Questions tagged [optimization]

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is $g(f)$ strictly convex. My attempt, since $f$ is…
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Quickest general strategy for village meeting

Here is a problem, which neither I nor my friends (very experienced in solving things like this) can't solve. But it was used for a competition several years ago and one guy solved it there, as far as I know. Unfortunately the solution (and the guy)…
klm123
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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 unconstrained minimization of the least-squares…
Tim
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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 diameter of the circle (you are on opposite sides of…
Sasha
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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 currency, which constantly increases at the given…
PhiNotPi
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What is the definition of a first order method?

The term "first order method" is often used to categorize a numerical optimization method for say constrained minimization, and similarly for a "second order method". I wonder what is the exact definition of a first (or second) order method. Does…
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Fitting $\cos(x)$ with a polynomial

Consider the graph of $\cos(x)$ on the interval $[-\pi/2,\pi/2]$. It looks very close to a parabola. I would like to find the parabola (or possibly family of parabolas) that "fit" the $\cos$ function the best on this interval. By "fit" the best, I…
Bonnaduck
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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 of that assertion, I began to wonder: What is…
Milo Brandt
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How do you prove ${n \choose k}$ is maximum when $k$ is $ \lceil \tfrac n2 \rceil$ or $ \lfloor \tfrac n2\rfloor $?

How do you prove $n \choose k$ is maximum when $k$ is $\lceil n/2 \rceil$ or $\lfloor n/2 \rfloor$? This link provides a proof of sorts but it is not satisfying. From what I understand, it focuses on product pairings present in $k! (n-k)!$ term…
curryage
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How to prove the sum of squares is minimum?

Given $n$ nonnegative 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 minimum when $x_1 = x_2 = \cdots = x_n$. But I can't…
Jingguo Yao
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Can the lion protect the sheep from three wolves?

Generally, in pursuit-evasion games, there's one prey and one or many pursuers. I'd like to know how extending the food chain would change the dynamics of such games. Specifically, let's consider a closed, circular shape arena in $\mathbb{R}^2$.…
Eric
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Cutting a cuboid to fit in a hemisphere

Today while making dinner consisting of instant noodles, I thought of the most ridiculous question I've ever asked this site. The Instant Noodle Problem Suppose you are a college student preparing one of those cuboid-shaped instant noodle packages.…
Graviton
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The generous lazy caterer

It is well-known that $n$ chords divide any convex shape into at most $\frac{n^2+n+2}2=T_n+1$ regions – the lazy caterer's sequence. For example, the pancake below is cut into seven pieces by three cuts: If these are meant for seven people,…
Parcly Taxel
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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 provide or recommend a derivation of the method at…
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Finding $n$ points $x_i$ in the plane, with $\sum_{i=1}^n \vert x_i \vert^2=1$, minimizing $\sum_{i\neq j}^n \frac{1}{\sqrt{\vert x_i-x_j \vert}}$

Let $x_1,..,x_n$ be points in $\mathbb R^2$ under the constraint $$\sum_{i=1}^n \vert x_i \vert^2=1.$$ So not all the points are on the circle, but their sum of the norms is constrained. I am looking for the minimizing configuration of the…