# 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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### 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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### 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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### 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 to ...
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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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### 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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### 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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In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size ...
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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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### 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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### 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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### 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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### Maximum total distance between points on a sphere

What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$? The sum of distances is measured by summing the ...
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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 natural ...
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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 an ...
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### Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
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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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### Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = v_1$...
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### Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x|$$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and ...