The particle swarm optimization (PSO) is a recently invented evolutionary computation technique which is gaining popularity owing to its simplicity in implementation and rapid convergence. In the case of single-peak functions, PSO rapidly converges to the peak; however, in the case of multimodal functions, the PSO particles are known to get trapped in the local optima. In this paper, we propose a variation of the algorithm called parallel swarms oriented particle swarm optimization (PSO-PSO) which consists of a multistage and a single stage of evolution. In the multi-stage of evolution, individual subswarms evolve independently in parallel, and in the single stage of evolution, the sub-swarms exchange information to search for the global-best. The two interweaved stages of evolution demonstrate better performance on test functions, especially of higher dimensions. The attractive feature of the PSO-PSO version of the algorithm is that it does not introduce any new parameters to improve its convergence performance. The strategy maintains the simple and intuitive structure as well as the implemental and computational advantages of the basic PSO.

Evolutionary algorithms (EAs) are increasingly being applied to solve the problems in diverse domains. These metaheuristic algorithms are found to be successful in many domains chiefly because of their domain-independent evolutionary mechanisms. Evolutionary computation is inspired by biological processes which are at work in nature. Genetic algorithm (GA) [

Particle swarm optimization (PSO) is a recently developed algorithm belonging to the class of biologically inspired methods [

Researchers have found that PSO has the following advantages over the other biologically inspired evolutionary algorithms: (1) its operational principle is very simple and intuitive; (2) it relies on very few external control parameters; (3) it can be easily implemented; and (4) it has rapid convergence. PSO has developed very fast, has obtained very good applications in wide variety of areas [

In the case of single-peak functions, PSO rapidly converges to the peak; however, in the case of multi-modal functions, the PSO particles are known to get trapped in the local optima. A significant number of variations are being made to the standard PSO to avoid the particles from getting trapped in the local optima [

However, in all the studies mentioned above, a number of

This paper is organized as follows. In Section

The population-based PSO conducts a search using a population of individuals. The individual in the population is called the particle, and the population is called the swarm. The performance of each particle is measured according to a predefined fitness function. Particles are assumed to “fly” over the search space in order to find promising regions of the landscape. In the minimization case, such regions possess lower functional values than other regions visited previously. Each particle is treated as a point in a

The notations used in PSO are as follows. The

Any successful meta-heuristic algorithm maintains a delicate balance between exploration (diversifying the search to wider areas of the search space) and exploitation (intensifying the search in narrow promising areas). Shi and Eberhart later introduced the inertia weight

Eberhart and Shi have further proposed a random modification for the inertia weight to make the standard PSO applicable to dynamic problems [

As opposed to the above linear decrement, Jie et al. [

Generally, the very same velocity and position update formulae are applied to each and every flying particle. Moreover, the very same inertia weight is applied to each particle in a given iteration. However, Yang et al. [

The aggregation degree is given by

The inertia weight is updated as

Another variation is the introduction of a constriction coefficient to replace

In all the PSO variations mentioned in the preceding sections, a number of

The algorithm consists of a multievolutionary phase and a single-evolutionary phase. In the multi-evolutionary phase, initially a number of sub-swarms are randomly generated so as to uniformly cover the decision space. The multiple sub-swarms are then allowed to evolve independently, each one maintaining its own particle-best (

Parallel swarms oriented PSO algorithm flowchart.

Randomly generate

Evaluate the fitness of the particles in each individual swarm. In the minimization problems, the fitness of a particle is inversely proportional to the value of the function.

Determine the particle-best (

Update the velocity and position of each particle in each swarm according to (

Allow the individual swarms to evolve independently through

Determine the global-best (

The individual swarms start interacting by using the

Update the positions of all the particles according to the following equation:

Repeat Steps

The PSO-PSO algorithm is implemented in MATLAB on a 24-core server with 256 GB RAM. Each CPU core is dedicated to the evolution of a single sub-swarm in the multi-stage computational phase of the algorithm. The results are presented in Figures

(a) 10D Rosenberg and (b) 10D Rastrigin.

(a) 20D Rosenbrock and (b) 20D Rastrigin.

(a) 30D Rosenbrock and (b) 30D Rastrigin.

The particle swarm optimization (PSO) is increasingly becoming widespread in a variety of applications as a reliable and robust optimization algorithm. The attractive features of this evolutionary algorithm is that it has very few control parameters, is simple to program, and is rapidly converging. The only drawback reported so far is that at times it gets trapped in local optima. Many researchers have addressed this issue but by introducing a number of new parameters in the original PSO. This destroys the simplicity of the algorithm and leads to an undesirable computational overhead. In this study, we have proposed a variation of the algorithm called parallel swarms oriented PSO (PSO-PSO) which consists of a multi-stage and a single stage of evolution. The two interweaved stages of evolution demonstrate better performance on test functions, especially of higher dimensions. The attractive feature of PSO-PSO version of the algorithm is that it does not introduce any new parameters to improve its convergence performance. The PSO-PSO strategy maintains the simple and intuitive structure as well as the implemental and computational advantages of the basic PSO. Thus, the contribution of this study is the improvement of the performance of the basic PSO without increasing the complexity of the algorithm.