2 Definitions and Notations

Index sets are essential for mastering the complexity of large models. All elements in a model, such as variables, parameters, and constraints – as will be shown later on – can appear in groups, in the same way they are indexed in mathematical notation.

The convention in algebraic notation to denote a set of similar expressions is to use indexes and index sets. For example, a summation of n (numerical) terms

a1 + a2 + ⋅⋅⋅ + an- 1 + an
(1)

is commonly abbreviated using the notation

  n
∑
    ai
 i=1
(2)

The expression (1) is called three-dots notation and expression (2) is called sigma-notation. Both expression are equivalent, but (2) is much shorter and more general. The later was introduced by Joseph Fourier in 1820, according to [1, p. 22]. There exist different variants of expression (2) :

 ∑           ∑n          ∑            ∑
     ai  ,      ai  ,         ai  ,      ai  with   I = {1 ...n}
1≤i≤n        i=1                      i∈I
                        i∈{1...n}
(3)

All four expressions in (3) are equivalent and they have all their advantages and disadvantages. The last notation in (3) is more general, because the index set I can be an arbitrary set defined outside the summation expression.

  2.1 Exercises in the Sigma Notation
  2.2 Formal Definition
  2.3 Extensions
   2.3.1 A list of (indexed) expressions
   2.3.2 Index Sets as Expressions
   2.3.3 Compound Index Notation
  2.4 Exercise 1:
   2.4.1 Indexed index sets
   2.4.2 Ordering