Basics Of Particle Swarm Oprimisation

 The basic Particle Swarm Optimization or PSO is a computer paradigm such is based upon social and cognitive behaviour of humans. Mainly, the algorithm belongs to the category of swarm intelligence and gains its broad guideline from the socio-cognitive perspectives. Particle Swarm Optimization has received growing attention in past few years; a few significant papers can be cited from the literature that studies the application of the same in difficult optimization problems. It was first introduced and developed by Eberhart and Kennedy. Their motive was to envisage the effect of collaboration of

species onto achieving their goals as a group. In contrast, to Genetic Algorithm (GA) and Evolutionary Strategies (ESs) which exploits the competitive characteristics of biological evolution (e.g. survival of the fittest), Particle Swarm Optimization exploits cooperative and social aspects. The traditional artificial intelligence paradigms focuses mainly on problem-variables like parameter space, the particle-swarm approach embeds problem- solving attempts in a social network. It can be regarded as a system where multiple candidate solution co-exists and collaborate simultaneously. Each solution candidate resembles a 'particle', which flies in the problem search space looking for the optimal position to land. The search process is same just as the bird swarm searches for food. Gradually, as time passes, particle with his quest adjusts its position according to its own experience (i.e. local search) and as well as from the experience of neighbouring particles (i.e. global search). Particle's experience builds up as it tracks and memorizes the best position encountered so far. Both of these attributes have been utilized in PSO system for exploring the search space effectively.

Moreover, two factors demonstrating the particle status on the search space are its position and velocity. The position and velocity vector of ith particle, in d-dimensional search space is represented as

                                      x_i  = ( x_i,₁ , x_i, ₂ , . . . , x_{i, d} )......................Eq(11)

                                       v_i  = (v_i,₁ , v_i, ₂ , . . . , v_{i, d} ) ................Eq(12)

 Assume the best position of each particle corresponding to best fitness value obtained by that particle at time t is given by

                                       p_i  = ( p_i,1 , p_{i, 2} , . . . , p_{i, d} ) ................Eq(13)

And the fittest particle found so long at time t is:

                                      p_g  = ( p_{g ,1}, p_{g , 2} , ... , p_{g , d} ) ................Eq(14)

The new position and velocity of the particle for next fitness evaluation can be calculated from Eqs. 10 and 11.

                                 v_{i, j}  = v_{i, j}  + ?₁ × rand (.) × Δx_c + ?₂  × Rand (.) × Δx_n............................Eq(15)

                                         x_{i, j}  = x_{i, j}  + v_{i, j}..................................Eq(16)

Here Δx_c = p_{i, j}  - p_{i, j}

 And

Δx_n = p_{g , j}  - x_{i, j}

In velocity relation, Δx_c and Δx_n are self best positional differences and neighborhood best positional differences respectively. While ?₁ and ?₂ are positive (+ive) constant parameters named as acceleration coefficients that are causes for controlling the maximum step size whereas rand as X and Rand as Y are two separately generated uniformly spread random numbers in the range [0, 1]. The algorithmic steps of essential Particle Swarm Optimization Algorithm are represented in Programme no.3.

In mathematical presentation of velocity, its first part shows the previous velocity for roaming across search space, is provided by the essential momentum to particles. The second part is cognitive component, resembling the individual thinking of all particles. This encourages the particle to move towards their best positions found so long. The third part of Eq. 15 termed as the social component indicates the collaborative effect of the particles in determining global optimal solution. The social component all the time pulls the particles towards the global best particle found so far.

Initialize population parameters, topology, and constants

Repeat

Do I = 1 to population size

Current Eval_i < eval (x_i)

If Current Eval < pbest_i then do

If Current Eval_i < Pbest_gbest then gbest = i

Pbest_i = Current Evali

For d=1 to Dimension

P_id = x_id

Next d

End if

g = index of best neighbor

For d = 1 to Dimension

V_i,d  = v_i,d + ?₁ × rand (.) × Δx_c + ?₂  × Rand (.) × Δx_n

x_i,d   = x_i,d   + v_i,d 

Next d

Next i

Until termination criterion

                     Programme no.3: Algorithmic Steps of Particle Swarm Optimization Algorithm