[MINIMUM,FVAL,EXITFLAG,OUTPUT] = FMINCMAES(FUN,PARS,[OPTIONS],[CONSTRAINTS], ...) Evolution Strategy with Covariance Matrix Adaption
CMAES implements an Evolution Strategy with Covariance Matrix
Adaptation (CMA-ES) for nonlinear function minimization.
The CMA-ES (Evolution Strategy with Covariance
Matrix Adaptation) is a robust search method which should be
applied, if derivative based methods, e.g. quasi-Newton BFGS or
conjucate gradient, (supposably) fail due to a rugged search
landscape (e.g. noise, local optima, outlier, etc.). On smooth
landscapes CMA-ES is roughly ten times slower than BFGS. For up to
N=10 variables even the simplex direct search method (Nelder & Mead)
is often faster, but far less robust than CMA-ES.
Calling:
fmincmaes(fun, pars) asks to minimize the 'fun' objective function with starting
parameters 'pars' (vector)
fmincmaes(fun, pars, options) same as above, with customized options (optimset)
fmincmaes(fun, pars, options, fixed)
is used to fix some of the parameters. The 'fixed' vector is then 0 for
free parameters, and 1 otherwise.
fmincmaes(fun, pars, options, lb, ub)
is used to set the minimal and maximal parameter bounds, as vectors.
fmincmaes(fun, pars, options, constraints)
where constraints is a structure (see below).
fmincmaes(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
fmincmaes(..., 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] = fmincmaes(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=fmincmaes('defaults');
options.PopulationSize sets the population size (20-40).
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.
References
Hansen, N. and S. Kern (2004). Evaluating the CMA Evolution Strategy on
Multimodal Test Functions. Eighth International Conference on Parallel
Problem Solving from Nature PPSN VIII, Proceedings, pp. 282-291, Springer.
Hansen, N. and A. Ostermeier (2001). Completely Derandomized Self-Adaptation
in Evolution Strategies. Evolutionary Computation, 9(2), pp. 159-195.
Hansen, N., S.D. Mueller and P. Koumoutsakos (2003). Reducing the Time
Complexity of the Derandomized Evolution Strategy with Covariance Matrix
Adaptation (CMA-ES). Evolutionary Computation, 11(1).
Contrib:
Nikolaus Hansen, 2001-2007. e-mail: hansen@bionik.tu-berlin.de [cmaes]
Version: Nov. 26, 2018
See also: fminsearch, optimset
(c) E.Farhi, ILL. License: EUPL.