7 Introduction to Optimisation
241
where x is the optimisation variables vector, f : Ω → R n obj , with n obj > 1, the
objective function and c : Ω → R m the constraints function. As before the set of
feasible points is denoted by D.
To extend the methodologies presented for global optimisation to the multiobjective case, it is necessary to introduce some definitions.
Definition 7.2.6 A point x 1 ∈ D Pareto dominates x 2 ∈ D if
f i (x 1 ) ≤ f i (x 2 ),
i = 1, . . . , n obj
and there is at least one component j ∈ {1, . . . , n obj } such that
f j (x 1 ) < f j (x 2 ).
This is indicated by
x 1 x 2 .
Definition 7.2.7 A point x ∗ ∈ D is Pareto optimal if it isn’t dominated by any
x ∈ D,
x x
∗ .
In other words, a solution is said to be Pareto optimal, or equivalently nondominated,
if there is no other point in the feasible space for which a decrease in one objective
will not cause a simultaneous increase of at least one of the other objectives.
Definition 7.2.8 For a multiple objective optimisation problem, the Pareto optimal
set is defined as
P
∗
= {x ∈ D | ∃x
∈ D x
x}.
Definition 7.2.9 The union of the objective values of all Pareto optimal points is
called Pareto front or equivalently
PF
∗
= {f (x) ∈ R
n obj | x ∈ P
∗
}.
The Pareto front is the set of all solutions in the feasible space that are not
dominated by any other possible solution. The minima in the sense of Pareto will
lie on the boundary of the feasible region or in the tangent points of the objective
functions. Generally it is not possible to derive analytically the equation of the front.
Approximation techniques have been developed during the years to approach the
Pareto frontier by successive iterations or to solve in parallel a sequence of singleobjective optimisation problems.
A comprehensive survey of multi-objective optimisation techniques is given
in [44–46], the last two focusing mainly on global evolutionary multi-objective
241
where x is the optimisation variables vector, f : Ω → R n obj , with n obj > 1, the
objective function and c : Ω → R m the constraints function. As before the set of
feasible points is denoted by D.
To extend the methodologies presented for global optimisation to the multiobjective case, it is necessary to introduce some definitions.
Definition 7.2.6 A point x 1 ∈ D Pareto dominates x 2 ∈ D if
f i (x 1 ) ≤ f i (x 2 ),
i = 1, . . . , n obj
and there is at least one component j ∈ {1, . . . , n obj } such that
f j (x 1 ) < f j (x 2 ).
This is indicated by
x 1 x 2 .
Definition 7.2.7 A point x ∗ ∈ D is Pareto optimal if it isn’t dominated by any
x ∈ D,
x x
∗ .
In other words, a solution is said to be Pareto optimal, or equivalently nondominated,
if there is no other point in the feasible space for which a decrease in one objective
will not cause a simultaneous increase of at least one of the other objectives.
Definition 7.2.8 For a multiple objective optimisation problem, the Pareto optimal
set is defined as
P
∗
= {x ∈ D | ∃x
∈ D x
x}.
Definition 7.2.9 The union of the objective values of all Pareto optimal points is
called Pareto front or equivalently
PF
∗
= {f (x) ∈ R
n obj | x ∈ P
∗
}.
The Pareto front is the set of all solutions in the feasible space that are not
dominated by any other possible solution. The minima in the sense of Pareto will
lie on the boundary of the feasible region or in the tangent points of the objective
functions. Generally it is not possible to derive analytically the equation of the front.
Approximation techniques have been developed during the years to approach the
Pareto frontier by successive iterations or to solve in parallel a sequence of singleobjective optimisation problems.
A comprehensive survey of multi-objective optimisation techniques is given
in [44–46], the last two focusing mainly on global evolutionary multi-objective