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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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Minimize differences across dimensions

This is an example to illustrate the problem. Imagine you have to make two groups of 5 students each from a classroom of 50 students. Students differ on three grades: math, physics, and language (all have the same unit of measure). The goal is to…
mat
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Find maximum $\theta$ such that $|x + \theta a| \leq b$

I have an optimisation problem: $$ \max_{\theta} \quad \theta \\ \text{such that} \qquad |x + \theta a| \leq b $$ where $x, a \in \mathbb{R}^{n}$. We know that $|x| \leq b$. The norm referred to here is the $\ell_{1}$-norm. Is there a simple way to…
Nick Bishop
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An Optimization Problem with Two Variables

I am working on a problem where I am at crossroads with ensuring whether or not the function $$f(x,y)=x\cdot \exp(-yx^{c})(1+2y) $$ has a unique maximum subject to $x\ge 1, 1\le y\le x$. (Here, $c$ is any positive constant.) Any help will be greatly…
GA-Student
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Optimization of an expression using the intersections of curves

I have a question regarding a problem which I was able to solve(I don't know the answer though!), but I'm not very clear with the method I've used. The problem goes like this- (x,y) is a pair of two real numbers which satisfies the equation…
Abhinav Tahlani
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First iteration with golden ratio method.

So, if I have $[a,b] = [-1,2]$ and function $f(x) = x^2$, then on the first iteration we get: $x_1 = -1 +0.38 \cdot 3 = 0.14$ and $x_2 = -1 + 0.62 \cdot 3 = 0.86$. What is the formula for the calculation? And why is there a $3$? The second iteration…
user
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operator symbol "max" in optimization

The section 4.2 "Poor Conditioning" in the book Deep Learning defines the condition number of the function $f(x) = A^{-1}x$ as \begin{align} \underset{i,j}{\max}~ \Bigg| \frac{\lambda_i}{ \lambda_j} \Bigg|. \end{align} and explains the ratio…
JJJohn
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Find the optimal model which is good on multiple environments.

My question might be a well-known problem, or already discussed in this forum, but I don't know what keyword to search for, so I leave a question in here. Sorry if it is duplicated. So, let me explain the problem with an example, Suppose that I…
Mark Yoon
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Compute linear combination of complex vectors such that norm is equal to 1?

Let $A\in\mathbb{C}^{m,n}$. I am looking to compute $x\in\mathbb{C}^n$ such that for $Ax=y$ it holds: $|y_i|=1,~i=1,...,m$. In practice, the columns of $A$ are transfer functions of $n$ sources, sampled at $m$ points in space. I would like to…
ArtPe
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Optimization Problem with a ratio objective function

Objective function: $$ \left\{ \begin{array}{cccc} y_1 =& 10 & + 18x_1 & + 17x_2\\ y_2 =& 100& + 40x_1& + 41x_2 \end{array} \right. \qquad \text{and} \qquad k = \frac{y_1}{y_2} $$ Constraint to: $$ \left\{ \begin{array}{ccc} 15x_1 &+ 10x_2 &\leq 60…
NoName123
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Minimization of overlap of several sets of ranges

I'm solving an optimization problem involving several sets of ranges: there are n sets of exclusion ranges, I need to find a set of segments, that has the least overlap with the exclusion ranges, while keeping the distance between individual…
DrLeprikon
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Can I write a optimization problem in this form?

The optimization problem looks like this now: $minimize\;\frac{1}{N}\sum_{s=1}^N max\{L-Br_s^Tx,0\}$ $s.t.\;\sum_i x_i=1$ $x\ge0$ Is it ok to put the max part inside the objective function? If not, how should I correct it?
MMMMMay
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Optimizing an encounter rate in a population of interacting agents

I'm coding an algorithm in C that would need a little bit of optimization I think. Imagine I have a population of N individuals, separated in two classes. Some of the individuals are from class A, and the others from class B. Individuals meet each…
Sulli
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Linear Programming example

My question: how do we know that this is the unit cube and why is it denoted as [0,1]^n ? Why 2n inequalities and 2^n extreme points?
Effective Learning
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Why the level set function is decreasing as it goes up?

My question is about the geometrical solution for LP: how do we know that the level set function is decreasing or increasing? Where did we get the -1/2 from?
Effective Learning
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Is a pair of max functions equal to one max with two arguments?

Perhaps a trivial question but I'm wondering if the following equivalence always holds. Consider the maximization problem (P1): $$\max_x\max_{y\in Y(x)} f(x,y)$$ And the problem (P2): $$\max_{\{(x,y):y\in Y(x)\}} f(x,y)$$ Are (P1) and (P2)…
jonem
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