# Tagged Questions

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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### Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
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### The “pepperoni pizza problem”

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...
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### What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. [You might imagine ...
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### AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
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### What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). ...
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### Where to build a bridge to cross a river in the shape of an annulus

There is a river in the shape of an annulus. Outside the annulus there is town "A" and inside there is town "B". One must build a bridge towards the center of the annulus such that the path from A ...
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### The Stupid Computer Problem : can every polynomial be written with only one $x$?

When I was a child, I wanted to be a mathematician so I asked my parents to buy me a computer to make super complex calculations. Of course, they were not crazy enough to buy an expensive super ...
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### A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Consider the following system of real matrices, \begin{align} {\bf P} &={\bf F}({\bf I}_{n}-{\bf K} {\bf H}^\top){\bf P}{\bf F}^\top+{\bf Q}, \;\;\;\;\;\;\;\;\text{where}\;\;\;\;{\bf ...
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### Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
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### Run away from lions in a cage

I came across an interesting problem: There is a round cage and you are in it. Also two lions are in this cage too. The start position is that the distance between you and both lions is the ...
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### How do Lagrange multipliers work to find the lowest value of a function subject to a constraint?

I have been using Lagrange multipliers in constrained optimization problems, but I don't see how they actually work to simultaneously satisfy the constraint and find the lowest possible value of an ...
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### Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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### Math Wizardry - Formula for selecting the best spell

Imagine we have a wizard that knows a few spells. Each spell has 3 attributes: Damage, cooldown time, and a cast time. Cooldown time: the amount of time (t) it takes before being able to cast that ...
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### Derivation of the method of Lagrange multipliers?

I've always used the method of Lagrange multipliers with blind confidence that it will give the correct results when optimizing problems with constraints. But I would like to know if anyone can ...
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### Largest circle between $y=x^n$ and $y=\sqrt[n]{x}$

Something I have been wondering about for a while. Let us look at the area between $x^n$ and $\sqrt[n]{x}$ when $x\in [0,1]$. Where $n$ is a positive integer. Below is an image. With a given n, how ...
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### Why does a distance and its square reach their minimum at the same point?

There is a question in my calculus textbook that asks to find a point on the parabola $y^2 = 2x$ that is closest to point $(1,4)$. They want us to first use the distance formula, but then proceeded ...
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### What does “curly (curved) less than” sign $\succcurlyeq$ mean?

I am reading "Convex Optimization" by Stephen Boyd. He is using a curved greater than and curved less than equal to signs. $f(x^*) \succcurlyeq \alpha$ or $f(x*) \preccurlyeq \alpha$ Can someone ...
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### What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
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### If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$?

Let each of $A, B, C, D, E$ be an angle that is less than $180^\circ$ and is greater than $0^\circ$. Note that each angle can be neither $0^\circ$ nor $180^\circ$. If $A+B+C+D+E = 540^\circ,$ what is ...
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### How fat is a triangle?

The slimness factor of a geometric shape in 2 dimensions is the ratio between the side-length of its smallest containing square and its largest contained square. This is an important factor in ...
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### Problem on EU commission

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
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### How to prove the sum of squares is minimum?

Given $n$ positive values. Their sum is $k$. $$x_1 + x_2 + \cdots + x_n = k$$ The sum of their squares is defined as: $$x_1^2 + x_2^2 + \cdots + x_n^2$$ I think that the sum of squares is ...
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### How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
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### 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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### Gerrymandering/Optimization of electoral districts for one particular party

I'm asking this on behalf of Zach Weiner (actually it's my own initiative in order to promote this site). Original text is here, and is as follows: Hey-- This is Zach from SMBC, and I have a math ...
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### Arnol'd's trivium problem #68

I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my ...
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### Minmax problem for polygons

Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ sides. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles whose vertices are some vertices of $P$. I ...
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### What is the best strategy for Cookie-Clicker-esque games?

Today, I stumbled across the game Cookie Clicker, which I recommend you avoid until you have at least a few hours of time to waste. The basic idea behind the game is this: You have a large stash of ...
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### Does a polynomial that's bounded below have a global minimum?

Must a polynomial function $f \in \mathbb{R}[x_1, \ldots, x_n]$ that's lower bounded by some $\lambda \in \mathbb{R}$ have a global minimum over $\mathbb{R}^n$?
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### Do dynamic programming and greedy algorithms solve the same type of problems?

I wonder if dynamic programming and greedy algorithms solve the same type of problems, either accurately or approximately? Specifically, As far as I know, the type of problems that dynamic ...
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### Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
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### Maximize $x_1x_2+x_2x_3+\cdots+x_nx_1$

Let $x_1,x_2,\ldots,x_n$ be $n$ non-negative numbers ($n>2$) with a fixed sum $S$. What is the maximum of $x_1x_2+x_2x_3+\cdots+x_nx_1$?
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### Time-optimal control to the origin for two first order ODES - Trying to take control as we speak!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...
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### Bath towel on the rope: minimize the area of self-intersection of a folded rectangle

This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o). There is this rectangle with sides $a<b$. The ...
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### The Farmyard problem

Problem: There is a farmer who has a $1\text{ mile}\times 1\text{ mile}$ square piece of land. He knows that there is a completely straight pipe underneath some part of his property, but it could ...
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### How can I find 3 positive numbers that have a sum of 1 and the sum of their squares is minimum?

How can I find 3 positive numbers that have a sum of 1 and the sum of their squares is minimum? So far I have: $$x+y+z=1$$ $$z=1-(x+y)$$ $$f(x,y)=xyz=xy(1-x-y)$$ But I'm stuck from here. Hints?
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### $|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
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### Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
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### Searching for tighter bounds

I have to solve an equation $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where $$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$ The $z_i$ are all positive and add ...
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