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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").

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Optimization about airflow

When a person coughs, the trachea (windpipe) contracts and the velocity of the airflow in the windpipe is proportional to the product of the increase in the air pressure and the cross sectional area of the trachea. This can be modelled by v = APπr^2…
Amanda
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Trapezoid Optimization

Problem: A child wants to build a tunnel using three equal boards, each 4 feet wide, one for the top and one for each side. The two sides are to be tilted at equal angles θ to the floor. What is the maximum cross-sectional area A that can be…
Herb
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Optimization problem: combination of values

Problem: Given an unlimited amount of empty boxes and a limited amount of items N, each with a specific value but not necessarily an unique value. A box has to be filled with at least 1 item and the total value of the items in the box has to be at…
Pascal
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Prove that the solution of a system is the same as minimum of the sum of squares

I was reading through some results on google to learn simple optimization problems in MATLAB, and found a PDF where a method is described which I don't understand. Here is a picture of it, or see PDF page 5, Part III. Why is the solution of the…
user188029
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Given $r_{i}>0;\: i=1\ldots K$ when $\frac{\sum_{i=1}^{K}l_{i}}{max(\frac{l_{1}}{r_{1}},\ldots,\frac{l_{K}}{r_{k}})}$ is maximized?

Given $r_{i}>0;\: i=1\ldots K$ , We want to determine $l_{i}>0;\: i=1,\ldots,K$, such that function $f(l_{1},\ldots,l_{K})=\frac{\sum_{i=1}^{K}l_{i}}{\max\left(\frac{l_{1}}{r_{1}},\ldots,\frac{l_{K}}{r_{K}}\right)}$ is maximized. How can we show…
Mojtaba Aajami
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What is the difference between a local maximum and an unconstrained local maximum?

I can see that the definition of local maximum and unconstrained local maximum is written differently, but to me they look like they are defining the same thing. Furthermore, based on Fig 4.1, it looks like both $x^*$ and $y^*$ meet the definition…
mauna
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Newtons method for optimization

How can I solve this question? Use Newton's method for a system to write $x^2+y^2=25$ and $x^2-y=2$ in the form $J*\delta=-f$. Define the matrix $J$ and vectors delta, $f$. Dont perform iterations.
sfe
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Importance of checking corners in this algorithm?

I was asked to analytically find the constrained minimum of a function using the following algorithm: Find stationary points in the interior of the region. Find stationary points on the boundary and the corners. Investigate the points to find the…
user3680
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Optimization, point on parabola closest to another point

The problem is as follows: Find the point on the parabola $2x=y^{2}$ closest to $(1,0$). I was highly surprised because I ended up with the correct answer doing something completely different than the solutions manual suggested, and now I am…
Akitirija
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Finding the shape which gives the max area with constraint.

The problem is: Given $x$ feet of material with one side being $y$ long, what shape gives the maximum area that can be enclosed. My solution is having the $y$ side a straight line, and having a curve that is semi-circular as possible to the end…
El Santi
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When is $A^t+B^{1-t}$ for $t \in[0,1]$ maximized?

When is $A^t+B^{1-t}$ for $t \in[0,1]$ maximized. Suppose that $A,B \in \mathbb{R}^{+}$. This is very similar to convex combination but only in exponents.
Boby
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confused between convex and non-linear optimziation

I have an optimization function which contains an objective function which contains sum of decision variables, division of sum of decision variables and also product of sum of decision variables. The constraints include equality constraints of the…
Waqas
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How to solve the following minimization problem?

For $n$ scalars $a_1,...,a_n$, find the solution of $$\min_{x \in \mathbb R}\sum^n_{i=1} |x - a_i|$$ I denoted $$\delta(x) = \sum^n_{i=1} |x - a_i| $$ and found $$ \delta'(x) = \sum^n_{i=1} \frac{x-a_i}{|x - a_i|}$$ However, this function is not…
Lucas Alanis
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Calculus optimization word problem

"A printed page is being designed to contain up to 96 square inches of printed material. The margins are 1” on the left and right and 1.5” on the top and bottom. Find the outer dimensions of the page of least area which will meet the stated…
Waffle
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How is called this an optimization problem of this kind, or which techniques could I use to solve it?

I have an optimization problem which is a multivariable problem(34 variables), I need to find the minimum cost but my solution must be only concerning to the value of 3 variables out of the 34; the other values must be zero. Concerning this problem…
user3070616
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