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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How to find outage probability in wireless communication

I wanted to find the outage probability when rate of each user < rate _ threshold over the Rayleigh fading channel. I have found as shown in the following code. However, a research paper uses another equation to find outage probability like 1- exp…
noor
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Traveling Salesman where the cost depend on visiting a city before some other city.

I'm looking for some inspiration about how to approach this problem. I'm currently assuming this is some variation of the traveling salesman problem, but there might be better fitting optimization problems. I have a set of cities and a traveling…
dani
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Formulate optimization with intermediate variables

I have the following problem Let $x\in\mathbb{R}^{6\times1}$ be unknown decision variables, $b\in\mathbb{R}^{3 \times 1}$ is a known vector, and $A$ is a known matrix. We formulate the following optimization problem \begin{array}{lc} …
Nick
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Max- Min Optimization problem

I am a noob in mathematic, so I would need your help in solving the optimization problem below \begin{array}{l} \max\limits_{\bf l} \min \left( \left| {\bf g}_1 {\bf Ml} \right|^2, \left| {\bf g}_2 \bf Ml \right|^2 \right) \\ \text{ s.t. } \left\|…
anhledc
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Optimize a function

I have an optimization problem: $$f(x)=\frac{\sqrt{\pi}}{\varphi \cosh (x \pi \sqrt{2}) \exp \left(x \mathbb{1}_{\{x>0.5\}}\right)+0.625}+0.05\left|\sin \left(\pi^3(x+0.1)^{1.1}\right) \exp (-\pi|0.8-x|)\right|$$ where $\varphi =…
Overkill123
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Proof that value of penalty function decreases as penalty parameter increases.

I'd like to prove the following: Given an optimization problem, $\min_{x} f(x)+\rho*P(x)$, as $\rho$ increases, the penalty function evaluated at the optimal value $x^*$, $P(x^*)$ decreases. $x\in \mathcal{R}^n$. I tried breaking this into…
somewhere
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Efficient salami placement

Given $S$ and a set of $n$ slices of salami ${C_1, C_2, ..., C_n}$, what is the most efficient way to place the slices of salami on $S$ such that the total area of $S$ covered by the slices of salami is maximized? Efforts in solving the problem: One…
rumathe
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The local minimum of the SQP (sequential quadratic programming) algorithm

Consider the constrained optimization problem \begin{eqnarray} goal~~&&\min f(x)\\ s.t.~~&&g_1(x)\leq0\\ &&g_2(x)\leq0\\ &&\cdots\\ &&g_n(x)\leq0 \end{eqnarray} where $x$ is a vector variable. An efficient method to solve this problem is…
ClePIR
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A problem on combinatorial optimization

I am thinking about this problem: Given $\beta_1,\beta_2,\ldots,\beta_n \in \mathbb{R}$, where $n$ is very large. Denote the sum of them by $S=\sum_{i=1}^n \beta_i$. For a fixed real number $\mu \in (0,1)$, consider the following optimization…
fyi
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Optimization Problem in a game

I'm playing a game that has an optimization problem built into it that I am struggling to determine the correct solution for. In this game, you have a probability to win determined by the function $f(x)$ posted below (e.g., when $x = 300$ you have a…
TheoCS
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Minimizing an algebraic sequence

Define a sequence where; $I_s$ are positive integer values, $I_1$ is a constant value and $I_2k_1=I_1$ $I_3k_2=I_1+I_2$ $I_4k_3=I_1+I_2+I_3$ this goes on for some $k_i$ value All variable "$I$" are bounded such that $I \in\{1,20\}$ and need to…
NewBeginner
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Find the point on graph of $xy=12$ that is closest to the point $(5,0)$

This is from a Derivatives chapter in the section on Optimization. Find the point on graph of $xy=12$ that is closest to the point $(5,0)$ I believe I have to use the distance formula. So, so far I have: $y=12/x$ $d^2=f(x)=(x-5)^2 + ((12/x) -…
user5826
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Determine the largest $k$ to guarantee the existence of a subset that satisfies a particular condition

Color any $67$ numbers in the set $M = \{1,2,\dots, 100 \}$ $\color{red}{\rm red}$ and the rest $\color{blue}{\rm blue}$. If there exists a subset $A_{i,k} = \{i, i +1, \dots, i + 3k -1\}$ of $M$, with exactly $2k$ numbers in red and $k$ numbers…
CBot
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Chemical reactions and solutions of a constrained optimisation problem

I have to find a solution for this problem: given $N$ materials of density $\rho_k$, find the mixture of them giving a compound of density $\rho$. From a mathematical point of view, we have to find $q_k$ and $\rho_k$ for…
Riccardo.Alestra
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Minimizing a unary quotient over a box

I'd like to find a method to solve the problem $$\min \Biggl\{ \frac{x^2+b_1x+c_1}{x^2+b_2x+c_2}:\quad 0\leq x\leq \bar{x} \Biggr\}$$ where the variable $x\in \mathbb{R}$ is bounded on $[0,\bar{x}]$ and the quotient is supposed to be well defined…
kaienfr
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