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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optimize the expression respecting to a distribution

Given that $p$ is a proper distribution, $p_i \geq 0$, $\sum_i p_i = 1$ Maximize the following expression with respect to $p_i$ $$\sum_i \log(p_i + c_i)$$ where $c_i$ are known positive constants. Is there any closed-form solution to the problem?
Vuong Bui
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Transportation problem with complication

I have a transportation problem with complication. Essense of complication -- shipping is carried out with a trucks. Each truck has limited capacity. Each truck has the same capacity. Let's call capacity of eack truck -- K. How to reduce this…
Ivan Afonichkin
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Maximum and minimum of a three-variable system

Let $a,b,c$ be reals such that $a-b+c=3, a^2+b^2+c^2=4$. Find the maximum and minimum of $\sqrt{2}a + \sqrt{2}b + 3c$. Don't use coordinates or Lagrange multipliers in order to solve this. Are there any algebraic ways to solve this that don't…
user19405892
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How to write this constraint in an optimization problem?

Let $x_i$ be a binary variable in the set $\{0,1\}$ for all $i$ in the set $\{1,\ldots,n\}$, i.e., $x_i\in\{0,1\},\,\forall\,i\in\{0,\ldots,n\}$. I would like to write the constraint: $$\displaystyle\prod_{\substack{i=1\\ …
drzbir
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How to minimize this sum?

Given $n$ positive number that $x_1+x_2+...+x_n=k$. I want to minimize $\sum \sqrt{x_i^2+1}$. I guess that this sum is minmum when $x_1=x_2=...=x_n$, but I cant prove it. can anybody help me? Thanks.
raha
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Optimization with constrain deviation minimization

If we are going to invest 40000$ to bank that yields 7% (expected return), bonds that yield 9% and stock that yields 14% with the following constrains: Expected return on investment has to be at least 5000$ Investment on stock has to be at least…
ELEC
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Cutting the wire - optimalization

We cut the 120-cm wire into 3 pieces and from these pieces form three square frames. Can we maximize the sum of areas of these frames?
mrnobody
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Is this problem a geometric program?

$$ \begin{equation} \begin{aligned} & \underset{x,y}{\text{minimize}} & & f_0(x,y) \label{eq:6}\\ & \text{subject to} & & f_1(x)\geq 0, \label{eq:7} \\ & & & f_2(y)=0,\\ & & & x \in \{0,1\}^n, y\in\mathbb{R}^n. \label{eq:9}…
zebda
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How to model optimization problem where the variables are in a finite subset of $\mathbb{R}$?

I have a nonlinear optimization problem of the form : $$ \begin{align} & \underset{ \mathbf{x} }{ \text{ maximize } } & & f(\mathbf{x}) \\ & \text{ subject to } & & g(\mathbf{x}) \leqslant 0,\\ & & & x_{ i } \in \{ 0,\cdots,r \}. \forall…
drzbir
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Easier explanation of the worst case minimization problem?

The question is as follows (the classic nine balls question): "You have nine balls. One of them is heavier, and you are given a balance to find out which one is heavier. You are allowed to use this balance twice and it tells you which side is…
SDG
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Question about optimization?

Suppose you have a local minimizer x of a function f(x). Knowing that x is a local minimizer, can you say it is a Kuhn Tucker point?
ThinkConnect
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Having no circuit with negative total length is equivalent to having a shortest path

Suppose there are two points $s$ and $t$, and that edges can have negative length and that from each vertex there is at least one directed path to $t$, how do I show that the statements "there is no circuit with negative total length" and "there…
user235238
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Maximize sum of numbers subject to an upper bound of the sum of each of these numbers raised to an arbitrary power

I want to find a maximum for $\sum_{i = 1}^n x_i$ subject to $\sum_{i = 1}^n x_i^{\alpha} \leqslant c$ where $c > 0$, $x_i > 0$ and $\alpha$ is a positive number greater than zero and not equal to 1. This is an extension to my previous question…
Gaussian0617
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Do extreme points change after written as standard form?

When working with linear optimization, we often need to rewrite the problem in the standard form: minimize $z=c^Tx$ subject to $Ax=b$ and $x\ge 0$ During that process, sometimes we introduce new variables ($x = x'-x''$) or use slack variables. My…
3x89g2
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How are KKT necessary conditions deduced?

I am trying to understand and remember the conditions here but I cannot yet understand them. You could take any example like here to explain or more theretical slant with proper references. My lecturer is just stating that they are the necessary…
hhh
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