Swarm Intelligence Algorithms

Conclusion 173

subject to

φ

i

(x) = 0, (i = 1, 2, ..., M), (13.21)

ψ

j

(x) ≤ 0, (j = 1, 2, ..., N), (13.22)

where x is the vector of D design variables, and φ

i

(x) and ψ

j

(x) are the equal-

ity constraints and inequality constraints, respectively. The penalty-based

method transforms the objective f(x) into a modified objective Θ in the

following form:

Θ(x) = f(x)[objective] + P (x)[penalty], (13.23)

where the penalty term P (x) can take different forms, depending on the actual

ways or variants of constraint-handling methods. For example, a static penalty

method uses

P (x) = λ

h

M

X

i=1

φ

2

i

(x) +

N

X

j=1

max{0, ψ

j

(x)}

2

i

. (13.24)

Since λ > 0 is fixed, independent of the iteration ...