D	Mean distance of each particle to other particles
$F(·)$	Penalty function to be minimised or maximised
$H(·)$	Function stretching for multimodal function optimisation
$H_p(·)$	Penalty factor in a penalty function
$K(·)$	Constriction factor
$N_i$	Neighbourhood of the particle i
$R_1$	Cognitive uniformly distributed random vector used to compute the particle’s velocity
$R_2$	Social uniformly distributed random vector used to compute the particle’s velocity
S	Search space, defined by the domain of the function to be optimised,
	that contains all the feasible solutions for the problem
$\alpha$	Diagonal matrix whose diagonal values are within the range of $[0,1]$
$ϵ$	Absolute different between the last and the current best fitness value, or the algorithm accuracy
$\gamma (·)$	Power of a penalty function
$\widehat{y}$	Position of the best particle in the swarm or in the neighbourhood (target particle)
$\omega (·)$	Inertia weight parameter used to compute the velocity of each particle
$\rho$	Diagonal matrix that represents the architecture of the swarm
$\sigma$	Scale parameter of Cauchy mutation
$\tau$	Index of the global best particle in the swarm
$\theta (·)$	Multi-stage assignment function in a penalty function
${\phi }_1$	Cognitive real acceleration coefficient used to compute the particle’s velocity
${\phi }_2$	Social real acceleration coefficient used to compute the particle’s velocity
${\phi }_3$	Deviation real acceleration coefficient used to compute the particle’s velocity
${\overrightarrow{P}}_t^j$	Prior best position that maximises the FDR measure
$\overrightarrow{V}$	Particle’s velocity
${\overrightarrow{c}}^j$	Position of the centroid of the group j
$\overrightarrow{g}$	Global best position of a particle in the swarm
${\overrightarrow{p}}_t^i$	Personal best position of particle i
$\overrightarrow{x}$	Position vector of a solution found in the search space
${\overrightarrow{x}}_{max}$	Upper limit of the dimension d in the search space
${\overrightarrow{x}}_{min}$	Lower limit of the dimension d in the search space
$\overrightarrow{y^{*}}$	Set of feasible solutions that forms the Pareto front
d	Number of dimensions of the search space
$e_f$	Evolutionary factor used in the APSO
$f(·)$	Objective function to be minimised or maximised
g	Set of inequality function constraints
h	Set of equality function constraints
$h_p(·)$	Dynamic modified penalty value in a penalty function
l	Number of particles in the swarm or in the neighbourhood
m	Number of inequality constraints
n	Non-linear modulation index
p	Number of equality constraints
$q_i(·)$	Relative violated function of the constraints in a penalty function
s	Number of particles in the swarm
t	The number of the current iteration
$w_g$	Parameter, in the form of a diagonal matrix, to add variability to the best position in the swarm