7 Introduction to Optimisation
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where f : Ω → R n obj is the objective function and c : Ω → R m the constraints
function.
The set of points satisfying the constraints is called the feasible region
D = {x ∈ Ω | c(x) ≤ 0}.
The problem can be rewritten as
min
x∈D
f (x).
Depending on the nature of the objective function and constraints (linear or nonlinear, single or multi-objective) of the search space (continuous or discrete), the
optimisation problems can be divided in different classes
• Continuous or discrete: the optimisation variables belong to a feasible set that
is a subset of the real space
x ∈ D ⊆ R
n .
In some problems the variable x represents integer values. Such problems are
defined as integer programming problems, and the variables are in a feasible
set such that
x ∈ D ⊆ Z
n .
A subset of integer programming problems is the binary programming problems where
x ∈ D = {0, 1}
n .
If some of the variables in the problem are not restricted to be integer variables,
the problem is called mixed-integer programming problem
x = (x r , x d ) ∈ D ⊆ R
n r × Z
n d , with n r + n d = n.
• Constrained or unconstrained: if there are no constraints on the design
variables (m = 0), the problem is unconstrained. For constrained optimisation, instead m > 0. Unconstrained problems arise also as reformulations of
constrained optimisation problems, in which the constraints are added to the
objective function as penalisation terms.
• Linear or non-linear: if the objective function and all the constraints are linear
functions of x, the problem is called linear programming problem. Otherwise
if some of the constraints or the objectives are non-linear functions, the problem
is a non-linear programming problem.