Mathematics Stack Exchange
Mathematics Stack Exchange

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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Hungarian algorithm , Kuhn paper, definition of transfer and theorem 1 proof

http://tom.host.cs.st-andrews.ac.uk/CS3052-CC/Practicals/Kuhn.pdf Is the paper. I am looking at the definition of transfer, essential, inessential and the proof of theorem 1. Consider qualification matrix Q i=1,2 j=1,2 with every entry is 1. Then…
topoquestion
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To show that $f(y)$ has only one maximum in $y\in[0,1]$

I have function $$f(y)=\frac{1}{2} y \log \left(\frac{a^2 b \left(\frac{2}{y}-2\right)}{a b \left(\frac{2}{y}-2\right)+a+1}+1\right)$$ where $a,b>0$ and $y\in[0,1]$. I want to show that $f(y)$ has a maximum at $y_{\max}$ in $y\in[0,1]$. If…
Frey
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Non linear programming problem using Kuhn-Tucker method

Solve using Kuhn-Tucker method $ z=x_1^2+x_2^2 $subject to i) $ x_1+x_2\le 4$ ii)$ 2x_1+x_2\le 5$ where $ x_1\ge 0 ,x_2\ge 0$
user159480
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Maxima-Minima with multivariable function.

Let's have: $$f(x,y)=\max\{x^2+2y,y^2+4x\}\forall x,y\in\mathbb R$$ We need to find the least value of $f(x+y)$, Now let us choose $*$ to denote any of $\ge,\le,=,<,>$. Here: $$x^2+2y*y^2+4x\implies…
RE60K
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Set boundary for least-square calculation

When calculating least-squares we use the form $$Xw=y$$ This gives us an approximation, and it can happen that the resulting $w$ will be such that $Xw \lt y$ (entry-wise). Is there a way how to make every entry in $w$ big enough so that $Xw \ge…
hugo
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Grouping constrained optimization

I am looking for an efficient solution to solve the following problem. Can anybody help? S is a finite set of elements $k_i$ V is a subset of S, e.g. $v_4$={$k_1$,$k_3$} E is a finite ensemble of V, e.g. E = { v1={$k_1$}, v2={$k_1$,$k_2$},…
Sebastien
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Implementation of Lagrange Multiplier to solve constrained optimization problem.

I'm trying to solve an optimization problem. I have a list of around 4000 geo coordinates data, and I want to cluster them into 30 groups based on the distance, so that the closer properties belongs to one group. I started with FCM algorithm In…
qshng
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How to find optimal subset $I$ such that $(\sum_{i \in I} a_i)^x / \sum_{i \in I} b_i$ is maximized?

Suppose we are given pairs $(a_i,b_i)$ of positive numbers and $x \geq 1$. The goal is find the optimal subset of indices $I$ that maximizes: $$\frac{(\sum_{i \in I} a_i)^x}{\sum_{i \in I} b_i}$$ This is a combinatorial optimization so I'm hoping to…
user2566092
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Semidefiniteness of the Hessian and optimization

This question is for sure a duplicate, but different users seem to give different answers. The question is: suppose you find that the Hessian matrix for a function $f(\textbf{x})$ is semidefinite positive on the whole domain. Are all stationary…
Luigi
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How to maximize shipping box volume

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible. I first looked at USPS standard post rates. I noted that if the "combined length and girth" of…
ravron
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Naive Question: How to convert max{t,0} to min{.. , ..}

Perhaps it is a very basic question, I want the following in $min$ form: $\max\{A,B\}$ What is the equivalent $\min\{.,.\}$ formulation? Thanks.
Mohsin
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Solve $\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$

I have to solve $$\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$$ For $a_1 \leq a_2 \leq ...\leq a_n$ My intuition says that this x is a point in the middle of the $a_i's$ but I am not sure that it is correct and how can I prove this. Thanks!
Giiovanna
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Find $y$ to minimize $\sum (x_i - y)^2$

I have a finite set of numbers $X$. I want to minimize the following expression by finding the appropriate value for y: $$\sum\limits_{i=1}^n (x_i - y)^2$$
Uwat
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Hints on solving a constrained optimization problem

Here's a simple constrained optimization problem ($X,P \in\mathbb{R}^{m\times n}$): $Minimize_{X}~ \|P-X\|_{F}^{2}$ $subject~to~\|X\|_{0}= k$. The optimal $X$ can be got by setting all but the $k$ largest elements of $P$ to zero. That seems…
nemo
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To find the maximum and minimum value of x such that it satisfies a polynomial

Find the maximum and minimum value of $x$, where: $x+y+z=4$ $x^2+y^2+z^2 =6$ I thought I could use these values to form a equation having $x,y,z$ as roots and the sum of roots and $\sum{xy}$ but could not get any idea about the product of roots…
Sukrut Nigwekar
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