Hello there,

if I were you I would stick to Clerc's alternative constriction formulation, which basically gets rid of those parameters while still granting convergence and good exploration.

Check these two publications... first one should be enough, read second one for mathematical in-depth analysis.

Poli, R., Kennedy, J., & Blackwell, T. (2007). Particle swarm optimization. Swarm intelligence, 1(1), 33-57.

Clerc, M., & Kennedy, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. Evolutionary Computation, IEEE Transactions on, 6(1), 58-73.

Cheers,

R.T.

C1 = C2= 2

summation of C1 & C2 should be less than or equal to 4

inertia weight may be between 0.9 to 0.4 with a linearly decreasing rate

Check this work. A full-factorial analysis of the parameters influencing the swarm dynamincs, such as the coeffcient, the initializzation and the number of particles, has be done.

From this study, using a deterministic versione of PSO (no random coefficient), the most promising set up correspond to use: a number of particle equal to 4 times the number of variables of the problem, a intilization following an Hammersley sequence sampling with an intial velocity ditribution, and the coefficient proposed by Clerc in 2007

Conference Paper On the use of synchronous and asynchronous single-objective ...

In my opinion, this depends on the nature of the problem on which PSO is being applied. For details go through the review papers on PSO, and check the CEC, or equivalent conference (workshop) proceedings.

Dear Ramadan Zeineldin

Try the following:

c1=2.05

c2=2.05

w=0.729

For reference you can see:

M. N. Alam, B. Das, V. Pant, A comparative study of metaheuristic optimization approaches for directional overcurrent relays coordination, Electric Power Systems Research 128 (2015) 39-52.

Dear Ramadan Zeineldin

For basic PSO the value of parameters are considered as follow:

c1=2, c2=2, 

Inertia weight (W) may be linearly varied between 0.9 to 0.4. 

But for the different developed PSO variants,  the value of parameters is varied.

You can go through the following references:

[1].   Vinay Kumar Jadoun, Nikhil Gupta, K. R. Niazi, and Anil Swarnkar, "Dynamically Controlled Particle Swarm Optimization for Large Scale Non-convex Economic Dispatch Problems" the International Transactions on Electrical Energy Systems, John Wiley & Sons, Ltd., 2014.   

[2].   Vinay Kumar Jadoun, Nikhil Gupta, K. R. Niazi, and Anil Swarnkar, “Nonconvex Economic Dispatch Using Particle Swarm Optimization with Time Varying Operators,” the Advances in Electrical Engineering, Hindawi Publishing Corporation, vol. 2014, Article ID 301615, 13 pages, 2014. doi:10.1155/2014/301615.

Dear Ramadan Zeineldin,

Generally, it is well-known that there is no parameter configuration for metaheuristic algorihtms like PSO that performs better for any problem. Indeed, it should be noted that parameter tuning of the optimization algorithms itself is an optimization problem; as discussed in the literature. However, you can choose the more appropriate settings form the state-of-the-art studies. For example, please visit this link

http://www.sciencedirect.com/science/article/pii/S0096300309002860

You may need to vary the parameters in their ranges. For all types of problem fixed parameters might not lead to the global optimum solution.

It is a very interesting questions. Your question does not mention the problem domain or the instances used. With evolutionary algorithms, many research has been published on automatically adapting the parameters to different problems instances. Some researchs have demonstrated also that the parameters can vary during the search. They would be too many references to mention here.

More recently, this concept has been extended to any algorithms. Holger Hoos and his team have developed a framework called "Programming by optimisation". I would recommend you to read it. 

Automated computation can test and assess the effectiveness of many more parameters setting that us human can do. I would recommend you to look in this direction and perhaps you will find some interesting results. To my knowledge, PSO has yet to be used for such optimisation. I can be incorrect, so please others can correct my assumption.

Patricia 

PSO codes in MATLAB

https://www.researchgate.net/publication/296636431_Codes_in_MATLAB_for_Particle_Swarm_Optimization?ev=prf_pub

Code Codes in MATLAB for Particle Swarm Optimization

https://www.researchgate.net/publication/280244004_A_comparative_study_of_metaheuristic_optimization_approaches_for_directional_overcurrent_relays_coordination?ev=prf_pub

Article A comparative study of metaheuristic optimization approaches...

MATLAB programming codes for PSO

https://www.researchgate.net/publication/296636431_Codes_in_MATLAB_for_Particle_Swarm_Optimization?ev=prf_pub

Code Codes in MATLAB for Particle Swarm Optimization

https://www.researchgate.net/publication/297245624_Particle_Swarm_Optimization_Algorithm_and_its_Codes_in_MATLAB

Research Particle Swarm Optimization: Algorithm and its Codes in MATLAB

A practical video course on PSO and its implementation in MATLAB is available for free, via following URL:

http://udemy.com/pso-in-matlab

After this video course, you may find answer to your question. One of good configurations is Constriction Coefficients, which is described and implemented in the mentioned video tutorial.

how penalty factor decided for constraint type multiobjective problem. 

I have discussed in detail the choice of values in PSO in our survey of the state-of-the-art in PSO (as of April 2018) in sections 4.1 and 4.4. Read the entirety of section 4 for a detailed understanding.

We have made the article free for access on RG. Please find the link below and appropriately cite if you find the survey useful in your work.

Article Particle Swarm Optimization: A Survey of Historical and Rece...

Cite as:

Sengupta,S., Basak, S., Peters II, R.A., Particle Swarm Optimization: A survey of historical and recent developments with hybridization perspectives, arXiv:1804.05319 [cs.NE]

These parameters do not have a specific value and depend entirely on the behavior of the function and the constraint used.

Hi Ramadan Zeineldin ,

I recommend you read the paper:

D. Bratton and J. Kennedy, "Defining a Standard for Particle Swarm Optimization," 2007 IEEE Swarm Intelligence Symposium, Honolulu, HI, 2007, pp. 120-127. doi: 10.1109/SIS.2007.368035 .