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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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Inequality contrained optimization problem (contingency table)

I am concerned with the following optimization problem. I first state the problem and then briefly discus its background. $$f(a_1,...,a_n)=\frac 1n \sum_1^n (a_i-b_i)^2$$ should be maximized subject to constraints: $$\sum a_i=1$$ $$\sum b_i=1$$ $$…
tomka
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Finding the "best" way to map set of points to another set

I've got a set of points (currently 4, but I can increase the number for better accuracy), and I want to find the optimal transformation so that they can be mapped to another set of points. For example, I have (2,2), (2, 4), (4, 2), (4, 4) which…
andrw
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Factory Optimization

First I feel like a disclaimer here is needed that this is NOT a homework problem but I am going to ask the question in something that looks like one. I don't know how to ask my question any other way. Imagine you have a factory. The factory…
Matthew Kemnetz
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Optimization of wave drag formula ( gradient calculation of double summation)

i want to ask a question about gradient calculation of double summation term wave drag formula The formula (objective function to minimize !) shown above calculates the wave drag of an aircraft, S is cross-sectional area S" is the 2nd derivative of…
ccdeus
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For any positive integer $n$, what is the value of $t^*$ that maximises the following expression?

For any positive integer n, what is the value of t* that maximises the following expression? $$\displaystyle \sum_{j=1}^{n-t^*}\left(\frac{t^*-j+2}{t^*+j}\right)$$ where $t^*$ is some integer in the set $\{0,1,2...,n-1\}$. Clearly $t^*$ = $f(n)$…
awwlaz
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A basic question on contours/level sets of a function in quadratic form

Consider $$f(x)=\frac{1}{2}x^TQx-x^Tb$$ where $Q$ is an $n \times n$ symmetric matrix The contours of $f$ are $n-$dimensional ellipsoids with axes in the directions of the $n$-mutually orthogonal eigenvectors of $Q$. What is the meaning of this…
user21982
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Problem: Optimally distribute a fixed amount of work among a set of workers of different performance

Consider there is amount of work $W$ to be done and set of $N$ workers who has different performance: $p=\{p_1,p_2,\dots,p_N\}$. Consider that the work is divisable and can be distributed among the workers: $W=w_1+w_2+\dots+w_N$. Let…
mbaitoff
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How to minimize this non-linear function?

minimize the following function: $$\sum^n_{v=1}\left(S_{1v} - t \frac{(1-p_v)\sin r_v }{1-p_v\cos r_v }\right)^2 + \left(S_{2v} - t \frac{(1-p_v)\sin (6r_v) }{1-p_v\cos (6r_v) }\right)^2$$ subject to inequality constraints: $$\begin{bmatrix}-t \\…
user94677
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Prove a x* is the only stationary point of a given fuction, is a local min but not global min

With n $\geq$ 2 and f: $\mathbb{R}^n \rightarrow \mathbb{R}$ $$f(x) = (1+x_n)^3\sum\limits_{i=1}^nx^2_i+x^2_n$$ Show that x*=0 is the only stationary point and that x* is a strict local minimum of f, but not global minimum. I tried to find the…
Iago Victor
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A solution to a minimax problem is a saddle point?

Let $X\subset \mathbb{R}^n, Z\subset \mathbb{R}^m$ and $\phi:X\times Z\to \mathbb{R}$. We define a saddle point of $\phi$ as follows: Definition. $(x^*, z^*)\in X\times Z$ is said to be a saddle point if for any $x\in X, z\in Z$, we have $\phi(x^*,…
Kaira
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Linear Programming - Basic solution

I'm reading some notes Linear Programming on linear programming. I'm considering the standard minimum problem. Why is it so obvious that, if $-\textbf{c}\geq \textbf{0}$ and $\textbf{b}\geq \textbf{0}$, then minimum occurs at $\textbf{y}=\textbf{0}$…
New_to_this
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Optimization of an infinite horizon problem

I have a doubt regarding the next optimization problem with infinite horizon. My objetive function $U(x_t)\in C^{\prime}$ is a function $U:R^+ \rightarrow R^+ $ such that $U^{\prime}(x_t)>0$, $ U^{\prime\prime}(x_t)<0$, $ \lim_{_{x}\to\infty}…
Ibai
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Taking Derivatives : Summation of Logarithms vs Logarithm of Summations

In the context of differentiation, I have a question about comparing the complexity for the "summation of logarithms" vs the "logarithm of summations". To illustrate my question, I will use the example of Gaussian Mixture Models (GMM) and the EM…
stats_noob
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How to Define a Bell Curve

For integers 0 to 100, I am trying to write a series of constraints that define a bell curve shape centered around 50 and mass =0.5. Informally, I used my own logic to come up with a set of constraints to make this happen: Suppose I have variables…
stats_noob
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Minimize the maximum value when partitioning integer $s$ into $n$ buckets

I would like some hints about how solve this kind of problems. I am not asking a detailed solution, just how the problem is classified (if it is) and what part of math I should take a look. Given positive integer $n$ and $s$, we want to construct a…
john_smith
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