A Study on the Convergence of Family Particle Swarm Optimization

The sociological concept of family has been introduced in the particle swarm optimization (PSO) and the family PSO (FPSO) has been proposed, in which the particle swarm consisted of different families, each family consisted of different members, and there were different constraint relationships between family members. To further study the sensitivity of FPSO to the control parameters, this paper proposed a special model of FPSO and analyzed the convergence of FPSO theoretically.This model offered a new view to research the particle trajectory and divided the position sequence of particle into the even and odd subsequences. By mathematical analysis, the condition of two subsequences convergence was obtained and the related convergent theories and corollaries were proved. Simulations for benchmark functions showed that the convergence behavior of model and experimental results provided a valuable guideline for selecting control parameters.


Introduction
Particle swarm optimization (PSO) is an evolutionary computation algorithm which is motivated by the preying behavior of bird flocking [1].Due to the simple but efficient characteristics, the PSO has been successfully applied to biomedical image segmentation [2], gene selection and classification [3], and data mining [4], and so forth.
A great deal of theoretical research has been done to study the convergence performance of PSO [5][6][7].From these studies it has been concluded that the PSO is sensitive to the choice of control parameters, specifically to the inertia weight and acceleration coefficients.Wrong initialization of these parameters may lead to divergent.To further understand the behavior of particle swarm, some theoretical studies have been done to analyze the trajectory of a single particle in PSO.Ozcan and Mohan [8,9] concluded that the trajectory of a particle in a simple PSO system was a sinusoidal wave where the initial conditions and parameter choices determined its amplitude and frequency.Van Den Bergh and Engelbrecht [10] developed a model of PSO considering the influence of the inertia weight.Clerk and Kennedy [11] provided a theoretical analysis of particle behavior in which a set of coefficients to control the system's convergence tendencies were analyzed.
Most of the theoretical studies were based on the simplified PSO models, in which a swarm consisted of one particle of one dimension.The personal best position and the global best position of particles were assumed to be constant throughout the process.Obviously, interactive effects among particles were not taken into account effectively.
To study the interactive effects among particles and enlarge an individual's cognitive ability, the sociological concept, the so-called family [12,13] was introduced in the PSO and the family PSO (FPSO) [14,15] was proposed.When family is considered as a unit, the relationship of family members will be very important.There are different constraint relationships between family members.For example, an equal relationship exists between husband and wife or between siblings; a generational relation exists between parents and children.The different types of relationships among family members mean the different family communication strategies [12,13].So the particle swarm consisted of different families, each family consisted of different members and there were different constraint relationships between family members in the FPSO [14,15].
In this paper, we further analyzed the FPSO theoretically and proposed the special model of FPSO.This model divided the position sequence of particle into the even and odd subsequences.The condition of two subsequences convergence was obtained and the related convergent theories and corollaries were proved.These theories and corollaries demonstrated that the particle trajectory is remarkably different for different parameter sets.

Overview of the PSO
PSO is a population-based stochastic optimization technique.In the PSO algorithm, an individual particle  is composed of three vectors: its position   , the best position found by itself   , and its velocity V  .Particles are originally initialized in a uniform random manner throughout the search space.Then, their positions are changed according to their own experience and that of the entire swarm.
The velocity and position are defined by the following rules: where  1 =  1  1 ,  2 =  2  2 ;  1 and  2 are called acceleration coefficients;  1 and  2 are uniformly distributed pseudorandom numbers in the range of 0-1;   () is the personal best position, and () is the best position found by the swarm at the th iteration.The constriction factor  is defined by Clerc and Kennedy [11].

The Performance Analysis of the FPSO
3.1.Description of the FPSO.In the FPSO [14,15], the particle swarm consists of different families.Every family has more than one member.Every member in the family provides the information got by the previous experience to other family members.
The velocity and position of particle  are updated by the following rules (suppose particle  belongs to the th family): where   () is the best position found by the th family and () is the best position found by all families at the th iteration.
The particle position update equation can by derived by the following transform: Substituting V  () =   () −   ( − 1) into (4), By substitution, we obtain that the position of the th particle is updated by the following second-order nonhomogeneous linear differential equation: Let we have 3.2.The Special Model of FPSO.The constraint relationship among the parameters ,  1 , and  2 can be obtained through studying the coefficients of   () and   ( − 1) in (8).To analyze the parameters of FPSO, there are two kinds of noticeable parameter settings: one is  = 0 and another is When  = 0, the previous velocity will not influence the new velocity and the memory of the previous flight direction will be erased.Equation (8) can be simplified as shown below: can be simplified as shown below: In this section, 1 +  −  1 −  2 = 0 will be particularly analyzed.

The Convergent Property of the Special FPSO Model
Proof.
(1) If  = 1, we trivially have (2) Assume that the equation is true for  = : that is, Then we need to show that the equation continues to hold for  =  + 1.
First, we prove the limit lim Second, we prove the limit lim →∞   (2 − 1) exists.
Proof.If particles  and  belong to the th family, then   () =   ().Thus, This means if particles  and  belong to the same family, the even and odd sequences of {  ()} ∞ =0 and {  ()} ∞ =0 will converge to the same point, respectively.Let lim Proof.One has This means if particles  and  belong to the different families, the distances of the even and odd sequences of {  ()} ∞ =0 and {  ()} ∞ =0 will converge to the invariable values, respectively.
Proof.One has This means if   =    and   =    , the even and odd sequences of {  ()} ∞ =0 will converge to the same point.

Corollary 6.
If the family best position and the global best position are assumed to be constant throughout the process, that is, In particular, Proof.One has This means if the family best position and the global best position are assumed to be constant throughout the process, the even and odd sequences of {  ()} ∞ =0 will converge to the weighted average of the family best position and the global best position; if   = , they will converge to the global best position; if   =  = 0, they will converge to zero.

Corollary 7. When 𝐹
oscillate with the amplitude gradually decreasing to zero.

Experiments
In order to analyze the parameters, extensively adopted benchmark functions were used in the experiments, as listed in Table 1.The functions  1 - 2 were simple unimodal problems;  3 - 5 were highly complex multimodal problems with many local minima.A detailed description of these functions could be found in [16].

Random Examples of the Even and Odd Properties of FPSO.
In order to facilitate the comparison, the population size was four particles that included two families and every family had two particles in the experiments.Because different benchmark functions had different domains, the positions and velocities of particles were initialized in the experiments, respectively.In our experiments,  1 and  2 were generated randomly.The choices of parameters  satisfied the following equation: 1 +  =  1 +  2 .Simulations of particle trajectory for these parameters were given in Figures 1-7.

Examples of the
In Figures 10-13, part (a) represented the family best position and the global best position that were the same and equaled zero; part (b) represented the family best position and the global best position that were variable with the iteration times.
Simulations for benchmark functions demonstrated that the particle trajectory could be divided into the even and odd subtrajectories.Experimental results demonstrated that the particle trajectory of FPSO was remarkably affected by the different parameter choices.These results provided a valuable guideline for selecting control parameters.
Future research is needed to find better parameter sets in the convergence domain, the effect of the randomness and the interaction among particles in a family, and so forth.
The even and odd trajectories
The even and odd trajectories
Convergent Property of FPSO.The following examples represented how the convergent property of FPSO was affected by the different choices of the algorithm parameters  1 ,  2 , and .Simulations of particle trajectory according to related parameters were given in Figures10-13 .

)
Corollary 4. If particle  belongs to the  1 th family and particle  belongs to the  2 th family, then

Table 1 :
Mathematical representation of test functions.