Extra:Notation

From Glossary

Jump to: navigation, search

[edit]

Notation

General

$LaTeX: C^n$ : class of functions with $LaTeX: n$ continuous derivitives.
$LaTeX: \textstyle \nabla f(x)$ : gradient of $LaTeX: f$ at $LaTeX: x$
$LaTeX: \textstyle \nabla_x f(x,y)$ : gradient of $LaTeX: f$ at $LaTeX: (x,y)$
$LaTeX: H_f(x)$ : Hessian of $LaTeX: f$ at $LaTeX: x$
$LaTeX: \textstyle \sum_{i \in I} x_i$ , $LaTeX: \textstyle \sum_{i=1}^n x_i$ , $LaTeX: \textstyle \sum_{i} x_i$ : sum of $LaTeX: x_i$ over an appropriate index
$LaTeX: |x|$ : absolute value
$LaTeX: \lfloor x \rfloor$ : integer part, or floor, of $LaTeX: x$
$LaTeX: x \vee y$ : the join of $LaTeX: x$ and $LaTeX: y$
$LaTeX: x \wedge y$ : the meet of $LaTeX: x$ and $LaTeX: y$
$LaTeX: x^- = \min\{0, x\}$ : negative part of $LaTeX: x$
$LaTeX: x^+ = \max\{0, x\}$ : positive part of $LaTeX: x$
$LaTeX: \| x \|, \| x \|_p$ : ( $LaTeX: p$ -)norm of $LaTeX: x$
$LaTeX: \textstyle O(f) = \{ g : |g(x)| \le M |f(x)| \mbox{ for some $M$ independent of $x$}\}$ : class of functions whose value is bounded by a uniform multiple of the magnitude of $LaTeX: f$ , where the domain of $LaTeX: x$ is selected appropriately
$LaTeX: \sqrt[n]{x}$ : $LaTeX: n$ ^th root of $LaTeX: x$

Sets and Set Operations

$LaTeX: \sim S$ : complement of $LaTeX: S$
$LaTeX: S\backslash T$ : $LaTeX: S$ without $LaTeX: T$
$LaTeX: S \cap T, \cap_{i \in I} S_i, \cap_i S_i, \cap S_i$ : intersection of $LaTeX: S$ and $LaTeX: T$ or of a sequence of sets $LaTeX: S_i$
$LaTeX: S \times T, \textstyle\prod_{i \in I} S_i, \prod_i S_i, \prod S_i$ : (cross) product of $LaTeX: S$ with $LaTeX: T$ or of a sequence of sets $LaTeX: S_i$
$LaTeX: S \cup T, \cup_{i \in I} S_i, \cup_i S_i, \cup S_i$ : union of $LaTeX: S$ and $LaTeX: T$ or of a sequence of sets $LaTeX: S_i$
$LaTeX: \mbox{cl}(S)$ : closure of $LaTeX: S$
$LaTeX: \mbox{ext}(S)$ : extreme points of $LaTeX: S$
$LaTeX: \mbox{int}(S)$ : interior of $LaTeX: S$
$LaTeX: \mbox{ri}(S)$ : relative interior of $LaTeX: S$
$LaTeX: \arg\min\{f(x) : x \in X\} = \{x^* \in X : f(x^*) \le f(x) \mbox{ for all } x \in X \}$ : argument minimum of $LaTeX: f$ over $LaTeX: X$
$LaTeX: \arg\max\{f(x) : x \in X\} = \{x^* \in X : f(x^*) \ge f(x) \mbox{ for all } x \in X \}$ : argument maximum of $LaTeX: f$ over $LaTeX: X$
$LaTeX: \mbox{dom}(f)$ : effective domain of $LaTeX: f$
$LaTeX: \emptyset$ : empty set
$LaTeX: 2^S$ : power set, all subsets, of $LaTeX: S$
$LaTeX: \mathbb{R}^n$ : reals cross itself $LaTeX: n$ times -i.e. real $LaTeX: n$ -vectors
$LaTeX: \mathbb{R}^n_+, \mathbb{R}^n_{++}$ : componentwise non-negative and positive elements of $LaTeX: \mathbb{R}^n$
$LaTeX: \mathbb{Z}^n$ : integers cross itself $LaTeX: n$ times -i.e. integer $LaTeX: n$ -vectors
$LaTeX: \mathbb{Z}^n_+, \mathbb{Z}^n_{++}$ : componentwise non-negative and positive elements of $LaTeX: \mathbb{Z}^n$
$LaTeX: [a, b]$ : closed line segment
$LaTeX: (a, b)$ : open line segment
$LaTeX: S_n$ : simplex in $LaTeX: \mathbb{R}^n$

Vectors and Matrices

$LaTeX: x_i$ : $LaTeX: i$ ^th element of vector $LaTeX: x$
$LaTeX: e, e_i$ : vector of ones, $LaTeX: i$ ^th unit vector
$LaTeX: A_{(i,j}}$ : the element of matrix $LaTeX: A$ in the row $LaTeX: i$ and column $LaTeX: j$
$LaTeX: A_{(:,j)}, A_{(i,:)}$ : $LaTeX: j$ ^th column of $LaTeX: A$ , $LaTeX: i$ ^th row of $LaTeX: A$
$LaTeX: A^{-1}$ : inverse of $LaTeX: A$
$LaTeX: A^T$ : transpose of $LaTeX: A$
$LaTeX: \mbox{diag}(x)$ : diagonal matrix whose main diagonal is the vector $LaTeX: x$
$LaTeX: I$ : identity matrix

Abbreviations

AI	artificial intelligence
ANN	artificial neural network
B&B	branch and bound
BFGS	Broyden-Fletcher-Goldfarb-Shanno
BLP	Bilevel program
BTRAN	backward transformation
CP	complementarity problem
DFP	Davidon-Fletcher-Powell
DP	dynamic program
FIFO	first in, first out (pertains to a list processing rule)
FTRAN	forward transformation
GA	genetic algorithm
GLM	Generalized Lagrange Multiplier method
GRASP	Greedy Randomized Adaptive Search Procedures
GRG	generalized reduced gradient method
GUB	generalized upper bound
IIS	irreducible inconsistent subsystem
IP	integer program
IPM	interior point method
LCP	linear complementarity problem
LMR	Lagrange Multiplier Rule
LP	linear program
LIFO	last in, first out (pertains to a list processing rule)
MILP	mixed integer linear program
MINLP	mixed integer nonlinear program
MIP	mixed integer program
MIQP	mixed integer quadratic program
MOLP	multiple objective linear program
MOMP	multiple objective mathematical program
MPEC	Mathematical Program with Equilibrium Constraints
MPS	Mathematical Programming System
NLP	nonlinear program
OBJ	objective function
OR	Operations Research
PARTAN	parallel tangents
QAP	quadratic assignment problem
QP	quadratic program
QQCP	quadratic program with quadratic constraints
RHS	right-hand side
RO	robust optimization
SLP	sequential linear programming
SOS	specially ordered set
SP	stochastic program
SQP	sequential quadratic programming
SUMT	sequential unconstrained minimization technique
TSP	traveling salesman problem
VRP	vehicle routing problem
VUB	variable upper bound

Retrieved from "http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php/Extra:Notation"

Extra:Notation

From Glossary

Notation

Views

Personal tools

Search