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").

22512 questions
1
vote
1 answer

Optimal value of this non-convex optimization problem

Are there any optimization techniques for problems such as: \begin{alignat*}{2} &\text{Find: } &&\min \sum_i^n x_i e^{-x_i} \\ &\text{Subject to: } &&\sum_i^n x_i^2 e^{-x_i} = C\\ & &&x_i \ge 0,\ \forall i \end{alignat*} Would appreciate any helpful…
Television
  • 560
1
vote
0 answers

How to find minima of such functional?

How can I find minima of the following functional? $$J(u)=\int_0^1 (u^2(t)-x^2(t)) \,\mathrm dt \to min$$ $$\dot x(t)= u(t),\ t\in\ [t_0,T],\;x(0)=0 $$
Javidan
  • 121
1
vote
1 answer

Appropriate length of chromosome in Genetic Algorithms

In order to find the appropriate length of chromosomes in GA programming, the author of this book states: Suppose six decimal places for the variables' values is desirable. It is clear that to achieve such precision each domain Di = [ai,bi]…
RezAm
  • 129
1
vote
1 answer

Confusion related to the simplification of an equation

I was going through the optimization tutorial of Stephen Boyd. However, I didn't understand how this expression was dervied $sup\{{a_i^Tu \;|\; ||u||_2 <=r}\}$ = $r||a_i||_2$. I have attached the screenshot for the context
user34790
  • 4,192
1
vote
1 answer

closest integer value

Find the integer which is closest to the value of $\frac{1}{\sqrt[6]{5^6+1}- \sqrt[6]{5^6-1}}$ I have tried putting them between n and n+1 and tried manipulating n but I cant seem to find a strategy.
SuperMage1
  • 2,486
1
vote
0 answers

Submodular! Can Submodular cross the minimization?

$$f(w,b)=min_{x\ge y\ge w}\{ c_ry-c_rw+\alpha c_ux-\alpha c_uy+\alpha g(x,b)+(1-\alpha)g(y,b)\}$$ where $0\le\alpha\le1$, $g(z,b)$ is complex and there is $c_u$ inside of $g(z,b)$. What we know: 1. $g(z,b)$ is submodular in $(z,c_u)$ whereas…
MrJuice
  • 11
1
vote
2 answers

Optimisation - Chain Rule

So far, the only way I could think to do this is to use the chain rule by splitting up the cost function: dC/dt = dC/dx * dx/dt But, im not sure what I could use as dx/dt as there is no function for this. Any help is appreciated Thankyou!!
J-Dorman
  • 129
1
vote
0 answers

Policy for finding the best "object", given two properties of such object

first question here, so I hope that I'm doing everything correctly. Let's say that I have a certain number of objects. Each object is defined by two properties, $x$ and $y$. $x$ and $y$ both vary from $0$ to $100$, and the higher the value, the…
Rant423
  • 11
1
vote
2 answers

Minimizing $f(x,y,z)=y$

In the problem of minimizing $f(x,y,z)=y$ over the constraint set $z=y^3 - x^2$ and $z=x^2$, I have managed to solve the problem directly and obtain that the minimum occurs at $x=0, y=0, z=0$, yielding a value of $f(0,0,0)=0$, but when I write the…
Timur Lame
  • 73
  • 5
1
vote
0 answers

Simple optimization problem 2

Approaching this question, I first calculated the TOTAL cost of the trip (that is, $D(v) + F(v))$. This gave me $\dfrac{1}{v} (5v^{\frac 5 2} + 2059V + 150 000)$. Am I right in saying I now have to calculate this cost in cents per km? And would I…
Lara Jane
  • 11
1
vote
1 answer

Optimizing crafting choice for game

I have a game in which the goal is to craft objects and I need to formulate an optimization problem, but I don't know where to start. Rules In this game you have a backpack B where all your objects are stored. An object can be either craftable or…
ThatGuy
  • 11
1
vote
1 answer

The critical points and saddle points of the function $f\left(x,y\right)=x^{3}+y^{2}-3xy$, a confliction with a program

The critical points and saddle points of the function $$f\left(x,y\right)=x^{3}+y^{2}-3xy$$ are $A=\left(0,0\right)$ and $B=\left(\frac{3}{2},\frac{9}{4}\right)$. The hessian is $$ H=\begin{bmatrix}\begin{array}{rr} 6x & -3\\ -3 &…
mert
  • 674
  • 3
  • 11
1
vote
3 answers

How do you take the derivative of a function like this: $f(g(x),h(x))$?

I have a function $f(x,y)$ of two variables. I also have the relationship $x = a + by$. So, I can also write the function as $f(a+by, y)$. I want to maximize this function in $y$. So I need to take the derivative and set it equal to zero, so that I…
AnUser
  • 13
1
vote
1 answer

How to Minimize this Function?

Let $f\in C([0,1])$ and $K\subset C([0,1])$ be the set of constant functions on $[0,1]$. Let $\|u\|=\sup\{|u(x)|:\ x\in [0,1]\}$. Define $F:C([0,1])\rightarrow \mathbb{R}$ by $$F(g)=\|f-g\|$$ Consider the problem of minimize $F$ in $K$ and let $c\in…
Tomás
  • 22,559
1
vote
0 answers

How to enforce matrix invertibility in optimisation problem?

I want to find a matrix $A\in\mathbb{R}^{n\times n}$ and vector $x\in\mathbb{R}^{n}$ satisfying: \begin{align} Ax & \leq b,\\ Ac & = d, \end{align} with $b\in\mathbb{R}^{n},c\in\mathbb{R}^{n}$ and $d\in\mathbb{R}^{n}$ known vectors. My idea is to…
MathsStudent
  • 179
1 2 3
…
97 98