Particle Swarm Optimization

patricle_swarm(M, F)

perform the particle swarm optimization algorithm (PSO), starting with the initial particle positions $x_0$^{[Borckmans2010]}. The aim of PSO is to find the particle position $g$ on the Manifold M that solves

To this end, a swarm of particles is moved around the Manifold M in the following manner. For every particle $k$ we compute the new particle velocities $v_k^{(i)}$ in every step $i$ of the algorithm by

where $x_k^{(i)}$ is the current particle position, $\omega$ denotes the inertia, $c$ and $s$ are a cognitive and a social weight, respectively, $r_j$, $~j=1,2$ are random factors which are computed new for each particle and step, $\operatorname{retr}^{-1}$ denotes an inverse retraction on the Manifold M, and $\operatorname{T}$ is a vector transport.

Then the position of the particle is updated as

where $\operatorname{retr}$ denotes a retraction on the Manifold M. At the end of each step for every particle, we set

and

i.e. $p_k^{(i)}$ is the best known position for the particle $k$ and $g^{(i)}$ is the global best known position ever visited up to step $i$.

Input

• M – a manifold $\mathcal M$
• F – a cost function $F\colon\mathcal M\to\mathbb R$ to minimize

Optional

• n - (100) number of random initial positions of x0
• x0 – the initial positions of each particle in the swarm $x_k^{(0)} ∈ \mathcal M$ for $k = 1, \dots, n$, per default these are n random_points
• velocity – a set of tangent vectors (of type AbstractVector{T}) representing the velocities of the particles, per default a random_tangent per inital position
• inertia – (0.65) the inertia of the patricles
• social_weight – (1.4) a social weight factor
• cognitive_weight – (1.4) a cognitive weight factor
• retraction_method – (ExponentialRetraction()) a retraction(M,x,ξ) to use.
• inverse_retraction_method - (LogarithmicInverseRetraction()) an inverse_retraction(M,x,y) to use.
• vector_transport_mthod - (ParallelTransport()) a vector transport method to use.
• stopping_criterion – (StopWhenAny(StopAfterIteration(500), StopWhenChangeLess(10^{-4}))) a functor inheriting from StoppingCriterion indicating when to stop.
• return_options – (false) – if activated, the extended result, i.e. the complete Options are returned. This can be used to access recorded values. If set to false (default) just the optimal value x_opt if returned

... and the ones that are passed to decorate_options for decorators.

Output

• g – the resulting point of PSO

OR

• options - the options returned by the solver (see return_options)

patricle_swarm!(M, F; n=100, x0::AbstractVector=[random_point(M) for i in 1:n], kwargs...)

perform the particle swarm optimization algorithm (PSO), starting with the initial particle positions $x_0$^{[Borckmans2010]} in place of x0.

Input

• M – a manifold $\mathcal M$
• F – a cost function $F\colon\mathcal M\to\mathbb R$ to minimize

Optional

• n - (100) number of random initial positions of x0
• x0 – the initial positions of each particle in the swarm $x_k^{(0)} ∈ \mathcal M$ for $k = 1, \dots, n$, per default these are n random_points

for more optional arguments, see particle_swarm.

ParticleSwarmOptions{P,T} <: Options

Describes a particle swarm optimizing algorithm, with

Fields

a default value is given in brackets if a parameter can be left out in initialization.

• x0 – a set of points (of type AbstractVector{P}) on a manifold as initial particle positions
• velocity – a set of tangent vectors (of type AbstractVector{T}) representing the velocities of the particles
• inertia – (0.65) the inertia of the patricles
• social_weight – (1.4) a social weight factor
• cognitive_weight – (1.4) a cognitive weight factor
• stopping_criterion – (StopWhenAny(StopAfterIteration(500), StopWhenChangeLess(10^{-4}))) a functor inheriting from StoppingCriterion indicating when to stop.
• retraction_method – (ExponentialRetraction) the rectraction to use
• inverse_retraction_method - (LogarithmicInverseRetraction) an inverse retraction to use.

Constructor

ParticleSwarmOptions(x0, velocity, inertia, social_weight, cognitive_weight, stopping_criterion[, retraction_method=ExponentialRetraction(), inverse_retraction_method=LogarithmicInverseRetraction()])

construct a particle swarm Option with the fields and defaults as above.

See also

particle_swarm