[MINIMUM,FVAL,EXITFLAG,OUTPUT] = fminmarkov(FUN,PARS,[OPTIONS],[CONSTRAINTS], ...) Markov Chain Monte Carlo optimizer This minimization method uses a Markov Chain Monte Carlo, optionally with constraints on function parameters. This is an implementation of the Goodman and Weare 2010 Affine invariant ensemble Markov Chain Monte Carlo (MCMC) sampler. MCMC sampling enables bayesian inference. The problem with many traditional MCMC samplers is that they can have slow convergence for badly scaled problems, and that it is difficult to optimize the random walk for high-dimensional problems. This is where the GW-algorithm really excels as it is affine invariant. It can achieve much better convergence on badly scaled problems. It is much simpler to get to work straight out of the box, and for that reason it truly deserves to be called the MCMC hammer. (This code uses a cascaded variant of the Goodman and Weare algorithm). Calling: fminmarkov(fun, pars) asks to minimize the 'fun' objective function with starting parameters 'pars' (vector) fminmarkov(fun, pars, options) same as above, with customized options (optimset) fminmarkov(fun, pars, options, fixed) is used to fix some of the parameters. The 'fixed' vector is then 0 for free parameters, and 1 otherwise. fminmarkov(fun, pars, options, lb, ub) is used to set the minimal and maximal parameter bounds, as vectors. fminmarkov(fun, pars, options, constraints) where constraints is a structure (see below). fminmarkov(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 fminmarkov(..., 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] = fminmarkov(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=fminmarkov('defaults') options.MinFunEvals sets the minimum number of function evaluations to reach options.StepSize: unit-less stepsize (default=2.5). options.ThinChain: Thin all the chains by only storing every N'th step (default=10) options.BurnIn: fraction of the chain that should be removed. (default=0) 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: Goodman & Weare (2010), Ensemble Samplers With Affine Invariance, Comm. App. Math. Comp. Sci., Vol. 5, No. 1, 65-80 Foreman-Mackey, Hogg, Lang, Goodman (2013), emcee: The MCMC Hammer, arXiv:1202.3665 Contrib: By: Aslak Grinsted 2015 https://github.com/grinsted/gwmcmc Version: Nov. 26, 2018 See also: fminsearch, optimset (c) E.Farhi, ILL. License: EUPL.