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A. Riccardi et al.
These constraints are assigned different constants based on the importance of their
respective objective functions (e.g. minimum reliability levels, maximum price), and
multiple solution of a single-objective problem are found for different satisfaction
levels of each constraint. A deeper discussion into multi-objective strategies is
provided in Sect. 7.2.3.
7.2 Continuous Optimisation
7.2.1 Local Optimisation
Local optimisation algorithms are exact methods that guarantee the convergence to
the local optimum in a neighbourhood of search. They are the most investigated
optimisation techniques and have their roots in the calculus of variations and the
work of Euler and Lagrange. The development of linear programming falls back to
the 1940s, and it was the base of the modern optimisation theory that rapidly grew
and then was developed in the last 70 years.
As already defined in the previous section, the general optimisation problem is
defined as
min
x∈D
f (x)
where D = {x ∈ Ω | c(x) ≤ 0}, f : Ω → R and c : Ω → R m are sufficiently
smooth functions. It must be pointed out that local optimisation techniques restrict
their field of application to single-objective optimisation problems with continuous
variables. To extend the use to multi-objective optimisation problems, one of the
aggregate techniques presented above must be taken into consideration.
Before introducing the optimality results, some definitions need to be stated.
Definition 7.2.1 The real function L : Ω × R m → R defined as
L (x, λ) = f (x) − λ
T c(x)
is the Lagrangian, and the coefficients λ ∈ R m are called Lagrange multipliers.
Definition 7.2.2 Given a point x in the feasible region, the active set A (x) is
defined as
A (x) = {i ∈ I | c i (x) = 0},
where I = {1, . . . , m} is the index set of the constraint.
Definition 7.2.3 The linear independence constraint qualification (LICQ) holds if
the set of active constraint gradients {∇c i (x), i ∈ A (x)} is linearly independent,
that is,
A. Riccardi et al.
These constraints are assigned different constants based on the importance of their
respective objective functions (e.g. minimum reliability levels, maximum price), and
multiple solution of a single-objective problem are found for different satisfaction
levels of each constraint. A deeper discussion into multi-objective strategies is
provided in Sect. 7.2.3.
7.2 Continuous Optimisation
7.2.1 Local Optimisation
Local optimisation algorithms are exact methods that guarantee the convergence to
the local optimum in a neighbourhood of search. They are the most investigated
optimisation techniques and have their roots in the calculus of variations and the
work of Euler and Lagrange. The development of linear programming falls back to
the 1940s, and it was the base of the modern optimisation theory that rapidly grew
and then was developed in the last 70 years.
As already defined in the previous section, the general optimisation problem is
defined as
min
x∈D
f (x)
where D = {x ∈ Ω | c(x) ≤ 0}, f : Ω → R and c : Ω → R m are sufficiently
smooth functions. It must be pointed out that local optimisation techniques restrict
their field of application to single-objective optimisation problems with continuous
variables. To extend the use to multi-objective optimisation problems, one of the
aggregate techniques presented above must be taken into consideration.
Before introducing the optimality results, some definitions need to be stated.
Definition 7.2.1 The real function L : Ω × R m → R defined as
L (x, λ) = f (x) − λ
T c(x)
is the Lagrangian, and the coefficients λ ∈ R m are called Lagrange multipliers.
Definition 7.2.2 Given a point x in the feasible region, the active set A (x) is
defined as
A (x) = {i ∈ I | c i (x) = 0},
where I = {1, . . . , m} is the index set of the constraint.
Definition 7.2.3 The linear independence constraint qualification (LICQ) holds if
the set of active constraint gradients {∇c i (x), i ∈ A (x)} is linearly independent,
that is,