[MINIMUM,FVAL,EXITFLAG,OUTPUT] = fminpso(FUN,PARS,[OPTIONS],[CONSTRAINTS], ...) particle swarm optimization
fminpso finds a minimum of a function of several variables using the particle swarm
optimization (PSO) algorithm originally introduced in 1995 by Kennedy and
Eberhart. This algorithm was extended by Shi and Eberhart in 1998 through the
introduction of inertia factors to dampen the velocities of the particles.
In 2002, Clerc and Kennedy introduced a constriction factor in PSO, which was
later on shown to be superior to the inertia factors. Therefore, the algorithm
using a constriction factor was implemented here.
The objective function has syntax: criteria = objective(p)
Calling:
fminpso(fun, pars) asks to minimize the 'fun' objective function with starting
parameters 'pars' (vector)
fminpso(fun, pars, options) same as above, with customized options (optimset)
fminpso(fun, pars, options, fixed)
is used to fix some of the parameters. The 'fixed' vector is then 0 for
free parameters, and 1 otherwise.
fminpso(fun, pars, options, lb, ub)
is used to set the minimal and maximal parameter bounds, as vectors.
fminpso(fun, pars, options, constraints)
where constraints is a structure (see below).
fminpso(problem) where problem is a structure with fields
problem.objective: function to minimize
problem.x0: starting parameter values
problem.options: optimizer options (see below)
problem.constraints: optimization constraints
fminpso(..., args, ...)
sends additional arguments to the objective function
criteria = FUN(pars, args, ...)
Example:
banana = @(x)100*(x(2)-x(1)^2)^2+(1-x(1))^2;
[x,fval] = fminpso(banana,[-1.2, 1])
Input:
FUN is the function to minimize (handle or string): criteria = FUN(PARS)
It needs to return a single value or vector.
PARS is a vector with initial guess parameters. You must input an
initial guess. PARS can also be given as a single-level structure.
OPTIONS is a structure with settings for the optimizer,
compliant with optimset. Default options may be obtained with
o=fminpso('defaults')
option.PopulationSize sets the number of particles in the swarm (20-40).
option.SwarmC1 sets the local attractors strength (1-3)
option.SwarmC2 sets the global attractor strength (1-3).
options.MinFunEvals sets the minimum number of function evaluations to reach
An empty OPTIONS sets the default configuration.
CONSTRAINTS may be specified as a structure
constraints.min= vector of minimal values for parameters
constraints.max= vector of maximal values for parameters
constraints.fixed= vector having 0 where parameters are free, 1 otherwise
constraints.step= vector of maximal parameter changes per iteration
constraints.eval= expression making use of 'p', 'constraints', and 'options'
and returning modified 'p'
or function handle p=@constraints.eval(p)
An empty CONSTRAINTS sets no constraints.
Additional arguments are sent to the objective function.
Output:
MINIMUM is the solution which generated the smallest encountered
value when input into FUN.
FVAL is the value of the FUN function evaluated at MINIMUM.
EXITFLAG return state of the optimizer
OUTPUT additional information returned as a structure.
Reference:
Kennedy J., Eberhart R.C. (1995): Particle swarm optimization. In: Proc.
IEEE Conf. on Neural Networks, IV, Piscataway, NJ, pp. 1942-1948
Contrib: 2006 Brecht Donckels, BIOMATH, brecht.donckels@ugent.be [PSO]
(c) E.Farhi, ILL. License: EUPL.