Thanks for raising this useful question. However, comparison of algorithms is complicated for the following reasons.

(1) Benchmark test problems and/or application to be used

(2) Algorithm of each method to be used; there are many modifications in each of ACO, ABC, PSO, BFO, FA, SSO,etc  

(3) Values of algorithm parameters to be used (e.g., same for all test problems or tuned for each test problem)

(4) Method for handling of constraints if applicable 

(5) Comparison criteria

(6) When to compare (i.e., at what stage of the search)

Hence, think carefully the purpose of your comparative study, review papers which compared several optimization algorithms and then plan your study. All the best for your study.

Objective function/ Fitness function is totally depends on your problem and results that you want to measure from this comparisons.....You may follow my paper below where I have introduced a new canonical pso algorithm which gives better results than some existing algorithms.....The entire result analysis we have done using MATLAB...

https://www.hindawi.com/journals/isrn/2014/414013/

you may also  check the below links.....

https://pdfs.semanticscholar.org/b2e2/dba23716d8223fa56171334abb95ac6014ad.pdf

https://www.researchgate.net/post/How_do_you_define_a_fitness_function_in_a_Genetic_Algorithm

http://www.tmt.unze.ba/zbornik/TMT2010/030-TMT10-034.pdf

http://research.ijcaonline.org/volume119/number10/pxc3903831.pdf

I think that it is very important to also pay attention on the design of the algorithms. Sometimes, special design of operators in the algorithm may affect the quality of the algorithm, for example the chromosome design used in the paper I attach below.

My suggestion is you select algorithms, which are compared when they are used to solve a certain problem. The selected algorithms should be reported to perform well by any other papers, in order to obtain algorithms with good designs.

Thanks for raising this useful question. However, comparison of algorithms is complicated for the following reasons.

(1) Benchmark test problems and/or application to be used

(2) Algorithm of each method to be used; there are many modifications in each of ACO, ABC, PSO, BFO, FA, SSO,etc  

(3) Values of algorithm parameters to be used (e.g., same for all test problems or tuned for each test problem)

(4) Method for handling of constraints if applicable 

(5) Comparison criteria

(6) When to compare (i.e., at what stage of the search)

Hence, think carefully the purpose of your comparative study, review papers which compared several optimization algorithms and then plan your study. All the best for your study.

I suggest using performance profiles, as those proposed in the paper Benchmarking optimization software with performance profiles by Dolan and Moré, 2002. By using them you can compare time to solution and the quality of the solutions.

It may be worth emphasising that those three kinds of algorithms are intended to solve three different kinds of problems.  Gradient and Newton methods find local optima of continuous problems for which all functions are easily differentiable.  Derivative free methods are intended to find local minima for continuous differentiable problems. Derivative free methods are for continuous nondifferentiable

Thanks All for useful answers.

My application is data mining-extraction of patterns from datasets. Can I use the various functions like benchmark to evaluate and then use for data mining based on these benchmark results.

Dear Lokesh Gagnani

I think the section 6.4 (evaluation and comparison of the algorithms) of the following paper has valuable information for you. In this paper 5 comparison criteria are used.

http://www.sciencedirect.com/science/article/pii/S2212420916300073

There is a good work in which five different algorithms have been compared. I would like you to have a look on that as it may be very helpful in this regard.

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...

major differences between ACO and PSO

Communication Mechanism:

ACO uses an indirect communication mechanism among ants, called stigmergy, which means interaction through the environment.

The communication among particles in PSO is rather direct without altering the environment.

Problem Types

ACO was originally used to solve

combinatorial (discrete) optimization problems, but it was later modified to adapt continuous problems.

PSO was originally used to solve continuous problems, but it was later modified to adapt binary/discrete optimization problems.

Problem Representation

ACO’s solution space is typically

represented as a weighted graph, called construction graph.

PSO’s solution space is typically represented as a set of n-dimensional points.

Algorithm Applicability

ACO is commonly more applicable to problems where source and destination are predefined and specific.

PSO is commonly more applicable to problems where previous and next particle positions at each point are clear and uniquely defined.

Algorithm Objective

ACO’s objective is generally searching for an optimal path in the construction graph.

PSO’s objective is generally finding the location of an optimal point in a Cartesian coordinate system.

Pertaining to this What is the difference in the optimized value as 1.6599e-020 and 7.0254e-281?Which value can be considered as optimized as I received these value by using sphere benchmark function in FA and PSO respectively.

The smaller one, meaning that the 7 * 10^(-281) is smaller than 1^(-20)...

Thank you for raising this query on optimized values of objective function. In addition to the value of objective function, one should consider optimal values of decision variables. It is possible that differences in these are negligible for alternate solutions obtained. This indicates that the alternate solutions are essentially same.