Notation

This book uses consistent notation throughout. Below is a reference guide for the symbols and conventions used.

$ℝ$	Real numbers
$ℝ^{n}$	$n$ -dimensional real vectors
$ℝ_{+}$ , $ℝ_{+}^{n}$	Non-negative reals / non-negative real vectors
$ℤ$	Integers
$ℤ_{+}$ , $ℤ_{+}^{n}$	Non-negative integers / non-negative integer vectors
$ℚ$	Rational numbers
${0, 1}^{n}$	Binary vectors of dimension $n$

$x, y, z$	Decision variable vectors (bold lowercase)
$a, b, c$	Parameter vectors (bold lowercase)
$0$	Zero vector
$A, B$	Matrices (capital italic)
$x_{i}$	$i$ -th component of vector $x$
$a_{i j}$	Element in row $i$ , column $j$ of matrix $A$
$x^{⊤}$	Transpose of vector $x$
$A^{⊤}$	Transpose of matrix $A$
$c^{⊤} x$	Dot product (inner product) of $c$ and $x$
$∥ x ∥$	Norm of vector $x$

A standard linear program is written as:

\begin{matrix} \begin{aligned} \max & c^{⊤} x \\ s.t. & A x \leq b \\ x \geq 0 \end{aligned} \end{matrix}

$\max$ , $\min$	Maximize, minimize
s.t.	Subject to (constraints follow)
$\leq$ , $\geq$ , $=$	Constraint types
$x^{*}$	Optimal solution
$z^{*}$	Optimal objective value

LP	Linear Programming
IP	Integer Programming
ILP	Integer Linear Programming
BIP	Binary Integer Programming
MILP	Mixed-Integer Linear Programming
NLP	Nonlinear Programming
MINLP	Mixed-Integer Nonlinear Programming