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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How can one solve this optimization problem by rewriting the matrix?

Given this problem: $$maximize \, \, x^T\begin{bmatrix} 3 & 4 \\ 0 & 3 \end{bmatrix}x $$ $$ subject \,to \, \, ||x||^2 = 1$$ I want to solve this using Langrange Conditions. What I wish to do is find the eigenvalues of a symmetric matrix $Q$, and…
Bob Pen
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Maximising Question

I need to maximise $P$ under these conditions. First, $$P=N[1.4W-0.31L]\,\text{ and }\,W\le19,\,7\le{L}\le11,\,N\in\{16,17\}.$$ I did this using elemantary methods. In order to maximise $P$, we need to maximise $[1.4W-0.31L]$, and I thought since…
ciceksiz kakarot
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Optimizing a generic function

I have a function $f_\theta$ that is generic. More particularly, it is the function that describes the conversion of pressure to altitude. $$ y_i=\frac{\mathrm{T}_{0}}{\mathrm{~L}}\left(\left(\frac{x_i}{\mathrm{P}_{0}}\right)^{-\mathrm{L} \mathrm{R}…
truvaking
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A Function That Converge To The Min After Many Iterations Using Steepest Decent

I am searching for a function to show the "zig-zag" pattern of the steepest decent, I want to show that SGD converges after less iterations. Which functions should look at?
newhere
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Maximum Optimization Problem with Coins

So, on this online game, there's this currency system where you want to break down these large coins into smaller denominations because of their desirability, but you can only do so by purchasing these items. You are allowed to combine small coins…
Chips
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How to solve an optimisation equation with unknown parameters?

Given an example equation: $$ z = Mx + Ny $$ where $M$, $N$ are unknown parameters and $x, y, z$ are features of a dataset. My initial guess is to use gradient descent and the least squares error to obtain $M$ and $N$ from the dataset. After that,…
user894011
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Solving constrained least squares problem with modified linear constraint

could anybody give me some suggestions on solving this least-squares problem? Find $x\in R^{n}$ that minimize $||Ax||_2$ subject to $||Bx||_2=1$, with $A\in R^{n×n} $ always full rank, $B\in R^{n×n}$ not necessary full rank. Thanks for your help
Eden Willy
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Math extrema problem

I have a math extremum problem, but I am unable to solve it. "Car gasoline use per hour is: $w = 6 - 0.15v + 0.0025v^2$. At what speed does the car have to travel, so that it would be cheapest. " I used derivative to find a global minimum and…
user63021
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Minimising surface area of a rectangular parallelepiped

We want to construct a rectangular parallelepiped with Volume $2L$, one of its sides measures $10$cm, let $x$ and $y$ be the two dimensions in dm, of this box. (Note: $1$dm = $10$cm) 1)Prove that the surface of this box is $S(x) = 2x+4+4/x$. 2)What…
Iridium
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Prove that $\min_{x,y} f(x,y) = \min_{x}(\min_{y}f(x,y))$

Given $f\colon X \times Y \to \mathbb{R}$, prove that \begin{align*} \min_{(x,y) \in X \times Y} f(x,y) = \min_{x \in X}\left( \min_{y \in Y} f(x,y) \right). \end{align*} Intuitively, I see that the left hand side is the minimum value over all…
sixtyTonneAngel
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Determine conjugate function

Let $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=e^x$. Determine $f^*(y)$. I try to use some inequalities to get supremum but it is impossible. Seemingly, I must consider some cases of $x$.
anhkhoavo1210
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What is this optimization method called? Sequence of non convex regularized optimization

I have the following non convex problem \begin{array}{ll} &\min_{x}f(x)& \\ &\text{s.t. }g(x)=0& \end{array} Where I have a particular $f$ and $g$. To solve this problem in my case I found that it was much easier to solve the sequence of the…
Badr Youbi Idrissi
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Local minimum free derivative algorithms?

I am looking for local minimum search algorithms free of derivative. More specifically, given a continuous and multi-modal $h:[0,1]\longrightarrow \mathbb{R}$ (note that we do not assume $h$ is smooth or Lipschitz), What free derivative algorithm…
user123043
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how to scalarize multi objective functions

i have a multiobjective functions that i need to optimise it using GA. how to scalarize the functions to get a single function and do the optimization? the functions and constraints are in the attached picture thenk you inadvanceenter image…
af sd
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Minimum to maximum in optimization

I have a simple doubt, we know that $$\min f(x)=-\max(-f(x)),$$ but suppose we have an optimization problem, say a linear programming problem, such that the objective is to minimize $f(x)$. If we want to write it as a maximization problem, we write…
Cherryblossoms
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