A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 63

600

GA with traditional crossover
GA with multiple crossover
Standard PSO
Hybrid algorithm

500

400

300

200

100

0

0 2000 4000 6000 8000 10000

Iteration

20

15

10

5

0

2000 3000 4000 5000 6000 7000 8000 9000 10000

Fig. 6 The evolutionary trajectory of the single-objective optimization algorithm on the Noise test
function

P ¼ ~

f x 2 X j~ x is Pareto-optimalf g

De nition 5 Pareto-optimal front: The Pareto-optimal front or in a more straight-

forward expression, Pareto front PF is de ned as:

PF ¼ f f R m P g:

ð~ x Þ2 j~ x 2

5.2.2 The Structure of the Hybrid Algorithm for Multi-objective
Optimization

It is necessary to make modi finding the

optimal solutions for multi-objective problems. In the single-objective algorithm of
PSO, the best particle of the entire swarm (

cations to the original scheme of PSO in

~

x gbest ) is utilized as a leader. In the multi-
objective algorithm, each particle has a set of different leaders that one of them is
chosen as a leader. In this topic paper, a leader selection method based upon density