A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 55

5 Optimization

Optimization is mathematical and numerical approaches to gain and identify the
best candidate among a range of alternatives without having to explicitly enumerate
and evaluate all possible alternatives (Ravindran et al. 2006 ). While maximum or
minimum solutions of objective functions are the optimal solutions of an optimi-
zation problem, optimization algorithms are usually trying to address a minimiza-
tion problem. In this regard, the goal of optimization is to gain the optimal solutions
which are the points minimizing the objective functions. Based upon the number of
objective functions, an optimization problem is classi

ed as single-objective and
multi-objective problems. This study uses both single-objective and multi-objective
optimization algorithms to evaluate the capabilities of the hybrid of particle swarm
optimization and the genetic algorithm (Mahmoodabadi et al. 2013 ). To this end,
challenging benchmarks of the

field of optimization are chosen to evaluate the
optimization algorithm. The hybrid of particle swarm optimization and the genetic
algorithm is applied to these benchmarks and the obtained results are compared to
the obtained results of running a number of similar algorithms on the same
benchmark problems.

5.1 Single-Objective Optimization

5.1.1 De nition of Single-Objective Optimization Problem

A single-objective optimization problem involves just one objective function as
there are many engineering problems where designers combine several objective
functions into one. Each objective function can include one or more variables.
A single-objective optimization problem can be de ned as follow:

Find ! ¼½x 1 ; x 2 ; ...; x n 2 R n
Tominimize f

ð~ x Þ

By regarding p equality constraints g i ð ~ x Þ ¼ 0 ; i ¼ 1 ; 2 ; ...; p; and q inequality

constraints h j ð ~

~

x Þ 0 ; j ¼ 1 ; 2 ; ...; q, where x represents the vector of decision

variables and f ð~ x Þ denotes the objective function.

5.1.2 The Architecture of the Algorithm of Single-Objective
Optimization

In this section, a single-objective optimization algorithm is used which is based on a
hybrid of genetic operators and PSO formula to update the chromosomes and
particle positions (Mahmoodabadi et al. 2013 ). In elaboration, the initial population