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UAV Path Planning Using Evolutionary Algorithms 99
donor comprises the local information of all members of the triplet, provid-
ing a better starting-point for the mutation operation that result in a better
distribution of the trial-vectors. As it is reported in [42], the modified donor
scheme accelerated the DE convergence rate, without sacrificing the solution
precision or robustness of the DE algorithm.
The random number generation (with uniform probability) is based on
the algorithm presented in [43], which computes the remainder of divisions
involving integers that are longer than 32 bits, using 32-bit (including the
sign bit) words. The corresponding algorithm, using an initial seed, produces
a new seed and a random number. In each different operation inside the DE
algorithm that requires a random number generation, a different sequence
of random numbers is produced, by using a different initial seed for each
operation and a separate storage of the corresponding produced seeds. By
using specific initial seeds for each operation, it is ensured that the different
sequences differ by 100,000 numbers.

5.2 Radial Basis Function Network for DE Assistance

Despite their advantages, EAs ask for a considerable amount of evaluations.
In order to reduce their computational cost several approaches have been pro-
posed, such as the use of parallel processing, the use of special operators and
the use of surrogate models and approximations. Surrogate models are auxil-
iary simulations that are less physically faithful, but also less computationally
expensive than the expensive simulations that are regarded as “truth”. Sur-
rogate approximations are algebraic summaries obtained from previous runs
of the expensive simulation [44, 45]. Such approximations are the low-order
polynomials used in Response Surface Methodology [46, 47], the kriging esti-
mates employed in the design and analysis of computer experiments [48], and
the various types of Artificial Neural Networks [45]. Once the approximation
has been constructed, it is typically inexpensive to use.
DE has been demonstrated to be one of the most promising novel EAs, in
terms of efficiency, effectiveness and robustness. However, its convergence rate
is still far from ideal, especially when it is applied in optimization problems
with time consuming objective functions. In order to enhance the convergence
rate of DE algorithm, an approximation model is used for the objective func-
tion, based on a Radial Basis Functions Artificial Neural Network [49]. In
general a RBFN (Fig. 5), is a three layer, fully connected feed-forward net-
work, which performs a nonlinear mapping from the input space to the hidden
1
M
L
space (R → R ), followed by a linear mapping (R M → R ) from the hidden
to the output space (L is the number of input nodes, M is the number of
hidden nodes, while the output layer has a single node).
The corresponding output yy(xx), for an input vector xx=[xx 1 , xx 2 ,...,xx L ]
is given
M

yy (xx)= w i · ϕ i (xx). (26)
i=1