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A. Riccardi et al.
strategies. Evolutionary programming is the area of multi-objective optimisation
research that in the last years registered the fastest growth. This is due to the
intrinsic structure of the evolutionary algorithms, population based, well suited for
an extension to multi-objective problems.
The multi-objective approaches are divided in methods that use the concept of
Pareto dominance for the selection mechanism of the next iterates and methods that
develop a special handling of the objective functions for reformulating the problem
as single objective. The latter techniques are applicable to all the presented global
optimisation strategies, while the former are typically for evolutionary algorithms.
The aggregation of the multiple objectives into a common single objective
can be achieved by the different techniques presented below, outlining their main
advantages and disadvantages.
• Weighted sum approach: the objectives are aggregated into a single function
using weighting coefficients. The optimisation problem becomes
min
x∈Ω
n obj
i=1 w i f i (x)
subject to c(x) ≤ 0,
where w i ≥ 0 and it is usually assumed that
n obj
i=1
w i = 1.
By varying the values of the coefficients, different solutions on the Pareto front
are traced. To cover the entire front, a sequence of single-objective optimisation
problems needs to be solved, making the procedure very inefficient from a
computational point of view. Moreover, this technique has the drawback of not
generating proper Pareto optimal solutions in the presence of non-convex search
spaces [47]. Additionally, there is no a priori knowledge about how a change in
the weights will affect the position on the Pareto front of the new solution.
• Goal programming [48]: the designer has to assign targets to the objectives, and
the optimisation problem is transformed in the problem of minimising the sum
of the norms of the deviations from the targets
min
x∈Ω
n obj
i=1 f i (x) − T i 2
subject to c(x) ≤ 0.
Prerequisite in the application of such a technique is a deep knowledge about
the optimisation problem to be able to assign meaningful target values to
the objectives. The search space is explored by varying the T i targets, and
convergence to the Pareto front is achieved with a prior knowledge of the
problem, to assign the targets close to the objectives values of the Pareto optimal
points.
A. Riccardi et al.
strategies. Evolutionary programming is the area of multi-objective optimisation
research that in the last years registered the fastest growth. This is due to the
intrinsic structure of the evolutionary algorithms, population based, well suited for
an extension to multi-objective problems.
The multi-objective approaches are divided in methods that use the concept of
Pareto dominance for the selection mechanism of the next iterates and methods that
develop a special handling of the objective functions for reformulating the problem
as single objective. The latter techniques are applicable to all the presented global
optimisation strategies, while the former are typically for evolutionary algorithms.
The aggregation of the multiple objectives into a common single objective
can be achieved by the different techniques presented below, outlining their main
advantages and disadvantages.
• Weighted sum approach: the objectives are aggregated into a single function
using weighting coefficients. The optimisation problem becomes
min
x∈Ω
n obj
i=1 w i f i (x)
subject to c(x) ≤ 0,
where w i ≥ 0 and it is usually assumed that
n obj
i=1
w i = 1.
By varying the values of the coefficients, different solutions on the Pareto front
are traced. To cover the entire front, a sequence of single-objective optimisation
problems needs to be solved, making the procedure very inefficient from a
computational point of view. Moreover, this technique has the drawback of not
generating proper Pareto optimal solutions in the presence of non-convex search
spaces [47]. Additionally, there is no a priori knowledge about how a change in
the weights will affect the position on the Pareto front of the new solution.
• Goal programming [48]: the designer has to assign targets to the objectives, and
the optimisation problem is transformed in the problem of minimising the sum
of the norms of the deviations from the targets
min
x∈Ω
n obj
i=1 f i (x) − T i 2
subject to c(x) ≤ 0.
Prerequisite in the application of such a technique is a deep knowledge about
the optimisation problem to be able to assign meaningful target values to
the objectives. The search space is explored by varying the T i targets, and
convergence to the Pareto front is achieved with a prior knowledge of the
problem, to assign the targets close to the objectives values of the Pareto optimal
points.