i=1}^n ...
2
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1answer
53 views

Maximizing and minimizing dot products

Given 2 vectors $u,v \in \mathbb{R^n}$ such that $\|u\| = 1$ and $\sum_{i=1}^n v_i= c$ where $c<1$, I would like to maximize $$\sum_{i=1}^n u_i v_i \log (v_i)$$ and minimize $$\sum_{i=1}^n u_i v_i ...
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1answer
53 views

Adding constraints in a constrained problem

Consider a simplified version of a problem I am looking at: $$\min_{x, y, z, t_1, t_2, t_3} x - x^2 - y + y^2 - z + z^2 + t_1$$ subject to: $$ -x + x^2 \leq a + t_1$$ $$ -y + y^2 \leq b - t_2$$ $$ -z ...
2
votes
1answer
42 views

Inside an elliptical paraboloid with an equation $z=\frac{x^2}{a^2}+ \frac{y^2}{b^2}$ bounded by $z=h$ draw an right-angle parallelepiped..

Inside an elliptical paraboloid with an equation $z=\frac{x^2}{a^2}+ \frac{y^2}{b^2}$ bounded by $z=h$ draw an right-angle parallelepiped.. with the largest possible volume. What confuses my most is ...
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1answer
23 views

Constrained Optimization

I am taking an intro real analysis course and we just covered constrained optimization.I remember the Lagrange multiplier method from multi variable calc, but I want to understand the reasoning behind ...
4
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2answers
131 views

Radius of a largest circle inscribed under $y=\frac{1}{(1+x^2)^n}$, closed form

The curve $y=\frac{1}{1+x^2}$ has an obvious connection to circles, because it's the derivative of the arctangent function. Besides, if we inscribe a circle under it, its radius is exactly ...
0
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1answer
45 views

How to solve a binary LP.

I have the optimization problem given below max $\sum_{i=1}^{N}\sum_{j=1}^{M} x_{ij}R_{ij}$ s.t $\quad 1)\quad \sum_{j=1}^{M} x_{ij}=1 \quad \forall i$ $\quad 2)\quad x_{ij} \in {0,1}$ $\quad ...