Metaheuristic Optimization in Seismic Structural Design and Inspection Scheduling of Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 163

Figure 4. Visualization of the particle's movement in a two-dimensional design space

drawing. It can be seen how the particle's move-
ment is affected by: (i) it's velocity v j ( t ); (ii) the
personal best ever position of the particle, x Pb, j ,
at the right of the figure; and (iii) the global best
location found by the entire swarm, x Gb , at the
upper left of the figure.
In the above formulation, the global best loca-
tion found by the entire swarm up to the current
iteration ( x Gb ) is used. This is called a fully con-
nected topology (fully informed PSO), as all
particles share information with each other about
the best performer of the swarm. Other topologies
have also been used in the past where instead of
the global best location found by the entire swarm,
a local best location of each particle's neighbour-
hood is used. Thus, information is shared only
among members of the same neighbourhood.
The term w of Eq. (7) is the inertia weight,
essentially a scaling factor employed to control
the exploration abilities of the swarm, which
scales the current velocity value affecting the
updated velocity vector. The inertia weight was
not part of the original PSO algorithm (Kennedy

& Eberhart,1995), as it was introduced later by
Shi and Eberhart (1998) in a successful attempt
to improve convergence. Large inertia weights
will force larger velocity updates allowing the
algorithm to explore the design space globally.
Similarly, small inertia values will force the veloc-
ity updates to concentrate in the nearby regions
of the design space.
The inertia weight can also be updated during
iterations. A commonly used inertia update rule is
the linearly-decreasing, calculated by the formula:

w − w

w

t + =

w −

max min

⋅ t (9)

1 max

t

max

where t is the iteration number, w max and w min are the
maximum and minimum values, respectively, of
the inertia weight. In general, the linearly decreas-
ing inertia weight has shown better performance
than the fixed one.
Particles' velocities in each dimension i ( i = 1,
…, n ) are restricted to a maximum velocity v max i .
The vector v max of dimension n holds the maximum