Metaheuristic Optimization in Seismic Structural Design and Inspection Scheduling of Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 164

absolute velocities for each dimension. It is more
appropriate to use a vector rather than a scalar, as
in the general case different velocity restrictions
can be applied for different dimensions of the
particle. If for a given particle j the sum of ac-
celerations of Eq. (7) causes the absolute velocity
for dimension i to exceed v max i , then the velocity
on that dimension is limited to ± v max,i . The vector
parameter v max is employed to protect the cohesion
of the system, in the process of amplification of
the positive feedback. The basic PSO has only few
parameters to adjust. In Table 8 there is a list of
the main parameters, their typical values as well
as other information (Perez & Behdinan, 2007).

criterion presumes prior knowledge of the global
optimal value, which is feasible for testing or fine-
tuning the algorithm in mathematical problems
when the optimum is known a priori, but this
is certainly not the case in practical structural
optimization problems where the optimum is not
known a priori.
In our study, together with the maximum
number of iterations, we have implemented the
convergence criterion connected to the rate of
improvement of the value of the objective function
for a given number of iterations. If the relative
improvement of the objective function over the
last k f iterations (including the current iteration) is
less or equal to a threshold value f m , convergence is
supposed to have been achieved. In mathematical
terms, denoting as Gbest t the best value for the
objective function found by the PSO at iteration t ,
the relative improvement of the objective function
can be written for the current iteration t as follows

Convergence Criteria

Due to the repeated process of the PSO search,
convergence criteria have to be applied for the
termination of the optimization procedure. Two
widely adopted convergence criteria are the
maximum number of iterations of the PSO algo-
rithm and the minimum error requirement on the
calculation of the optimum value of the objective
function. The selection of the maximum number
of iterations depends, generally, on the complexity
of the optimization problem at hand. The second

Gbest − Gbest

t k − + 1 t

≤ f (10)

f

m

Gbest

t k − + 1

f

In Table 9 there is a list of the convergence
parameters of the PSO used in this study with
description and details.

Table 8. Main PSO parameters

Symbol Description Details

NP Number of particles A typical range is 10 - 40. For most problems 10 particles is sufficient

enough to get acceptable results. For some difficult or special prob-
lems the number can be increased to 50-100.

n Dimension of particles It is determined by the problem to be optimized.

w Inertia weight Usually is set to a value less than 1, i.e. 0.95. It can also be updated

during iterations.

x L , x U Vectors containing the lower and upper

bounds of the n design variables, respec-
tively

They are determined by the problem to be optimized. Different ranges
for different dimensions of particles can be applied in general.

v max Vector containing the maximum allowable

velocity for each dimension during one
iteration

Usually is set half the length of the allowable interval for the given
dimension: v max i = ( x U i - x L i )/2. Different values for different dimen-
sions of particles can be applied in general.

c 1 , c 2 Cognitive and social parameters Usually c 1 = c 2 =2. Other values can also be used, provided that 0 <

c 1 + c 2 < 4 (Perez & Behdinan, 2007)