Mathematics Stack Exchange
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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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Given a local minimizer, is it necessarily a Kuhn Tucker Point?

In the context of linear programming: A Kuhn Tucker Point is a point that satisfies certain conditions, making it a candidate for being a minimizer of a function (global or local). Now, given a local minimum, can we necessarily say it is a Kuhn…
ThinkConnect
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General technique for solving these optimization problems

There's three particular optimization problems I see quite often(in various applications). They seem to me to be specific cases of a more general problem, for which there perhaps exists a technique with which they can be solved. 1) Suppose…
user308485
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Fit a function f on dataset X such that f(X) fits a histogram

I have dataset $X=\{\boldsymbol{x_1},\boldsymbol{x_2},\dots,\boldsymbol{x_n}\}$ and $Y=\{y_1,y_2,\dots,y_n\}$ and want to learn a function $f$ such that $y = f(\boldsymbol{x})$ can be approximated as much as possible (by whatever cost measure). The…
Warbean
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Dirac delta function in a optimization

Define a vector $z:=\begin{bmatrix} z_1^\top & z_2^\top \end{bmatrix}^\top$. Can this optimization problem \begin{equation*} \min_z z_1^\top H z_1 + f^\top z_1 - \delta(z_2) \end{equation*} be solved? Where matrix $H$ is a positive definite matrix,…
Stephen Ge
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Equivalency of two optimization problem

On what conditions (constraints) on $x,y$ the following two problems are equivalent? P1: $\underset{x,y}\max ~\log(1+\frac{x}{y})$ P2: $\underset{x,y}\max ~ x-y$ the domain of the two problems is $x,y>0 , \frac{x}{y}\geq 1$
Sajad Daei
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How to maximize a product of integers?

I have been given a positive integer $B$. How can I find a positive integer $k$ and positive integers $n_1,n_2,\ldots,n_k$ such that $\sum_{i=1}^k n_i=B$ and $\prod_{i=1}^k n_i$ is as large as possible. I think this has something to do with…
student
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Find the positions of $A$ and $B$ which minimizes the length $AB$.

The line $AB$ joins the points $A(a, 0 )$, $B(0, b)$ on the $x$ and $y$ axes respectively and passes through the points $(8, 27)$. Find the positions of $A$ and $B$ which minimizes the length $AB$. Anyone have any idea, I drew it out and for…
user2250537
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Setting the right restriction in a simple linear optimization task

here is the task: One factory produces 3 types of cars: small, midsize and big. There are 6000 tons of steel and 60000 total time available. For each type of car produced, there must be 1000 cars of these types produced. Build an optimization model…
James
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difference between linear separable hyperplane and affine separable hyperplane

What is the differece between linear separable hyperplane and affine separable hyperplane? How does one represent a hyperplane graphically in R^2?
johnathan
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Deriving optimal time to change

I am working in economics and I am trying to build a model that take into account the fact that indivudal can take a decision once in their life time that changes the value of a parameter R. To be more clearer: Time t is discrete and goes from 0 to…
Tochoka
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Find the smallest possible value of $a^4+b^4+c^4-136abc$

Let $a$, $b$, and $c$ be real numbers such that $a+b+c=-68$ and $ab+bc+ca=1156$. The smallest possible value of $a^4+b^4+c^4-136abc$ is $k$. Find the remainder when $k$ is divided by $1000$. I would solve this question by first noting that…
Puzzled417
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Limit the objective function in optimization

Is it correct to limit the objective function of an optimization problem sometimes? I heard we shouldn't limit the objective function at all. In other words, a namely problem like this, is possible? $Min{ \space\space\space } Z $…
SAH
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Minimisation in two unknowns and absolute value

$$Error = \vert(\tan(\theta)*20 – (9.8/2V^2(\cos\theta)^2)*20^2 - 2.2)\vert$$ $\vert\bullet\vert $ indicate absolute value. Is it possible to manually minimise the above equation and if so how do I go about it? Here both $\theta$ and $V$ are the…
user300228
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confusion regarding minimum value

Let $a+2b+c=4$. Find the maximum value of $ab+bc+ac$. I tried using the arithmetic-geometric mean concept, but couldn't get to the answer. The Cauchy-Schwartz equation isn't of great help either.
Raunak Banerjee
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Optimizing weighted sum of Heaviside functions

I am looking for an algorithm to find all minimizers (in fact, one would be enough as from there it is easy to find all) of the following problem: Let $y_1,\dots,y_n \in \mathbb{R}$ and $x_1,\dots,x_n \in \mathbb{R}^k$ and let $a \in \mathbb{R}$.…
Johanna
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