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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duality and nonlinear optimization

If at a feasible point (x,y) we get p*=d* that means the point is optimal. Looking at the example: min (-x-y) subject to: xy<=4 …
hana paz
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Optimization: The Standardform and absolute values

I'm working on the following problem and am unsure if I'm doing it right: Let $y = (y_1,y_2,....,y_N)^T$ be an $N$- vector of non-negative real numbers. Consider the following problem: $$min_{x \in \mathbb{R^N}} \sum_{i=1}^N |x_i - y_i| +…
3nondatur
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Can gradient descent be written without time step?

I am trying to learn gradient descent for machine learning. In this highly cited research paper https://arxiv.org/pdf/1609.04747.pdf, the author presents the gradient descent as $$\theta = \theta - \eta \nabla_\theta J(\theta)$$ I have never seen…
Machine Learning Noob
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How to judge "hilliness"

With reference to Using a large set of elevation values for an area how can I find how 'hilly' it is? which is a reasonable question, perfectly intuitive. Clearly a 2 km radius near the fjords in Norway is more "hilly" than a similar radius around…
Steve Kangas
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linear conditional constraints

I would like to linearize the following for an optimization problem if $(r1>r2)$ then $s1>s2$ $r1$ and $r2$ are integer variables in $[1,30]$ $s1$ and $s2$ are variables to be optimized that are continuous decimals in $[0,1]$ a not working…
Nardine Basta
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Framing a travelling salesman problem variation

I have an optimisation problem I want to understand and tackle. I think it is a Travelling Salesman variation, where I want to find an answer to the question: "What is the most efficient route for a set of Visits?" The variables for this question…
user9483158
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The mapping of a cost vector to the set of solutions of the respective LP is concave

I would like to know if I understood the following exercise (a part of a larger one) right: Let $X \subset \mathbb{R}$ be an nonempty compact set. Prove that the function $V: c \in \mathbb{R}^n \mapsto \{ { min_{x \in \mathbb{R}^n} \ \ c^Tx \ \ \…
3nondatur
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Linear Optimization - Finding all possible cost functions for a given optimum

I'm having problems with the following task: Let $A = \begin{pmatrix}1 & 1\\\ 1/2 & 1\\\ -1 & 1 \end{pmatrix}$ , $b = \begin{pmatrix}4 \\\ 2\\\ -1\end{pmatrix}$ . Find all possible $ c \in \mathbb{R}^2$ sucht that $(2,1)^T$ is a solution for the…
3nondatur
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Min function in objective problem

How can I convert this Min function (bold highlighted) in objective problem into simple mathematical equation: Min(x,y)⁡[a[i]*(x[i]-y[i])+ b[i] min⁡(x[i], M-c[i])] s.t. c[i]-x[i]+y[i]≤M here x[i] and y[i] are decision variables.
fadiz
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How to solve for an unknown variable

For the following, I do not understand how to solve for $d$ Problem Statement: Choose $\mathbf{c}$ and $d$ to minimize $MSE=E(|y_k-a_{k-d}|^2)= E(|\mathbf{c}^T\mathbf{r}_k-a_{k-d}|^2)$ where $d\in\mathbb Z^+_0$ and $c_i\in\mathbb{R}$. Also $a_k$…
NAASI
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Minimizing a function $f(x)$ subject to $f(x) \geq 0$

How can I minimize a function $f(x)$ subject to $f(x) \geq 0$? \begin{align} \min_x f(x)\\ \text{subject to } f(x) \geq 0\\ x \in \mathbb{R}^n \end{align} Function $f(x)$ is continuous and twice-differentiable, but can be nonlinear, e.g., $$f(x) = C…
Sourav Mondal
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Optimizing $f: {\Bbb R}^n\to {\Bbb R}^n$ without first reducing it to an ${\Bbb R}^n\to {\Bbb R}$ problem

I'm exploring a philosophical question which lead me towards the idea of optimizing in multiple dimensions, not just for inputs, but for the evaluated result as well. A physical example might be for maximizing "energy and momentum." All of the…
Cort Ammon
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How to maximize the argument?

There is plenty of information on maximizing functions given some constraints. But what if I already have the function value and want to know the maximum possible value of an argument? If I know the surface, what is the highest point? I can't find…
Džuris
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Including optimization constraint in objective function

Does solving the following optimization problem : $ \min_A \text{Tr} [A \sigma] $ such that $ \text{Tr} [A^{-1} \rho] \leq 1 $ give the same solution as solving $ \min_A \text{Tr} [A \sigma]\text{Tr} [A^{-1} \rho] $?, where $\rho, \sigma \geq 0 $…
Dina Abdelhadi
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Extremum of function $2x^2-y^2-6x$ under condition $x^2+y^2 \le 9$

Find extremum of function $z(x,y) = 2x^2-y^2-6x$ under condition $A(x,y) = x^2+y^2 \le 9$. How can we manage when the condition is given by inequality? I only know that Lagrange multiplier works with condition given by equality.
janusz
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