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
2
votes
1 answer

Find topologically closest graph under constraints?

I have a series of spatial polygons on a 2D plane (Fig a). These polygons can be represented as a graph where neighbouring polygons are linked and the location of the nodes is dictated by the centroid of the polygon (e.g. Fig b). I want to map…
Hamley
  • 31
2
votes
3 answers

About "optimizing over some variables" in Boyd & Vandenberghe

In section 4.1.3 of Convex Optimization, Boyd & Vandenberghe wrote the following: we can always minimize a function by first minimizing over some of the variables, and then minimizing over the remaining ones. For example, $$\min\limits_{…
Dave
  • 576
2
votes
0 answers

Help with maximization problem (related to proof of Kelly Criterion)

I'm working through John Kelly's original paper that proves the "Kelly Criterion", but I'm getting hung up on one of the general proofs. Here's a PDF of the paper. http://turtletrader.com/kelly.pdf On page 6 of the PDF, Kelly shows the…
dml263
  • 21
2
votes
0 answers

How to minimize the trace of $G^TAG$ by using Lagrangian Multiplier method?

The problem is formulated as: $$\max_Gtr(G^TAG)$$ where $G\in\Re^{n\times m}$, $n\geq m$, $G^TG=I$ and $A\in\Re^{n\times n}$ is positive semi-definite. How to solve this problem by using the Lagrangian Multiplier method?
mining
  • 1,075
2
votes
2 answers

Minimum values of coefficients of a quadratic

Given a function $f(x)= ax^{2} + bx + c$ where $a
Saakshya Devat
  • 117
  • 6
2
votes
1 answer

Do heuristic methods work?

I'm an undergraduate engineering student. I took the course of Optimization this semester, and the heuristic methods. My professor basically says that these methods, PSO for example have no mathematical background and they do not work! I have found…
mehrdad
  • 399
  • 2
  • 14
2
votes
2 answers

How to find the minimum value of volume of a sphere constrained by a equation?

Given that $x-2y+3z=1$, find the minimum value of $x^2+y^2+z^2$.
Jill Clover
  • 4,787
2
votes
0 answers

Uniquely specifying argmax

Let $f$ be a continuous function on $A$ which is a compact subset of $\mathbb{R}^2$. Due to Weierstrass theorem, I know that there exists $(a,b) \in A$ such that $$f(a,b) = \max_{(x,y)\in A} f(x,y).$$ This implies that the set $$U = \arg…
RomainT.
  • 21
2
votes
1 answer

Checking quadratic programming solution

I want to check if $(0,1,2)$ is the solution to a following problem: $$x_3\to \min,$$ $$x_1^2+2x_1x_2+2x^2_2-2x_1-x_2-x_3+1\geq 0,$$ $$x_1^2+x_2^2-x_1+x_2-x_3\leq 0,$$ $$x_1^2+x_1-4x_2-x_3+6\leq 0,$$ $$x_1,x_2,x_3\geq 0.$$ Any ideas on how to…
user408281
  • 103
  • 5
2
votes
0 answers

Approximate 3 reals as corresponding integers with similar relationships

I have a complicated algorithm that calculates 3 values, $\text{Low} \lt \text{High} \lt \text{Total}$ as real values (actually double-precision floating point) and I need to approximate them as 3 integers ($\text{32-bit} \implies \text{value} \in…
Extrarius
  • 273
2
votes
1 answer

Does transformation of a function changes oprimal values?

Consider the following problem \begin{equation*} \begin{aligned} & \underset{x_1, x_2}{\text{maximize}} & & u(x_1, x_2) \\ & \text{subject to} & & p_1x_1 + p_2x_2 = y \end{aligned} \end{equation*} Suppose I found optimal values $(x_1^*, x_2^*)$…
tosik
  • 195
2
votes
1 answer

How to compile the objective function in this optimization problem

For a series of variables, I have got the constrain functions: $ax_{1}+bx_{2}+cx_{3}+dx_{4}=0$ $0 \le x_{1}, x_{2}, x_{3}, x_{4} \le 1$ Now I want to find the solution that the minimum value of $x_{1}, x_{2}, x_{3}, x_{4}$ is the greatest number in…
ZYSTEEL
  • 21
2
votes
0 answers

Dual of piecewise defined function

I am stuck with the following optimization problem: $$\min_w \sum_j (\max\{0,w^Tx_j\}-b_j)^2 $$ the $x_j \in \mathbb{R}^n$ and $b_j>0$ are given and i am searching the w minimizing this. My approach was to find a constrained dual formulation. But up…
Ulfgard
  • 221
2
votes
2 answers

How to minimize $f(x)$ with the constraint that $x$ is an integer?

I would like to find the integer x that minimizes a function. That is: $$ x_{min} = \min_{x \in \mathbb{Z}}{(n - e^x)^2} $$ The goal is to write a program that computes the integer $x$ such that $e^x$ is closest to $n$, preferably avoiding…
rityzmon
  • 179
2
votes
3 answers

Minimising a linear function with a regularizer

I have the following optimization problem. $$\underset{w}{\text{minimize}} \ w^\intercal x + \lambda \|w\|^2$$ I am told to solution is: $$w = -2 \lambda x$$ Why? Also scribbled on the paper is $$x + \lambda w = 0$$
TrueWheel
  • 143
  • 1
  • 5
1 2 3
…
97 98