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cmaes.m
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function [xmin, ... % minimum search point of last iteration
fmin, ... % function value of xmin
counteval, ... % number of function evaluations done
stopflag, ... % stop criterion reached
out, ... % struct with various histories and solutions
bestever ... % struct containing overall best solution (for convenience)
] = cmaes( ...
fitfun, ... % name of objective/fitness function
xstart, ... % objective variables initial point, determines N
insigma, ... % initial coordinate wise standard deviation(s)
inopts, ... % options struct, see defopts below
varargin ... % arguments passed to objective function
)
% cmaes.m, Version 3.61.beta, last change: April, 2012
% CMAES implements an Evolution Strategy with Covariance Matrix
% Adaptation (CMA-ES) for nonlinear function minimization. For
% introductory comments and copyright (GPL) see end of file (type
% 'type cmaes'). cmaes.m runs with MATLAB (Windows, Linux) and,
% without data logging and plotting, it should run under Octave
% (Linux, package octave-forge is needed).
%
% OPTS = CMAES returns default options.
% OPTS = CMAES('defaults') returns default options quietly.
% OPTS = CMAES('displayoptions') displays options.
% OPTS = CMAES('defaults', OPTS) supplements options OPTS with default
% options.
%
% XMIN = CMAES(FUN, X0, SIGMA[, OPTS]) locates an approximate minimum
% XMIN of function FUN starting from column vector X0 with the initial
% coordinate wise search standard deviation SIGMA.
%
% Input arguments:
%
% FUN is a string function name like 'frosen'. FUN takes as argument
% a column vector of size of X0 and returns a scalar. An easy way to
% implement a hard non-linear constraint is to return NaN. Then,
% this function evaluation is not counted and a newly sampled
% point is tried immediately.
%
% X0 is a column vector, or a matrix, or a string. If X0 is a matrix,
% mean(X0, 2) is taken as initial point. If X0 is a string like
% '2*rand(10,1)-1', the string is evaluated first.
%
% SIGMA is a scalar, or a column vector of size(X0,1), or a string
% that can be evaluated into one of these. SIGMA determines the
% initial coordinate wise standard deviations for the search.
% Setting SIGMA one third of the initial search region is
% appropriate, e.g., the initial point in [0, 6]^10 and SIGMA=2
% means cmaes('myfun', 3*rand(10,1), 2). If SIGMA is missing and
% size(X0,2) > 1, SIGMA is set to sqrt(var(X0')'). That is, X0 is
% used as a sample for estimating initial mean and variance of the
% search distribution.
%
% OPTS (an optional argument) is a struct holding additional input
% options. Valid field names and a short documentation can be
% discovered by looking at the default options (type 'cmaes'
% without arguments, see above). Empty or missing fields in OPTS
% invoke the default value, i.e. OPTS needs not to have all valid
% field names. Capitalization does not matter and unambiguous
% abbreviations can be used for the field names. If a string is
% given where a numerical value is needed, the string is evaluated
% by eval, where 'N' expands to the problem dimension
% (==size(X0,1)) and 'popsize' to the population size.
%
% [XMIN, FMIN, COUNTEVAL, STOPFLAG, OUT, BESTEVER] = ...
% CMAES(FITFUN, X0, SIGMA)
% returns the best (minimal) point XMIN (found in the last
% generation); function value FMIN of XMIN; the number of needed
% function evaluations COUNTEVAL; a STOPFLAG value as cell array,
% where possible entries are 'fitness', 'tolx', 'tolupx', 'tolfun',
% 'maxfunevals', 'maxiter', 'stoptoresume', 'manual',
% 'warnconditioncov', 'warnnoeffectcoord', 'warnnoeffectaxis',
% 'warnequalfunvals', 'warnequalfunvalhist', 'bug' (use
% e.g. any(strcmp(STOPFLAG, 'tolx')) or findstr(strcat(STOPFLAG,
% 'tolx')) for further processing); a record struct OUT with some
% more output, where the struct SOLUTIONS.BESTEVER contains the overall
% best evaluated point X with function value F evaluated at evaluation
% count EVALS. The last output argument BESTEVER equals
% OUT.SOLUTIONS.BESTEVER. Moreover a history of solutions and
% parameters is written to files according to the Log-options.
%
% A regular manual stop can be achieved via the file signals.par. The
% program is terminated if the first two non-white sequences in any
% line of this file are 'stop' and the value of the LogFilenamePrefix
% option (by default 'outcmaes'). Also a run can be skipped.
% Given, for example, 'skip outcmaes run 2', skips the second run
% if option Restarts is at least 2, and another run will be started.
%
% To run the code completely silently set Disp, Save, and Log options
% to 0. With OPTS.LogModulo > 0 (1 by default) the most important
% data are written to ASCII files permitting to investigate the
% results (e.g. plot with function plotcmaesdat) even while CMAES is
% still running (which can be quite useful on expensive objective
% functions). When OPTS.SaveVariables==1 (default) everything is saved
% in file OPTS.SaveFilename (default 'variablescmaes.mat') allowing to
% resume the search afterwards by using the resume option.
%
% To find the best ever evaluated point load the variables typing
% "es=load('variablescmaes')" and investigate the variable
% es.out.solutions.bestever.
%
% In case of a noisy objective function (uncertainties) set
% OPTS.Noise.on = 1. This option interferes presumably with some
% termination criteria, because the step-size sigma will presumably
% not converge to zero anymore. If CMAES was provided with a
% fifth argument (P1 in the below example, which is passed to the
% objective function FUN), this argument is multiplied with the
% factor given in option Noise.alphaevals, each time the detected
% noise exceeds a threshold. This argument can be used within
% FUN, for example, as averaging number to reduce the noise level.
%
% OPTS.DiagonalOnly > 1 defines the number of initial iterations,
% where the covariance matrix remains diagonal and the algorithm has
% internally linear time complexity. OPTS.DiagonalOnly = 1 means
% keeping the covariance matrix always diagonal and this setting
% also exhibits linear space complexity. This can be particularly
% useful for dimension > 100. The default is OPTS.DiagonalOnly = 0.
%
% OPTS.CMA.active = 1 turns on "active CMA" with a negative update
% of the covariance matrix and checks for positive definiteness.
% OPTS.CMA.active = 2 does not check for pos. def. and is numerically
% faster. Active CMA usually speeds up the adaptation and might
% become a default in near future.
%
% The primary strategy parameter to play with is OPTS.PopSize, which
% can be increased from its default value. Increasing the population
% size (by default linked to increasing parent number OPTS.ParentNumber)
% improves global search properties in exchange to speed. Speed
% decreases, as a rule, at most linearely with increasing population
% size. It is advisable to begin with the default small population
% size. The options Restarts and IncPopSize can be used for an
% automated multistart where the population size is increased by the
% factor IncPopSize (two by default) before each restart. X0 (given as
% string) is reevaluated for each restart. Stopping options
% StopFunEvals, StopIter, MaxFunEvals, and Fitness terminate the
% program, all others including MaxIter invoke another restart, where
% the iteration counter is reset to zero.
%
% Examples:
%
% XMIN = cmaes('myfun', 5*ones(10,1), 1.5);
%
% starts the search at 10D-point 5 and initially searches mainly
% between 5-3 and 5+3 (+- two standard deviations), but this is not
% a strict bound. 'myfun' is a name of a function that returns a
% scalar from a 10D column vector.
%
% opts.LBounds = 0; opts.UBounds = 10;
% opts.Restarts = 3; % doubles the popsize for each restart
% X = cmaes('myfun', 10*rand(10,1), 5, opts);
%
% searches within lower bound of 0 and upper bound of 10. Bounds can
% also be given as column vectors. If the optimum is not located
% on the boundary, use rather a penalty approach to handle bounds.
%
% opts=cmaes;
% opts.StopFitness=1e-10;
% X=cmaes('myfun', 'rand(5,1)', 0.5, opts);
%
% stops the search, if the function value is smaller than 1e-10. The second
% argument is re-evaluated for each restart.
%
% [X, F, E, STOP, OUT] = cmaes('myfun2', 'rand(5,1)', 1, [], P1, P2);
%
% passes two additional parameters to the function MYFUN2.
%
% See also FMINSEARCH, FMINUNC, FMINBND.
% ---------------------------------------------------------------
% CMA-ES: Evolution Strategy with Covariance Matrix Adaptation for
% nonlinear function minimization. To be used under the terms of
% either
% the GNU General Public License (http://www.gnu.org/copyleft/gpl.html).
% or
% the BSD 3-Clause License (see below).
%
% Author (copyright): Nikolaus Hansen, 2001-2012.
% e-mail: nikolaus.hansen AT inria.fr
%
% References: See below.
% ---------------------------------------------------------------
%
% GENERAL PURPOSE: 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. To see the
% advantage of the CMA, it will usually take at least 30*N and up to
% 300*N function evaluations, where N is the search problem dimension.
% On considerably hard problems the complete search (a single run) is
% expected to take at least 30*N^2 and up to 300*N^2 function
% evaluations.
%
% SOME MORE COMMENTS:
% The adaptation of the covariance matrix (e.g. by the CMA) is
% equivalent to a general linear transformation of the problem
% coding. Nevertheless every problem specific knowlegde about the best
% linear transformation should be exploited before starting the
% search. That is, an appropriate a priori transformation should be
% applied to the problem. This also makes the identity matrix as
% initial covariance matrix the best choice.
%
% The strategy parameter lambda (population size, opts.PopSize) is the
% preferred strategy parameter to play with. If results with the
% default strategy are not satisfactory, increase the population
% size. (Remark that the crucial parameter mu (opts.ParentNumber) is
% increased proportionally to lambda). This will improve the
% strategies capability of handling noise and local minima. We
% recomment successively increasing lambda by a factor of about three,
% starting with initial values between 5 and 20. Casually, population
% sizes even beyond 1000+100*N can be sensible.
%
% TODO:
% write dispcmaesdat for Matlab (and Octave)
% control savemodulo and plotmodulo via signals.par
cmaVersion = '3.62.beta';
% ----------- Set Defaults for Input Parameters and Options -------------
% These defaults may be edited for convenience
% Input Defaults (obsolete, these are obligatory now)
definput.fitfun = 'felli'; % frosen; fcigar; see end of file for more
definput.xstart = rand(10,1); % 0.50*ones(10,1);
definput.sigma = 0.3;
% Options defaults: Stopping criteria % (value of stop flag)
defopts.StopFitness = '-Inf % stop if f(xmin) < stopfitness, minimization';
defopts.MaxFunEvals = 'Inf % maximal number of fevals';
defopts.MaxIter = '1e3*(N+5)^2/sqrt(popsize) % maximal number of iterations';
defopts.StopFunEvals = 'Inf % stop after resp. evaluation, possibly resume later';
defopts.StopIter = 'Inf % stop after resp. iteration, possibly resume later';
defopts.TolX = '1e-11*max(insigma) % stop if x-change smaller TolX';
defopts.TolUpX = '1e3*max(insigma) % stop if x-changes larger TolUpX';
defopts.TolFun = '1e-12 % stop if fun-changes smaller TolFun';
defopts.TolHistFun = '1e-13 % stop if back fun-changes smaller TolHistFun';
defopts.StopOnStagnation = 'on % stop when fitness stagnates for a long time';
% TODO: stagnation has four parameters for the period: min = 120, const = 30N/lam, rel = 0.2, max = 2e5
% defopts.StopOnStagnation = '[120 30*N/popsize 0.2 2e5] % [min const rel_iter max] measuring period';
defopts.StopOnWarnings = 'yes % ''no''==''off''==0, ''on''==''yes''==1 ';
defopts.StopOnEqualFunctionValues = '2 + N/3 % number of iterations';
% Options defaults: Other
defopts.DiffMaxChange = 'Inf % maximal variable change(s), can be Nx1-vector';
defopts.DiffMinChange = '0 % minimal variable change(s), can be Nx1-vector';
defopts.WarnOnEqualFunctionValues = ...
'yes % ''no''==''off''==0, ''on''==''yes''==1 ';
defopts.LBounds = '-Inf % lower bounds, scalar or Nx1-vector';
defopts.UBounds = 'Inf % upper bounds, scalar or Nx1-vector';
defopts.EvalParallel = 'no % objective function FUN accepts NxM matrix, with M>1?';
defopts.EvalInitialX = 'yes % evaluation of initial solution';
defopts.Restarts = '0 % number of restarts ';
defopts.IncPopSize = '2 % multiplier for population size before each restart';
defopts.PopSize = '(4 + floor(3*log(N))) % population size, lambda';
defopts.ParentNumber = 'floor(popsize/2) % AKA mu, popsize equals lambda';
defopts.RecombinationWeights = 'superlinear decrease % or linear, or equal';
defopts.DiagonalOnly = '0*(1+100*N/sqrt(popsize))+(N>=1000) % C is diagonal for given iterations, 1==always';
defopts.Noise.on = '0 % uncertainty handling is off by default';
defopts.Noise.reevals = '1*ceil(0.05*lambda) % nb. of re-evaluated for uncertainty measurement';
defopts.Noise.theta = '0.5 % threshold to invoke uncertainty treatment'; % smaller: more likely to diverge
defopts.Noise.cum = '0.3 % cumulation constant for uncertainty';
defopts.Noise.cutoff = '2*lambda/3 % rank change cutoff for summation';
defopts.Noise.alphasigma = '1+2/(N+10) % factor for increasing sigma'; % smaller: slower adaptation
defopts.Noise.epsilon = '1e-7 % additional relative perturbation before reevaluation';
defopts.Noise.minmaxevals = '[1 inf] % min and max value of 2nd arg to fitfun, start value is 5th arg to cmaes';
defopts.Noise.alphaevals = '1+2/(N+10) % factor for increasing 2nd arg to fitfun';
defopts.Noise.callback = '[] % callback function when uncertainty threshold is exceeded';
% defopts.TPA = 0;
defopts.CMA.cs = '(mueff+2)/(N+mueff+3) % cumulation constant for step-size';
%qqq defopts.CMA.cs = (mueff^0.5)/(N^0.5+mueff^0.5) % the short time horizon version
defopts.CMA.damps = '1 + 2*max(0,sqrt((mueff-1)/(N+1))-1) + cs % damping for step-size';
% defopts.CMA.ccum = '4/(N+4) % cumulation constant for covariance matrix';
defopts.CMA.ccum = '(4 + mueff/N) / (N+4 + 2*mueff/N) % cumulation constant for pc';
defopts.CMA.ccov1 = '2 / ((N+1.3)^2+mueff) % learning rate for rank-one update';
defopts.CMA.ccovmu = '2 * (mueff-2+1/mueff) / ((N+2)^2+mueff) % learning rate for rank-mu update';
defopts.CMA.on = 'yes';
defopts.CMA.active = '0 % active CMA, 1: neg. updates with pos. def. check, 2: neg. updates';
flg_future_setting = 0; % testing for possible future variant(s)
if flg_future_setting
disp('in the future')
% damps setting from Brockhoff et al 2010
% this damps diverges with popsize 400:
% defopts.CMA.damps = '2*mueff/lambda + 0.3 + cs % damping for step-size';
% cmaeshtml('benchmarkszero', ones(20,1)*2, 5, o, 15);
% how about:
% defopts.CMA.damps = '2*mueff/lambda + 0.3 + 2*max(0,sqrt((mueff-1)/(N+1))-1) + cs % damping for step-size';
defopts.CMA.damps = '0.5 + 0.5*min(1, (0.27*lambda/mueff-1)^2) + 2*max(0,sqrt((mueff-1)/(N+1))-1) + cs % damping for step-size';
if 11 < 3
defopts.CMA.damps = '0.5 + 0.5*min(1,(lam_mirr/(0.159*lambda)-1)^2) + 2*max(0,sqrt((mueff-1)/(N+1))-1) + cs % damping for step-size';
defopts.mirrored_offspring = 'floor(0.5 + 0.159 * lambda)';
% TODO: this should also depend on diagonal option!?
defopts.CMA.active = 'floor(int8(lam_mirr>0)) % active CMA 1: neg. updates with pos. def. check, 2: neg. updates';
end
% ccum adjusted for large mueff, better on schefelmult?
% TODO: this should also depend on diagonal option!?
defopts.CMA.ccum = '(4 + mueff/N) / (N+4 + 2*mueff/N) % cumulation constant for pc';
defopts.CMA.active = '1 % active CMA 1: neg. updates with pos. def. check, 2: neg. updates';
end
defopts.Resume = 'no % resume former run from SaveFile';
defopts.Science = 'on % off==do some additional (minor) problem capturing, NOT IN USE';
defopts.ReadSignals = 'on % from file signals.par for termination, yet a stumb';
defopts.Seed = 'sum(100*clock) % evaluated if it is a string';
defopts.DispFinal = 'on % display messages like initial and final message';
defopts.DispModulo = '100 % [0:Inf], disp messages after every i-th iteration';
defopts.SaveVariables = 'on % [on|final|off][-v6] save variables to .mat file';
defopts.SaveFilename = 'variablescmaes.mat % save all variables, see SaveVariables';
defopts.LogModulo = '1 % [0:Inf] if >1 record data less frequently after gen=100';
defopts.LogTime = '25 % [0:100] max. percentage of time for recording data';
defopts.LogFilenamePrefix = 'outcmaes % files for output data';
defopts.LogPlot = 'off % plot while running using output data files';
%qqqkkk
%defopts.varopt1 = ''; % 'for temporary and hacking purposes';
%defopts.varopt2 = ''; % 'for temporary and hacking purposes';
defopts.UserData = 'for saving data/comments associated with the run';
defopts.UserDat2 = ''; 'for saving data/comments associated with the run';
% ---------------------- Handling Input Parameters ----------------------
if nargin < 1 || isequal(fitfun, 'defaults') % pass default options
if nargin < 1
disp('Default options returned (type "help cmaes" for help).');
end
xmin = defopts;
if nargin > 1 % supplement second argument with default options
xmin = getoptions(xstart, defopts);
end
return;
end
if isequal(fitfun, 'displayoptions')
names = fieldnames(defopts);
for name = names'
disp([name{:} repmat(' ', 1, 20-length(name{:})) ': ''' defopts.(name{:}) '''']);
end
return;
end
input.fitfun = fitfun; % record used input
if isempty(fitfun)
% fitfun = definput.fitfun;
% warning(['Objective function not determined, ''' fitfun ''' used']);
error(['Objective function not determined']);
end
if ischar(fitfun)
isCharFitFun = 1;
fitfun_s = fitfun;
elseif isa(fitfun, 'function_handle')
isHandleFitFun = 1;
% fitfun_s = inputname(1); %handle variable is evaluated into expression,
% losing its name, so this one won't work
fitfun_s = 'h_fitfun';
else
error('first argument FUN must be a string or a function handle');
end
if nargin < 2
xstart = [];
end
input.xstart = xstart;
if isempty(xstart)
% xstart = definput.xstart; % objective variables initial point
% warning('Initial search point, and problem dimension, not determined');
error('Initial search point, and problem dimension, not determined');
end
if nargin < 3
insigma = [];
end
if isa(insigma, 'struct')
error(['Third argument SIGMA must be (or eval to) a scalar '...
'or a column vector of size(X0,1)']);
end
input.sigma = insigma;
if isempty(insigma)
if all(size(myeval(xstart)) > 1)
insigma = std(xstart, 0, 2);
if any(insigma == 0)
error(['Initial search volume is zero, choose SIGMA or X0 appropriate']);
end
else
% will be captured later
% error(['Initial step sizes (SIGMA) not determined']);
end
end
% Compose options opts
if nargin < 4 || isempty(inopts) % no input options available
inopts = [];
opts = defopts;
else
opts = getoptions(inopts, defopts);
end
i = strfind(opts.SaveFilename, '%'); % remove everything after comment
if ~isempty(i)
opts.SaveFilename = opts.SaveFilename(1:i(1)-1);
end
opts.SaveFilename = deblank(opts.SaveFilename); % remove trailing white spaces
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
counteval = 0; countevalNaN = 0;
irun = 0;
while irun <= myeval(opts.Restarts) % for-loop does not work with resume
irun = irun + 1;
% ------------------------ Initialization -------------------------------
% Handle resuming of old run
flgresume = myevalbool(opts.Resume);
xmean = myeval(xstart);
if all(size(xmean) > 1)
xmean = mean(xmean, 2); % in case if xstart is a population
elseif size(xmean, 2) > 1
xmean = xmean';
end
if ~flgresume % not resuming a former run
% Assign settings from input parameters and options for myeval...
N = size(xmean, 1); numberofvariables = N;
lambda0 = floor(myeval(opts.PopSize) * myeval(opts.IncPopSize)^(irun-1));
% lambda0 = floor(myeval(opts.PopSize) * 3^floor((irun-1)/2));
popsize = lambda0;
lambda = lambda0;
insigma = myeval(insigma);
if all(size(insigma) == [N 2])
insigma = 0.5 * (insigma(:,2) - insigma(:,1));
end
else % flgresume is true, do resume former run
tmp = whos('-file', opts.SaveFilename);
for i = 1:length(tmp)
if strcmp(tmp(i).name, 'localopts');
error('Saved variables include variable "localopts", please remove');
end
end
local.opts = opts; % keep stopping and display options
local.varargin = varargin;
load(opts.SaveFilename);
varargin = local.varargin;
flgresume = 1;
% Overwrite old stopping and display options
opts.StopFitness = local.opts.StopFitness;
%%opts.MaxFunEvals = local.opts.MaxFunEvals;
%%opts.MaxIter = local.opts.MaxIter;
opts.StopFunEvals = local.opts.StopFunEvals;
opts.StopIter = local.opts.StopIter;
opts.TolX = local.opts.TolX;
opts.TolUpX = local.opts.TolUpX;
opts.TolFun = local.opts.TolFun;
opts.TolHistFun = local.opts.TolHistFun;
opts.StopOnStagnation = local.opts.StopOnStagnation;
opts.StopOnWarnings = local.opts.StopOnWarnings;
opts.ReadSignals = local.opts.ReadSignals;
opts.DispFinal = local.opts.DispFinal;
opts.LogPlot = local.opts.LogPlot;
opts.DispModulo = local.opts.DispModulo;
opts.SaveVariables = local.opts.SaveVariables;
opts.LogModulo = local.opts.LogModulo;
opts.LogTime = local.opts.LogTime;
clear local; % otherwise local would be overwritten during load
end
%--------------------------------------------------------------
% Evaluate options
stopFitness = myeval(opts.StopFitness);
stopMaxFunEvals = myeval(opts.MaxFunEvals);
stopMaxIter = myeval(opts.MaxIter);
stopFunEvals = myeval(opts.StopFunEvals);
stopIter = myeval(opts.StopIter);
if flgresume
stopIter = stopIter + countiter
end
stopTolX = myeval(opts.TolX);
stopTolUpX = myeval(opts.TolUpX);
stopTolFun = myeval(opts.TolFun);
stopTolHistFun = myeval(opts.TolHistFun);
stopOnStagnation = myevalbool(opts.StopOnStagnation);
stopOnWarnings = myevalbool(opts.StopOnWarnings);
flgreadsignals = myevalbool(opts.ReadSignals);
flgWarnOnEqualFunctionValues = myevalbool(opts.WarnOnEqualFunctionValues);
flgEvalParallel = myevalbool(opts.EvalParallel);
stopOnEqualFunctionValues = myeval(opts.StopOnEqualFunctionValues);
arrEqualFunvals = zeros(1, 10+N);
flgDiagonalOnly = myeval(opts.DiagonalOnly);
flgActiveCMA = myeval(opts.CMA.active);
noiseHandling = myevalbool(opts.Noise.on);
noiseMinMaxEvals = myeval(opts.Noise.minmaxevals);
noiseAlphaEvals = myeval(opts.Noise.alphaevals);
noiseCallback = myeval(opts.Noise.callback);
flgdisplay = myevalbool(opts.DispFinal);
flgplotting = myevalbool(opts.LogPlot);
verbosemodulo = myeval(opts.DispModulo);
flgscience = myevalbool(opts.Science);
flgsaving = [];
strsaving = [];
if strfind(opts.SaveVariables, '-v6')
i = strfind(opts.SaveVariables, '%');
if isempty(i) || i == 0 || strfind(opts.SaveVariables, '-v6') < i
strsaving = '-v6';
flgsaving = 1;
flgsavingfinal = 1;
end
end
if strncmp('final', opts.SaveVariables, 5)
flgsaving = 0;
flgsavingfinal = 1;
end
if isempty(flgsaving)
flgsaving = myevalbool(opts.SaveVariables);
flgsavingfinal = flgsaving;
end
savemodulo = myeval(opts.LogModulo);
savetime = myeval(opts.LogTime);
i = strfind(opts.LogFilenamePrefix, ' '); % remove everything after white space
if ~isempty(i)
opts.LogFilenamePrefix = opts.LogFilenamePrefix(1:i(1)-1);
end
% TODO here silent option? set disp, save and log options to 0
%--------------------------------------------------------------
if (isfinite(stopFunEvals) || isfinite(stopIter)) && ~flgsaving
warning('To resume later the saving option needs to be set');
end
% Do more checking and initialization
if flgresume % resume is on
time.t0 = clock;
if flgdisplay
disp([' resumed from ' opts.SaveFilename ]);
end
if counteval >= stopMaxFunEvals
error(['MaxFunEvals exceeded, use StopFunEvals as stopping ' ...
'criterion before resume']);
end
if countiter >= stopMaxIter
error(['MaxIter exceeded, use StopIter as stopping criterion ' ...
'before resume']);
end
else % flgresume
% xmean = mean(myeval(xstart), 2); % evaluate xstart again, because of irun
maxdx = myeval(opts.DiffMaxChange); % maximal sensible variable change
mindx = myeval(opts.DiffMinChange); % minimal sensible variable change
% can both also be defined as Nx1 vectors
lbounds = myeval(opts.LBounds);
ubounds = myeval(opts.UBounds);
if length(lbounds) == 1
lbounds = repmat(lbounds, N, 1);
end
if length(ubounds) == 1
ubounds = repmat(ubounds, N, 1);
end
if isempty(insigma) % last chance to set insigma
if all(lbounds > -Inf) && all(ubounds < Inf)
if any(lbounds>=ubounds)
error('upper bound must be greater than lower bound');
end
insigma = 0.3*(ubounds-lbounds);
stopTolX = myeval(opts.TolX); % reevaluate these
stopTolUpX = myeval(opts.TolUpX);
else
error(['Initial step sizes (SIGMA) not determined']);
end
end
% Check all vector sizes
if size(xmean, 2) > 1 || size(xmean,1) ~= N
error(['intial search point should be a column vector of size ' ...
num2str(N)]);
elseif ~(all(size(insigma) == [1 1]) || all(size(insigma) == [N 1]))
error(['input parameter SIGMA should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
elseif size(stopTolX, 2) > 1 || ~ismember(size(stopTolX, 1), [1 N])
error(['option TolX should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
elseif size(stopTolUpX, 2) > 1 || ~ismember(size(stopTolUpX, 1), [1 N])
error(['option TolUpX should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
elseif size(maxdx, 2) > 1 || ~ismember(size(maxdx, 1), [1 N])
error(['option DiffMaxChange should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
elseif size(mindx, 2) > 1 || ~ismember(size(mindx, 1), [1 N])
error(['option DiffMinChange should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
elseif size(lbounds, 2) > 1 || ~ismember(size(lbounds, 1), [1 N])
error(['option lbounds should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
elseif size(ubounds, 2) > 1 || ~ismember(size(ubounds, 1), [1 N])
error(['option ubounds should be (or eval to) a scalar '...
'or a column vector of size ' num2str(N)] );
end
% Initialize dynamic internal state parameters
if any(insigma <= 0)
error(['Initial search volume (SIGMA) must be greater than zero']);
end
if max(insigma)/min(insigma) > 1e6
error(['Initial search volume (SIGMA) badly conditioned']);
end
sigma = max(insigma); % overall standard deviation
pc = zeros(N,1); ps = zeros(N,1); % evolution paths for C and sigma
if length(insigma) == 1
insigma = insigma * ones(N,1) ;
end
diagD = insigma/max(insigma); % diagonal matrix D defines the scaling
diagC = diagD.^2;
if flgDiagonalOnly ~= 1 % use at some point full covariance matrix
B = eye(N,N); % B defines the coordinate system
BD = B.*repmat(diagD',N,1); % B*D for speed up only
C = diag(diagC); % covariance matrix == BD*(BD)'
end
if flgDiagonalOnly
B = 1;
end
fitness.hist=NaN*ones(1,10+ceil(3*10*N/lambda)); % history of fitness values
fitness.histsel=NaN*ones(1,10+ceil(3*10*N/lambda)); % history of fitness values
fitness.histbest=[]; % history of fitness values
fitness.histmedian=[]; % history of fitness values
% Initialize boundary handling
bnd.isactive = any(lbounds > -Inf) || any(ubounds < Inf);
if bnd.isactive
if any(lbounds>ubounds)
error('lower bound found to be greater than upper bound');
end
[xmean ti] = xintobounds(xmean, lbounds, ubounds); % just in case
if any(ti)
warning('Initial point was out of bounds, corrected');
end
bnd.weights = zeros(N,1); % weights for bound penalty
% scaling is better in axis-parallel case, worse in rotated
bnd.flgscale = 0; % scaling will be omitted if zero
if bnd.flgscale ~= 0
bnd.scale = diagC/mean(diagC);
else
bnd.scale = ones(N,1);
end
idx = (lbounds > -Inf) | (ubounds < Inf);
if length(idx) == 1
idx = idx * ones(N,1);
end
bnd.isbounded = zeros(N,1);
bnd.isbounded(find(idx)) = 1;
maxdx = min(maxdx, (ubounds - lbounds)/2);
if any(sigma*sqrt(diagC) > maxdx)
fac = min(maxdx ./ sqrt(diagC))/sigma;
sigma = min(maxdx ./ sqrt(diagC));
warning(['Initial SIGMA multiplied by the factor ' num2str(fac) ...
', because it was larger than half' ...
' of one of the boundary intervals']);
end
idx = (lbounds > -Inf) & (ubounds < Inf);
dd = diagC;
isBadSigma = 5*sigma*sqrt(dd(idx)) < ubounds(idx) - lbounds(idx);
if any(isBadSigma)
warning(['Initial SIGMA is, in at least one coordinate [%s], ' ...
'much smaller than the '...
'given boundary intervals. For reasonable ' ...
'global search performance SIGMA should be ' ...
'between 0.2 and 0.5 of the bounded interval in ' ...
'each coordinate. If all coordinates have ' ...
'lower and upper bounds SIGMA can be empty'], ...
num2str(find(isBadSigma'), '%d '));
end
bnd.dfithist = 1; % delta fit for setting weights
bnd.aridxpoints = []; % remember complete outside points
bnd.arfitness = []; % and their fitness
bnd.validfitval = 0;
bnd.iniphase = 1;
end
% ooo initial feval, for output only
if irun == 1
out.solutions.bestever.x = xmean;
out.solutions.bestever.f = Inf; % for simpler comparison below
out.solutions.bestever.evals = counteval;
bestever = out.solutions.bestever;
end
if myevalbool(opts.EvalInitialX)
fitness.hist(1)=feval(fitfun, xmean, varargin{:});
fitness.histsel(1)=fitness.hist(1);
counteval = counteval + 1;
if fitness.hist(1) < out.solutions.bestever.f
out.solutions.bestever.x = xmean;
out.solutions.bestever.f = fitness.hist(1);
out.solutions.bestever.evals = counteval;
bestever = out.solutions.bestever;
end
else
fitness.hist(1)=NaN;
fitness.histsel(1)=NaN;
end
% initialize random number generator
if ischar(opts.Seed)
randn('state', eval(opts.Seed)); % random number generator state
else
randn('state', opts.Seed);
end
%qqq
% load(opts.SaveFilename, 'startseed');
% randn('state', startseed);
% disp(['SEED RELOADED FROM ' opts.SaveFilename]);
startseed = randn('state'); % for retrieving in saved variables
% Initialize further constants
chiN=N^0.5*(1-1/(4*N)+1/(21*N^2)); % expectation of
% ||N(0,I)|| == norm(randn(N,1))
countiter = 0;
% Initialize records and output
if irun == 1
time.t0 = clock;
% TODO: keep also median solution?
out.evals = counteval; % should be first entry
out.stopflag = {};
outiter = 0;
% Write headers to output data files
filenameprefix = opts.LogFilenamePrefix;
if savemodulo && savetime
filenames = {};
filenames(end+1) = {'axlen'};
filenames(end+1) = {'fit'};
filenames(end+1) = {'stddev'};
filenames(end+1) = {'xmean'};
filenames(end+1) = {'xrecentbest'};
str = [' (startseed=' num2str(startseed(2)) ...
', ' num2str(clock, '%d/%02d/%d %d:%d:%2.2f') ')'];
for namecell = filenames(:)'
name = namecell{:};
[fid, err] = fopen(['./' filenameprefix name '.dat'], 'w');
if fid < 1 % err ~= 0
warning(['could not open ' filenameprefix name '.dat']);
filenames(find(strcmp(filenames,name))) = [];
else
% fprintf(fid, '%s\n', ...
% ['<CMAES-OUTPUT version="' cmaVersion '">']);
% fprintf(fid, [' <NAME>' name '</NAME>\n']);
% fprintf(fid, [' <DATE>' date() '</DATE>\n']);
% fprintf(fid, ' <PARAMETERS>\n');
% fprintf(fid, [' dimension=' num2str(N) '\n']);
% fprintf(fid, ' </PARAMETERS>\n');
% different cases for DATA columns annotations here
% fprintf(fid, ' <DATA');
if strcmp(name, 'axlen')
fprintf(fid, ['%% columns="iteration, evaluation, sigma, ' ...
'max axis length, min axis length, ' ...
'all principal axes lengths (sorted square roots ' ...
'of eigenvalues of C)"' str]);
elseif strcmp(name, 'fit')
fprintf(fid, ['%% columns="iteration, evaluation, sigma, axis ratio, bestever,' ...
' best, median, worst fitness function value,' ...
' further objective values of best"' str]);
elseif strcmp(name, 'stddev')
fprintf(fid, ['%% columns=["iteration, evaluation, sigma, void, void, ' ...
'stds==sigma*sqrt(diag(C))"' str]);
elseif strcmp(name, 'xmean')
fprintf(fid, ['%% columns="iteration, evaluation, void, ' ...
'void, void, xmean"' str]);
elseif strcmp(name, 'xrecentbest')
fprintf(fid, ['%% columns="iteration, evaluation, fitness, ' ...
'void, void, xrecentbest"' str]);
end
fprintf(fid, '\n'); % DATA
if strcmp(name, 'xmean')
fprintf(fid, '%ld %ld 0 0 0 ', 0, counteval);
% fprintf(fid, '%ld %ld 0 0 %e ', countiter, counteval, fmean);
%qqq fprintf(fid, msprintf('%e ', genophenotransform(out.genopheno, xmean)) + '\n');
fprintf(fid, '%e ', xmean);
fprintf(fid, '\n');
end
fclose(fid);
clear fid; % preventing
end
end % for files
end % savemodulo
end % irun == 1
end % else flgresume
% -------------------- Generation Loop --------------------------------
stopflag = {};
while isempty(stopflag)
% set internal parameters
if countiter == 0 || lambda ~= lambda_last
if countiter > 0 && floor(log10(lambda)) ~= floor(log10(lambda_last)) ...
&& flgdisplay
disp([' lambda = ' num2str(lambda)]);
lambda_hist(:,end+1) = [countiter+1; lambda];
else
lambda_hist = [countiter+1; lambda];
end
lambda_last = lambda;
% Strategy internal parameter setting: Selection
mu = myeval(opts.ParentNumber); % number of parents/points for recombination
if strncmp(lower(opts.RecombinationWeights), 'equal', 3)
weights = ones(mu,1); % (mu_I,lambda)-CMA-ES
elseif strncmp(lower(opts.RecombinationWeights), 'linear', 3)
weights = mu+0.5-(1:mu)';
elseif strncmp(lower(opts.RecombinationWeights), 'superlinear', 3)
% use (lambda+1)/2 as reference if mu < lambda/2
weights = log(max(mu, lambda/2) + 1/2)-log(1:mu)'; % muXone array for weighted recombination
else
error(['Recombination weights to be "' opts.RecombinationWeights ...
'" is not implemented']);
end
mueff=sum(weights)^2/sum(weights.^2); % variance-effective size of mu
weights = weights/sum(weights); % normalize recombination weights array
if mueff == lambda
error(['Combination of values for PopSize, ParentNumber and ' ...
' and RecombinationWeights is not reasonable']);
end
% Strategy internal parameter setting: Adaptation
cc = myeval(opts.CMA.ccum); % time constant for cumulation for covariance matrix
cs = myeval(opts.CMA.cs);
% old way TODO: remove this at some point
% mucov = mueff; % size of mu used for calculating learning rate ccov
% ccov = (1/mucov) * 2/(N+1.41)^2 ... % learning rate for covariance matrix
% + (1-1/mucov) * min(1,(2*mucov-1)/((N+2)^2+mucov));
% new way
if myevalbool(opts.CMA.on)
ccov1 = myeval(opts.CMA.ccov1);
ccovmu = min(1-ccov1, myeval(opts.CMA.ccovmu));
else
ccov1 = 0;
ccovmu = 0;
end
% flgDiagonalOnly = -lambda*4*1/ccov; % for ccov==1 it is not needed
% 0 : C will never be diagonal anymore
% 1 : C will always be diagonal
% >1: C is diagonal for first iterations, set to 0 afterwards
if flgDiagonalOnly < 1
flgDiagonalOnly = 0;
end
if flgDiagonalOnly
ccov1_sep = min(1, ccov1 * (N+1.5) / 3);
ccovmu_sep = min(1-ccov1_sep, ccovmu * (N+1.5) / 3);
elseif N > 98 && flgdisplay && countiter == 0
disp('consider option DiagonalOnly for high-dimensional problems');
end
% ||ps|| is close to sqrt(mueff/N) for mueff large on linear fitness
%damps = ... % damping for step size control, usually close to one
% (1 + 2*max(0,sqrt((mueff-1)/(N+1))-1)) ... % limit sigma increase
% * max(0.3, ... % reduce damps, if max. iteration number is small
% 1 - N/min(stopMaxIter,stopMaxFunEvals/lambda)) + cs;
damps = myeval(opts.CMA.damps);
if noiseHandling
noiseReevals = min(myeval(opts.Noise.reevals), lambda);
noiseAlpha = myeval(opts.Noise.alphasigma);
noiseEpsilon = myeval(opts.Noise.epsilon);
noiseTheta = myeval(opts.Noise.theta);
noisecum = myeval(opts.Noise.cum);
noiseCutOff = myeval(opts.Noise.cutoff); % arguably of minor relevance
else
noiseReevals = 0; % more convenient in later coding
end
%qqq hacking of a different parameter setting, e.g. for ccov or damps,
% can be done here, but is not necessary anymore, see opts.CMA.
% ccov1 = 0.0*ccov1; disp(['CAVE: ccov1=' num2str(ccov1)]);
% ccovmu = 0.0*ccovmu; disp(['CAVE: ccovmu=' num2str(ccovmu)]);
% damps = inf*damps; disp(['CAVE: damps=' num2str(damps)]);
% cc = 1; disp(['CAVE: cc=' num2str(cc)]);
end
% Display initial message
if countiter == 0 && flgdisplay
if mu == 1
strw = '100';
elseif mu < 8
strw = [sprintf('%.0f', 100*weights(1)) ...
sprintf(' %.0f', 100*weights(2:end)')];
else
strw = [sprintf('%.2g ', 100*weights(1:2)') ...
sprintf('%.2g', 100*weights(3)') '...' ...
sprintf(' %.2g', 100*weights(end-1:end)') ']%, '];
end
if irun > 1
strrun = [', run ' num2str(irun)];
else
strrun = '';
end
disp([' n=' num2str(N) ': (' num2str(mu) ',' ...
num2str(lambda) ')-CMA-ES(w=[' ...
strw ']%, ' ...
'mu_eff=' num2str(mueff,'%.1f') ...
') on function ' ...
(fitfun_s) strrun]);
if flgDiagonalOnly == 1
disp(' C is diagonal');
elseif flgDiagonalOnly
disp([' C is diagonal for ' num2str(floor(flgDiagonalOnly)) ' iterations']);
end
end
flush;
countiter = countiter + 1;
% Generate and evaluate lambda offspring
fitness.raw = repmat(NaN, 1, lambda + noiseReevals);
% parallel evaluation
if flgEvalParallel
arz = randn(N,lambda);
if ~flgDiagonalOnly
arx = repmat(xmean, 1, lambda) + sigma * (BD * arz); % Eq. (1)
else
arx = repmat(xmean, 1, lambda) + repmat(sigma * diagD, 1, lambda) .* arz;
end
if noiseHandling
if noiseEpsilon == 0
arx = [arx arx(:,1:noiseReevals)];
elseif flgDiagonalOnly
arx = [arx arx(:,1:noiseReevals) + ...
repmat(noiseEpsilon * sigma * diagD, 1, noiseReevals) ...
.* randn(N,noiseReevals)];
else
arx = [arx arx(:,1:noiseReevals) + ...
noiseEpsilon * sigma * ...
(BD * randn(N,noiseReevals))];
end
end
% You may handle constraints here. You may either resample
% arz(:,k) and/or multiply it with a factor between -1 and 1
% (the latter will decrease the overall step size) and
% recalculate arx accordingly. Do not change arx or arz in any
% other way.
if ~bnd.isactive
arxvalid = arx;
else
arxvalid = xintobounds(arx, lbounds, ubounds);
end
% You may handle constraints here. You may copy and alter
% (columns of) arxvalid(:,k) only for the evaluation of the
% fitness function. arx and arxvalid should not be changed.
fitness.raw = feval(fitfun, arxvalid, varargin{:});
countevalNaN = countevalNaN + sum(isnan(fitness.raw));
counteval = counteval + sum(~isnan(fitness.raw));
end
% non-parallel evaluation and remaining NaN-values
% set also the reevaluated solution to NaN
fitness.raw(lambda + find(isnan(fitness.raw(1:noiseReevals)))) = NaN;
for k=find(isnan(fitness.raw)),
% fitness.raw(k) = NaN;
tries = flgEvalParallel; % in parallel case this is the first re-trial
% Resample, until fitness is not NaN
while isnan(fitness.raw(k))
if k <= lambda % regular samples (not the re-evaluation-samples)
arz(:,k) = randn(N,1); % (re)sample
if flgDiagonalOnly
arx(:,k) = xmean + sigma * diagD .* arz(:,k); % Eq. (1)
else
arx(:,k) = xmean + sigma * (BD * arz(:,k)); % Eq. (1)
end
else % re-evaluation solution with index > lambda
if flgDiagonalOnly
arx(:,k) = arx(:,k-lambda) + (noiseEpsilon * sigma) * diagD .* randn(N,1);
else
arx(:,k) = arx(:,k-lambda) + (noiseEpsilon * sigma) * (BD * randn(N,1));
end
end