Editing Particle swarm optimization (section)

===Binary, discrete, and combinatorial===
As the PSO equations given above work on real numbers, a commonly used method to solve discrete problems is to map the discrete search space to a continuous domain, to apply a classical PSO, and then to demap the result.  Such a mapping can be very simple (for example by just using rounded values) or more sophisticated.<ref>Roy, R., Dehuri, S., & Cho, S. B. (2012). [http://sclab.yonsei.ac.kr/publications/Papers/IJ/A%20Novel%20Particle%20Swarm%20Optimization%20Algorithm%20for%20Multi-Objective%20Combinatorial%20Optimization%20Problem.pdf A Novel Particle Swarm Optimization Algorithm for Multi-Objective Combinatorial Optimization Problem]. 'International Journal of Applied Metaheuristic Computing (IJAMC)', 2(4), 41-57</ref>

However, it can be noted that the equations of movement make use of operators that perform four actions:
*computing the difference of two positions. The result is a velocity (more precisely a displacement)
*multiplying a velocity by a numerical coefficient
*adding two velocities
*applying a velocity to a position

Usually a position and a velocity are represented by ''n'' real numbers, and these operators are simply -, *, +, and again +. But all these mathematical objects can be defined in a completely different way, in order to cope with binary problems (or more generally discrete ones), or even combinatorial ones.<ref>Kennedy, J. & Eberhart, R. C. (1997). [http://ahmetcevahircinar.com.tr/wp-content/uploads/2017/02/A_discrete_binary_version_of_the_particle_swarm_algorithm.pdf A discrete binary version of the particle swarm algorithm], Conference on Systems, Man, and Cybernetics, Piscataway, NJ: IEEE Service Center, pp. 4104-4109</ref><ref>Clerc, M. (2004). [https://link.springer.com/chapter/10.1007/978-3-540-39930-8_8 Discrete Particle Swarm Optimization, illustrated by the Traveling Salesman Problem], New Optimization Techniques in Engineering, Springer, pp. 219-239</ref><ref>Clerc, M. (2005). Binary Particle Swarm Optimisers: toolbox, derivations, and mathematical insights, [http://hal.archives-ouvertes.fr/hal-00122809/en/ Open Archive HAL]</ref><ref>{{cite journal|doi=10.1016/j.amc.2007.04.096|title=A combinatorial particle swarm optimization for solving multi-mode resource-constrained project scheduling problems|journal=Applied Mathematics and Computation|volume=195|pages=299–308|year=2008|last1=Jarboui|first1=B.|last2=Damak|first2=N.|last3=Siarry|first3=P.|last4=Rebai|first4=A.}}</ref> One approach is to redefine the operators based on sets.<ref name=Chen10SPSO />