Mathematics Stack Exchange
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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").

22512 questions
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Constrained Optimization in One Dimension

This is a weird optimization problem I recently came across which I cannot solve. Suppose we start off with an empty container of M&Ms. We take and add M&Ms to the container every year for $n$ years. The number of M&Ms I take out each year, $B$, is…
Matt Brenneman
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The minimum of P.

For $x,\,y,\,z>0$ and $x^2y^2z^2+\left(1+x\right)\left(1+y\right)\left(1+z\right)\ge x+y+z+xy+yz+zx+3.$ Find Min…
I Love Math
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Find minimum value of the expression $x^2 +y^2$ subject to conditions

Find the values of $x$, $y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$. Tried Cauchy-Schwarz and AM - GM, unable to do.
user71408
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Biggest rectangle inscribed in a triangle

Bonjour! Being the low level matematician that I am (unfortunately) I can't seem I find a way to solve this practical problem. I have a triangle, which sides are 40 for the base and 27 for the two sides. I want to find out which is the biggest…
Brick Top
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Finding an optimal function subject to some constraints.

I'm looking for hints on a problem I am facing. Not sure where to look to learn to solve such a thing or determine properties of the solution. I need to find an $f(x)$ that maximizes: $\int_b^a{f(x)w(x)dx}$ $w(x)$ is a density function bounded…
CommonerG
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Find $\max \sum_{i=1}^9x_i$ when $\sum_{i=1}^9x_i^3 = 0$ and $|x_i|\leq1$

Given are the real numbers $x_1,x_2,\dots,x_9$ which satisfy the conditions $\sum_{i=1}^9x_i^3 = 0$ and $|x_i|\leq1$ Find the maximum value of $\sum_{i=1}^9x_i$ Intiutively the sum has its max. value when $x_1=-1$ and $x_2,x_3,\dots,x_9=1/2$ but I…
Zafer Cesur
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Richardson method optimal parameter

The question I am about to ask came to my mind when I was analyzing Richardson method with symmetric and positive definite matrices. But it is really about simple math I somehow can't defeat. Given real numbers: $\lambda_1\ge \lambda_2 \ge ...\ge…
xan
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Minimise $a + b + c + d$ s.t. $a^3+b^3=c^3+d^3$

a, b, c, and d are all different positive integers. I've tried googling optimisation and constrained optimisation, but I've not found anything applicable as yet. EDIT: I tried using Lagrange mulipliers, but hit a wall: $$f(a,b,c,d) =…
George Tomlinson
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Lagrangian (constrained) optimization - rate of change of optimal values

In constrained Lagrangian optimization what is a general way to figure out how the optimal point varies with respect to parameters of the constraints? For example, maximize $x\cdot y$ when $x + y \lt k$, and I wanted to find…
figuringout
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Maximum uniqueness

Consider the function $g:\left(0,1\right)\rightarrow\mathbb{R}$ defined by $$ g\left(x\right)=\left(1-x\right)\left(1-\frac{1}{1+f\left(x\right)}\right), $$ where $f\left(x\right)$ is a continuously differentiable function that is positive and…
Paul Smith
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How to find the maximum of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{a+c}$ given certain constraints.

Let $a,b,c\ge 0,$ and such $a+b+c=1$. Find the maximum of: $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}$$ My try:…
user94270
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Formulate A Minimizing Function that Prefers A Minimal Variable Over Another

I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{i\min} \leq a_i \leq a_{i\max}$ for all $i$). And most importantly, each $a_n$ is ranked…
Graviton
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Algorithm for best piecewise linear fit

Suppose your are given a finite set $P$ of $n$ points $(x_i,y_i)$ with $x_0< x_1 <\ldots < x_n$. Denote by $f_P$ the function given by piecewise linear joining of these points. Now suppose you are given $k
wood
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Optimizing with Many Langrangians

I'm working on a problem where I've got to minimize the following: $\sum\limits_{i=1}^n(a+c\sqrt{(y_i^2+1)}+dy_i-v_i)^2$ with the following constraints: $0\leq c \leq 4e$ $|d| \leq c \mbox{ and } |d| \leq 4e-c$ $a \leq c \leq f$ Where $e$ and $f$…
radikalus
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If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the greatest possible value of $xy$?

I first wrote $y$ in terms of $x$. in $5x + 3y = 100$, I subtracted $5x$ from both sides to get $3y = 100 - 5x$. Therefore, $y = \frac{100 - 5x}{3}$. I substituted this into $xy$ to get $x(\frac{100 - 5x}{3}$) – You want to find the maximum value of…
user1043968
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