Artificial Intelligence Applications



Particle Swarm Optimization




Introduction

 ·         Swarm Intelligence (SI) is an innovative distributed intelligent paradigm for solving optimization problems that originally took its inspiration from the biological examples by swarming, flocking.
 ·         Particle Swarm Optimization (PSO) incorporates swarming behaviors observed in flocks of birds, schools of fish, or swarms of bees, and even human social behavior, from which the idea is emerged.
 ·
 ·         PSO is a population-based optimization tool, which could be implemented and applied easily to solve various function optimization problems, or the problems that can be transformed to function optimization problems.

The flocking behavior of a group of birds 


 ·         As an algorithm, the main strength of PSO is its fast convergence, when compares it with many global optimization algorithms like Genetic Algorithms (GA), and other global optimization algorithms.

 ·         For applying PSO successfully, one of the key issues is finding how to map the problem solution into the PSO particle, which directly affects its feasibility and performance.

Canonical Particle Swarm Optimization

 ·         The canonical PSO model consists of a swarm of particles, which are initialized with a population of random candidate solutions.
 ·         They move iteratively through the d-dimension problem space to search the new solutions, where the fitness, f, can be calculated as the certain qualities measure.
 ·         Each particle has a position represented by a position-vector  xi  (i is the index of the particle), and a velocity represented by a velocity-vector vi.
 ·        Each particle remembers its own best position so far in a vector

  and its j-th dimensional value is 

 The best position-vector among the swarm so far is then stored in a 

     Vector

 and its j-th dimensional value is 

 ·         During the iteration time t, the update of the velocity from the previous velocity to the new velocity is determined by Eq.(1).
 ·         he new position is then determined by the sum of the previous position and the new velocity by Eq.(2).
Eq.(1)




  §  where
  §  w is called as the inertia factor,
  §  r1 and r2 are the random numbers, which are used to maintain the diversity of the population, and are uniformly distributed in the interval [0,1] for the j-th dimension of the i-th particle.
  §  C1 is a positive constant, called as coefficient of the self-recognition component,
  §  c2 is a positive constant, called as coefficient of the social component.

  §  From Eq.(1), a particle decides where to move next, considering its own experience, which is the memory of its best past position, and the experience of its most successful particle in the swarm.

  §  The new position is then determined by the sum of the previous position and the new velocity by Eq.(2).

Eq.(2)




The Parameters of PSO

  §  The inertia weight w, in Eq.(1), an initial value around 1.2 and gradually reducing towards 0 can be considered as a good choice for w.
  §  The parameters c1 and c2, in Eq.(1), As default values, usually, c1 = c2 = 2 are used, but some experiment results indicate that c1 = c2 = 1.49 might provide even better results.

  §  In the particle swarm model, the particle searches the solutions in the problem space with a range [−s, s] (If the range is not symmetrical, it can be translated to the corresponding symmetrical range.)
  §  In order to guide the particles effectively in the search space, the maximum moving distance during one iteration must be clamped (fixed) in between the maximum velocity [−vmax, vmax] given in



Eq.(3)






The value of v max is p × s, with 0.1 ≤ p ≤ 1.0 and is usually chosen to be s, i.e. p = 1.

article Swarm Optimization Algorithm



The end criteria are usually one of the following:

§  Maximum number of iterations: the optimization process is terminated after a fixed
number of iterations, for example, 1000 iterations.
§  Number of iterations without improvement: the optimization process is terminated after some fixed number of iterations without any improvement.
§  Minimum objective function error: the error between the obtained objective function value and the best fitness value is less than a pre-fixed anticipated threshold.










By : mogtaba altyib 
Modification by : Abdelhamid Salih   Ph.D


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