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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Linear Optimization: Maxima

I've just began attempting chapter 1 in a linear optimization book and it's fairly simple maximizing problems. such as; maximize 3x + 2y such that 2x - y <= 6, 2x+ y <= 10 and x,y >= 0 Simple enough to find the feasible region, and where the…
strugglingstudent
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Minimizing the sum of vectors

I have this problem: Given a set of unit vectors $\{ \vec{v_i} \}$, I want to determine another set, $W$, the element of which are in $\{ \vec{v_i} \}$(repeating allowed), so that the module of the sum of all the vectors in W is minimized. The size…
user151515
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Finding extrema of a function depending on parameter

The question might seem basic and perhaps I am overlooking something. I'm looking at the development of the extremum of $$ f(x)=\left|\frac{ A \mathbf{x} \sin\left[B\sqrt{x^2 -a^2+2 i \cos[\Phi ] a b+b^2}\right] }{\sqrt{x^2 -a^2+2 i \cos[\Phi ] a…
mcandril
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Maximize $\sum_i \mathrm{rate}_i$ s.t. $\mathrm{rate}_i$

Question related to optimization problems. $$\mathrm{maximize} \sum\limits_{i=1}^{M}\log\left(1+f_i(\mathbf{x})\right)$$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\mathrm{subject}\,…
Jika
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examination of the function

I need help.. Question An examination of the function $f:\mathbb{R}^2 \to \mathbb{R}$, $f(x,y) = (y-3 x^2)(y-x^2)$ will give an idea of the difficulty of finding conditions that guarantee that a critical point is a relative extremum Show that (a)…
user142512
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Ladder Optimization Problem

A fence 4 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Here are some hints for finding a…
ladipo
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Optimization issue, how to obtain the maximal value?

$ max f(\beta)=\frac{\beta}{1+\beta}\cdot \left(1- \frac{\binom{N+B}{B}\cdot\beta^B} {\sum_{i=0}^B {\binom{N+i}{i} \cdot \beta^i}} \right)$ where $\beta\in[0,\infty)$, $N$ and $B$ are identified positive integer (i.e., not variables). How to…
JLiu
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Converting a primal LP to a dual LP with a constant in the question

Could someone help me with the next Primal LP to Dual LP conversion? $$ min z = -3x_1 + x_2 - 20 \\ s.t. \quad -3x_1 + 3x_\le 6 \\ \quad\quad\qquad -8x_1 + 4x_2 \le 4 \\ \qquad\qquad x_1,x_2 \ge 0 $$ These problems usually aren't hard at…
Kasper
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Minimizing distance between 2 arrays (or points)

I would like to get a solution or receive guidance on how I can solve the optimisation problem below. Let's say I have two arrays of length N , say A and B, and I want to find 2 coefficients $k_1$ and $k_2$ that will make distance between A and B…
KillaKem
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Finding minimum value of trigonometric function

Find the minimum value of $$\displaystyle \frac{2\cos^{-1}(x)}{\pi(1 - x)} , x \in [-1,+1) $$
sherlock
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Classification of critical point

The critical point of this function are $(0,0),(-1/3,-2/9),(-1/3,2/9)$. And for $(0,0)$, I get the difference that $△f=f(0+a,0+b)-f(0,0)=a^4 + (3a+1)b^2$ will always greater than zero where a,b are small values around $(0,0)$, so the point $(0,0)$…
user133140
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Profit maximisation Problem for Firm

I was wondering if someone could help me with the following problem. To be honest, I don't really know where to start ... A profit maximising firm producing output using a single input according to a production function $f(\cdot):\mathbb{R}_+ \to…
user133993
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cost minimization

There are n cities $c_1,c_2,...c_n$(in decreasing order of popularity) where a company wants to open its N branches. There is cost $w_i$ for opening a branch in city $c_i$. If company has budget W , how it should open its branches such that maximum…
rrpp
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Algorithm for optimizing width length of classes of an ordered list of data points under certain conditions

I have the following problem: I have an ordered list of $n$ data points jiggling around $0$ with no apparent order. The order this list is in should not be affected by the following procedure. I want to divide this list into 5 classes under the…
vonjd
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Need help with Lagrange Multipliers

I need to maximize $U = BM$ with constraits: $6B +3M = 60$, $B>0$ and $M>0$. The Lagrange function is $L=U + \lambda (6B+3M-60) + KB + HM$. So $$\partial_{\lambda}L= 6B+3M-60=0$$ $$\partial_{K}L = B=0$$ $$\partial_{H}L = M=0$$ $$\partial_{M}L =…
hhh
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