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

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Lagrange multipliers: when is local extremum a global extremum?

Consider the following Olympiad problem from the IMO shortlist: Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that: $36 \leq 4 \left(a^3+b^3+c^3+d^...
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2answers
430 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - y)$...
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1answer
37 views

maximize 3-variable linear function [version 1.0]

This problem came up when I was trying to solve a bigger, probabilistic problem. So at the end it boils down to this: how can we maximize the function $f(x_2,x_3,x_4) = \frac{18}{100}x_2 + \frac{...
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3answers
62 views

Find minimum and maximum on range

$f(x,y)=x^{4}-x^{2}+y^{2}$ $B={(x,y)\in \mathbb R, x^{2}+y^{2}\leq 1 }$ I should find minimum and maximum of this function on the range B. I tried it with Lagrange Multiplier and I got these points ...
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2answers
46 views

Lagrange multipliers with trigonometric functions. Stucked figuring out x and y values.

I want to find the maximum of the function $f(x,y) = \cos^2(x) + \cos^2(y)$ with the constraint $x-y = \pi/4$. Here are my partial derivatives: $$f_x = -2\cos(x)\cdot\sin(x)$$ $$f_y = -2\cos(y)\cdot\...
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3answers
48 views

Can't find minimum using Lagrange multipliers

I want to find the minimum of the function $f(x,y) = x + y^2$ with the constraint $2x^2 +y^2 = 1$. Here are my partial derivatives: $$f_x = 1$$ $$f_y = 2y$$ $$g_x = 4x$$ $$g_y = 2y$$ I have the ...
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Lagrange Multipliers Method of solving Question

Find the maximum and minimum values of $f(x, y) = x^2 + y^2$ subject to the constraint $x^2 − 2x + y^2 − 4y = 0$ So I have to use lagrange multipliers $ \nabla f(x,y) = \lambda\nabla g(x,y) $ $$ ...
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0answers
159 views

Least-squares problem with quadratic equality constraint

I want to find the solution of a Lagrange equation whose inputs are matrices. First I have the equation Ax=0. By decomposing $A$ into $A_3$ (columns 9 to 11 of A), $A_9$ (the rest of the columns), ...
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3answers
653 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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233 views

Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n ...
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23 views

Extrema on (compact) vinculum

My textbook ask to find the extrema of $f(x,y) = 2x^2+y^2$ on $x^4-x^2+y^2-5=0$. It uses the lagrangian multipliers to find critic points.. Then it computes the function on these points then says "...
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Lagrange multipliers question with 2 constraints

Let $A=\{x\in \mathbb{R}^n|\sum x_i=n/3, \sum x_i^2=n \}$ $f(x)=\sum x_i^3$ Prove that max of f on A is of the form: $x=(a,a,.....,a,b,b...,b)$ (no need to find a or b). So with Lagrange ...
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66 views

How to obtain the optimal Lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found?

I am using a black-box solver to solve the following non-convex QCQP to global optimality. $$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$ where $Q_0$ is ...
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44 views

Generalities regarding the Lagrange Multiplier

Apparently the following general statement is true. "Let $\gamma:g(x,y)=0$ be a closed curve that doesn't cross itself. If the maximisation of a function $f(x,y)$ on $g(x,y)$ using Lagrange ...
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10 views

Lagrangian Relaxation of quadratically constrained quadratic program

I have the following problem: $$ \min_{w,\theta\ge0}\frac{1}{2}\|w-w_t\|^2+(\theta-\theta_t)^2 \text{ s.t. } w^\top(\hat n\hat z-nz)+\theta w^\top(z-\hat z)+1 \le 0,\theta-1\le 0 $$ Notice that $w$ is ...
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41 views

How to solve a binary generalized assignment problem

I have the following generalized assignment problem: Z=max $\sum_{i=1}^{N}\sum_{j=1}^{M} x_{ij}R_{ij}$ such that $\quad 1)\quad \sum_{j=1}^{M} x_{ij}=1 \quad \forall i$ $\quad\quad\...
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47 views

Optimizing a problem using Lagrange multipliers

$\newcommand{\norm}[1]{\|#1\|}$ I have the following problem: $$ \min_{w,\theta}\frac{1}{2}\norm{w-w_t}^2+\frac{1}{2}(\theta-\theta_t)^2 \text{ s.t. } w^\top(z(n-\theta)-\hat z(\hat n - \theta)) \ge 1 ...
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2answers
470 views

Using Lagrange multipliers to find the 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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32 views

Find the farthest and nearest point of an ellipsoid

The equation of ellipsoid is $$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$ The ellipsoid is arbitary rotated and the orientation angle are given and center is at (x',y',z'). The radius of the ...
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1answer
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Extrema of $f(x)=g(|x-a|^2,|x-b|^2,|x-c|^2):\Bbb{R}^n\to \Bbb{R}$ in $S=\{x: |x|=1\}\subset \Bbb{R}^n$ is a linear combination of $a,b,c$

Since I am getting pretty close to the final exams, I would really yield from having my practice challenged and corrected. Question: Let $a,b,c\in \Bbb{R}^n$ be independent vectors, and $g\in C^{1}(\...
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1answer
42 views

Is Lagrangian Multiplier Equivalent to Brute Force for binary decision variables

I have a set of variables $x_{i} \in \{1,k\} $ in a non linear optimization problem. As this variable has only two possibilities I have encoded this into a constraint. I assumed having equality ...
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0answers
40 views

Finding the minimum of $x_1 + \cdots + x_n$ on ellipsoid

Let $A$ be a positive definite matrix $n \times n$ and $u^T = [1 \cdots 1]$. Use Lagrange multipliers to find the minimum of $f(x) = u^Tx$ on $h(x) = \frac{x^TAx}{2} = 2$ This is what I did. $$L(x,...
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0answers
64 views

I'm walking towards my car - when should I try the remote, in an optimal sense?

I'm interested to learn about how discrete/'event' based elements are incorporated into optimisation problems. Hopefully this is an interesting problem in its own regard, it's inspired by a daily ...
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2answers
67 views

How do I solve the following equality-constrained quadratic program?

I am trying to minimize: $$(x_1-k_1)^2 + (x_2-k_2)^2 + (x_3-k_3)^2 +\ldots+ (x_n-k_n)^2$$ subject to following equality: $$B = 1 + x_1 + x_2 + x_3 + x_4+\ldots+x_n.$$ Is there a closed form ...
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normalization of constraints $ 0 \leq x \leq 1 $ in Lagrangian KKT

With Lagrangian we have an objective function and a set of equality constraints of form $ g_{i}(x_{j}) = 0 $ . With KKT we can have another set of inequality constraints of the form $ h_{i}(x_{j}) \...
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1answer
46 views

Directional derivative and lagrange multipliers

Find the points $(x,y)\in \mathbb R^2$ and unit vectors $\vec u$ such that the directional derivative of $f(x,y)=3x^2+y$ has the maximum value if $(x,y)$ is in the circle $x^2+y^2=1$ My attempt: ...
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1answer
400 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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34 views

Using the Lagrange method to find max/min of $f(x,y) = \frac{x^3}3 + y$

Problem Use the Lagrange method to find max/min of $f(x,y) = \frac{x^3}3 + y$ Subject to the constraint $x^2 + y^2 = 1$ My attempt The constraint gives us $g(x,y) = x^2 + y^2 - 1$ $\displaystyle\...
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1answer
25 views

Alternative solution to a Lagrange Method Optimization Problem

Find extrema of $f(x,y,z)=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$ subject to $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ by reducing variables and then using the Single Variable Method or by using ...
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Solving Binary Linear Programming Problem Using KKT

Execuse me, I know that if I searched a lot I could find the answer, However I have already did my research and I am running out of time. I need the detailed solution of the following linear problem (...
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30 views

For some $f:\Bbb{R}^n\to \Bbb{R}$, $A\subset \Bbb{R}^n$ and $B\subset A$, show that $\max_{A}(f)=\max_{B}(f)$

Let $A=\{(x_1,...,x_n)|{1\over n}(\sum_{i=1}^n{x_i})={1\over 3},{1\over n}(\sum_{i=1}^{n}{x_i^2}))=1\}\subset \Bbb{R}^n$, and let $B\subset A$ be a subsets of points from $A$ of the form $${(\...
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52 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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3answers
87 views

Hottest and coldest points on a heated circular plate (use Lagrange multipliers)

A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using ...
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1answer
43 views

Why are most Lagrange multipliers zero in the SVM solution?

I read everywhere that a non-zero Lagrange multiplier $\lambda_i$ signifies that the corresponding point $x_i$ is a support vector, but I can't see how a support vector and a non-support vector have a ...
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3answers
49 views

Find points that give the shortest distance between $y = x^2$ and $y-x+2=0$ using Lagrange multipliers

I am asked to find, using Lagrange multipliers, the points on $y = x^2$ and $y-x+2=0$ that give the shortest distance between the curves. Obviously, $d(x,y) = \sqrt{(x-x_0)^2 + (y-y_0)^2}$, but I am ...
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3answers
470 views

Constrained optimization max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$

max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$ I need to solve this by the Kuhn Tucker conditions without using concavity of the Lagrangian.
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1answer
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Position of vertices of right triangle inscribed on $x^2+4y^2=1$ with maximum area using Lagrange Multipliers

I am asked to find, using Lagrange multipliers, the position of the vertices of a right triangle inscribed on $x^2+4y^2=1$ that has the maximum area. The two legs of the triangle (which are not the ...
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2answers
152 views

Why can/should we use 5 instead of 10?

Problem: A pharmacy has a uniform annual demand for 200 bottles of a certain antibiotic. It costs \$10 per year for a storage place for one bottle, and $40 to place an order. How many times during ...
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7 views

Conditional extremes, solving $xa+yb < (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$ if.

Conditional extremes, solving $$xa+yb \leq (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$$ using lagrange multipliers.. If $\frac{1}{q}+\frac{1}{p}=1$ and $p,q>1$. This reminds me of Holders ...
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Is convexity of the objective function sufficient for a local maxima to be a global maximum?

In my problem, I have to maximize a convex function $f(x_1,x_2,\cdots,x_n)$ subject to two equality constraints $g_1=0$ and $g_2=0$. As usual, I constructed the Lagrangian $L=f+\lambda_1g_1+\...
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1answer
33 views

Word Problem Lagrange Method

I am studying for my exams and got very very stuck at a word problem on the Lagrange Methods, my biggest difficulty is to properly identify the function to be maximized (in this case) and so its ...
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61 views

Find the maximum distance from origin to the surface?

I am having trouble with this problem... I need to find the maximum distance from the origin to the surface $$f:=\frac{x^4}{2^4}+\frac{y^4}{3^4}+\frac{z^4}{(\sqrt2)^4}-1$$ I think I have to use ...
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2answers
39 views

How is f(4,4,4)=48 a local minimum? Can it be inferred that it is either a maxima or minima & only one extreme value within the constraint?

Disclaimer: In the definition (Stewart Calculus, 7E): "Method of Lagrange Multipliers" part (b)- Evaluate $f$ at all extreme points $(x,y,z)$ from step a. The largest of these values is the maximum ...
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1answer
48 views

Maximized surface area of box with fixed length

Assuming we have a box given that the sum of all intervals is $a$. What is the maximal surface area of the box? I know I need to use Lagrange multiplier but when I find the hessian matrix I get that ...
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1answer
30 views

Solving a quadratic convex optimization problem

There's this convex optimization problem which I got stuck after writing the Lagrange equation. I simply couldn't find a way to eliminate the Lagrange multiplier. $$\begin{array}{ll} \text{minimize} &...
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1answer
88 views

Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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1answer
72 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
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2answers
167 views

Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?

The global maximum and the global minimum of the function $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ can be found using Lagrange multipliers. $\nabla f = \lambda \nabla g$ $g(x,y,z)=x^2+2y^...
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0answers
36 views

Finding the Maximum and Minimum values w/constraint [duplicate]

I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is. Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ $x$$...
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1answer
18 views

Lagrange method with vectors?

How does one apply the Lagrange method to vectors? The problem I have (it's financial) is $$\max_w w^T r - w^T\Sigma ^{-1} w $$ under the condition that $w_1 + w_2 = 1$. $r$ is a known vector with 2 ...