[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 firstname.lastname@example.org(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: email@example.com [cmaes] Version: Nov. 26, 2018 See also: fminsearch, optimset (c) E.Farhi, ILL. License: EUPL.