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• Maurice Clerc, 2012, "Standard Particle Swarm Optimisation", hal-00764996.
• Maurice Clerc, 2006, "Particle Swarm Optimisation", ISTE Ltd.
• Martin Roberts, 2020,"How to generate uniformly random points on n-spheres and in n-balls", Extreme Learning.

• The code has been written and tested in Python 3.7.7.
• Particle Swarm Optimization (PSO) implementation for metaheuristic minimization.
• Variables can be real, integer, or mixed real/integer.
• Variables can be constrained to a specific interval or value setting the lower and the upper boundaries.
• Confidence coefficients depend on one single parameter.
• Search space can be normalized to improve convergency.
• An adaptive random topology is used to define each agent's neighbourhood (with an option to use the full swarm as neighbourhood).
• Unbiased velocity equation using hyperspherical uniform distribution (including the corner case where an agent is the neighbourhood best).
• Three velocity confinement methods (hyperbolic, random-back, and mixed hyperbolic/random-back).
• Possibility to specify the velocity limits.
• Possibility to specify an initial position for the agents.
• To improve the execution speed the algorithm has been designed without any loop on the agents.
• An arbitrary number of parameters can be passed (in a tuple) to the function to minimize.
• Option to run sequential tests with constant or random (uniformly distributed) number of agents.
• Usage:python test.py example.

example Name of the example to run (Parabola, Alpine, Tripod, and Ackley.)

func Function to minimize. The position of all agents is passed to the function at the same time.

LB,UB Lower and upper boundaries of the search space.

nPop,epochs Number of agents (population) and number of iterations.

K Average size of each agent's group of informants. IfK=0 the entire swarm is used as agent's group of informants.

phi Coefficient to calculate the self-confidence coefficient and the confidence-in-others coefficient.

vel_fact Velocity factor to calculate the maximum and the minimum allowed velocities.

conf_type Confinement type (on the velocities):HY= hyperbolic,RB= random-back,MX= mixed hyeperbolic/random-back.

IntVar List of indexes specifying which variable should be treated as integer. If all variables are real setIntVar=None, if all variables are integer setIntVar=all. Indexes are in the range(1,nVar).

normalize Specifies if the search space should be normalized (to improve convergency).

rad Normalized radius of the hypersphere centered on the best particle. The higher the number of other particles inside and the better is the solution.

args Tuple containing any parameter that needs to be passed to the function to minimize. If no parameters are passed setargs=None.

Xinit Initial position of each agent.

nVar Number of variables (dimensions of the search space).

nRun Number of runs for a specific case.

random_pop Specifies if the number of agents (for each run) is kept constant atnPop or randomly choosen from the uniform distributionnVar/2 <= nPop <= nVar/2.

Xsol Solution to the minimization point. Set toNone if not known.

There are four examples: Parabola, Alpine, Tripod, and Ackley (seetest.py for the specific equations and parameters). As illustration, a 3D plot of these function is shownhere.

• Parabola,Alpine, andAckley can have an arbitrary number of dimensions, whileTripod has only two dimensions.

• Parabola,Tripod, andAckley are examples where parameters (respectively, arrayX0, scalarskx andky, and arrayX0) are passed usingargs.

• The global minimum forParabola andAckley is atX0; the global minimum forAlpine is at zero; the global minimum forTripod is at[0,-ky] with local minimum at[-kx,+ky] and[+kx,+ky].