Mohamed Azab, I am using inertial factor = 0.9, acc1 = 2 and acc = 1 (trying to priorize gbest). Would you have any suggestion for these values?

Dear Artur Nogueira de São José ,

In the equation: v_new = w*v_old + r1.c1.(g_best - x_old) + r2.c2.(p_best - x_old)

w should linear decrease from [0.9 -> 0.4], r1, r2 in range [0, 2] depend on your problem. You should try a larger population size or a larger number of iterations.

If it is still don't decrease your fitness value, you should check your fitness function.

Another hint is after several first epoch, fitness value will decrease but after that, they will converge near-optimal or optimal value.

Dear Nguyen Van Thieu thank you very much for your help.

I implemented the linear decrease of the inertial weight, as you suggested and also set r1=r2=1 as I am not sure if I should priorize gbest or pbest. The fitness function still presents a non-decreasing behavior (I am testing two different functions; one of them is almost constant over the epochs and the other seems random with some increases), but I will work on that.

With your last phrase, you mean the solution can be stuck near a local minima?

Dear Artur Nogueira de São José ,

As you already know, we are not sure the solution comes from meta-heuristics (GA, PSO, ...) is the global optimal or not. So, my last phrase means they are the convergence characteristic of meta-heuristic algorithms. I've tried several algorithms and their behavior do like that (You can see the image below).

So unless you know the global optimal value for your particular problem, you can realize that algorithm converge on global or local optima.

If you still stuck in your problem and you want to debug. I suggest you try your PSO code on simple function such as square function to see their converge.

If you familiar with python, check my github's repository. I implemented almost well-known meta-heuristics and experiment them on different functions.

https://github.com/thieunguyen5991/metaheuristics

Change the inertia factor. It should be in this interval [0.8, 1.2].

acc1 + acc2

Dear Nguyen Van Thieu ,

I got your idea now; in fact, I have no prior knowledge about the global optimum location, so it is difficult to distinguish a local from a global solution...

Fortunately, I found the problem on my code: the historical data regarding the gbest, pbest and fitness values along the epochs was being lost and replaced by zeros (a small problem I had with local variables). So all the further calculations (including velocities) have been affected.

I am not familiar with Phyton but I will take a look at your repository.

Thanks a lot for your help and best regards.

Dear Mohammed Bilal Danoune , Nesrine Amor and Mohamed Azab

I finally could get the typical decaying behavior for the fitness function.

After that, I ran my code based on all your suggestions but it seems that the solution is getting stuck around local minima (I will work on that).

Thanks a lot.

Dear Artur Nogueira de São José can you please input details about your fitness function and set of constraints, so that we can provide you the best possible solution.

Dear Asif Iqbal , after a careful revision on my code, I realized that there was something wrong on the elitism process. After a few changes, I could finally get the classical decreasing behavior of the fitness function. Thanks a lot!

This issue with heuristic algorithms are often and is called stagnation (non-improving solution).

Stagnation

—Non-improving solution —

—Sometimes, solutions not improves although you relaxed stopping criteria to allow to continue search betters solution

This is also a big issue with heuristic optimization techniques. This issue can be handled using the following steps:

—Properly selection of parameters can solve such issues

—Also, different population size can overcome such issues

Thank you for your suggestions, Mahamad Nabab Alam !

Change the inertia factor, or reduce the inertia factor below 0.5.

Regards

Thank you for your suggestion, Abdelghani Rouini