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optim: upgrade license to GPLv3+ and mention on DESCRIPTION the other package licenses
author carandraug
date Fri, 30 Mar 2012 15:14:48 +0000
parents 557017e3e86e
children 28b882879c6e
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1 ## Copyright (C) 2002-2008 Etienne Grossmann <etienne@cs.uky.edu>
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2 ##
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3 ## This program is free software; you can redistribute it and/or modify it under
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4 ## the terms of the GNU General Public License as published by the Free Software
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5 ## Foundation; either version 3 of the License, or (at your option) any later
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6 ## version.
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7 ##
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8 ## This program is distributed in the hope that it will be useful, but WITHOUT
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9 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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10 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
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11 ## details.
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12 ##
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13 ## You should have received a copy of the GNU General Public License along with
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14 ## this program; if not, see <http://www.gnu.org/licenses/>.
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15
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16 ## [x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization
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17 ##
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18 ## Minimize 'f' using the Nelder-Mead algorithm. This function is inspired
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19 ## from the that found in the book "Numerical Recipes".
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20 ##
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21 ## ARGUMENTS
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22 ## ---------
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23 ## f : string : Name of function. Must return a real value
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24 ## args : list : Arguments passed to f.
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25 ## or matrix : f's only argument
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26 ## ctl : vector : (Optional) Control variables, described below
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27 ## or struct
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28 ##
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29 ## RETURNED VALUES
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30 ## ---------------
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31 ## x0 : matrix : Local minimum of f
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32 ## v : real : Value of f in x0
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33 ## nev : number : Number of function evaluations
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34 ##
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35 ## CONTROL VARIABLE : (optional) may be named arguments (i.e. "name",value
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36 ## ------------------ pairs), a struct, or a vector of length <= 6, where
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37 ## NaN's are ignored. Default values are written <value>.
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38 ## OPT. VECTOR
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39 ## NAME POS
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40 ## ftol,f N/A : Stopping criterion : stop search when values at simplex
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41 ## vertices are all alike, as tested by
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42 ##
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43 ## f > (max_i (f_i) - min_i (f_i)) /max(max(|f_i|),1)
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44 ##
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45 ## where f_i are the values of f at the vertices. <10*eps>
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46 ##
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47 ## rtol,r N/A : Stop search when biggest radius of simplex, using
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48 ## infinity-norm, is small, as tested by :
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49 ##
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50 ## ctl(2) > Radius <10*eps>
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51 ##
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52 ## vtol,v N/A : Stop search when volume of simplex is small, tested by
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53 ##
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54 ## ctl(2) > Vol
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55 ##
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56 ## crit,c ctl(1) : Set one stopping criterion, 'ftol' (c=1), 'rtol' (c=2)
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57 ## or 'vtol' (c=3) to the value of the 'tol' option. <1>
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58 ##
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59 ## tol, t ctl(2) : Threshold in termination test chosen by 'crit' <10*eps>
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60 ##
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61 ## narg ctl(3) : Position of the minimized argument in args <1>
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62 ## maxev ctl(4) : Maximum number of function evaluations. This number <inf>
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63 ## may be slightly exceeded.
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64 ## isz ctl(5) : Size of initial simplex, which is : <1>
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65 ##
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66 ## { x + e_i | i in 0..N }
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67 ##
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68 ## Where x == args{narg} is the initial value
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69 ## e_0 == zeros (size (x)),
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70 ## e_i(j) == 0 if j != i and e_i(i) == ctl(5)
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71 ## e_i has same size as x
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72 ##
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73 ## Set ctl(5) to the distance you expect between the starting
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74 ## point and the minimum.
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75 ##
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76 ## rst ctl(6) : When a minimum is found the algorithm restarts next to
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77 ## it until the minimum does not improve anymore. ctl(6) is
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78 ## the maximum number of restarts. Set ctl(6) to zero if
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79 ## you know the function is well-behaved or if you don't
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80 ## mind not getting a true minimum. <0>
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81 ##
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82 ## verbose, v Be more or less verbose (quiet=0) <0>
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83
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84 function [x,v,nev] = nelder_mead_min (f, args, varargin)
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85
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86 verbose = 0;
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87
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88 # Default control variables
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89 ftol = rtol = 10*eps; # Stop either by likeness of values or
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90 vtol = nan; # radius, but don't care about volume.
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91 crit = 0; # Stopping criterion ctl(1)
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92 tol = 10*eps; # Stopping test's threshold ctl(2)
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93 narg = 1; # Position of minimized arg ctl(3)
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94 maxev = inf; # Max num of func evaluations ctl(4)
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95 isz = 1; # Initial size ctl(5)
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96 rst = 0; # Max # of restarts
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97
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98
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99 if nargin >= 3, # Read control arguments
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100 va_arg_cnt = 1;
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101 if nargin > 3,
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102 ctl = struct (varargin{:});
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103 else
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104 ctl = varargin{va_arg_cnt++};
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105 end
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106 if isnumeric (ctl)
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107 if length (ctl)>=1 && !isnan (ctl(1)), crit = ctl(1); end
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108 if length (ctl)>=2 && !isnan (ctl(2)), tol = ctl(2); end
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109 if length (ctl)>=3 && !isnan (ctl(3)), narg = ctl(3); end
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110 if length (ctl)>=4 && !isnan (ctl(4)), maxev = ctl(4); end
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111 if length (ctl)>=5 && !isnan (ctl(5)), isz = ctl(5); end
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112 if length (ctl)>=6 && !isnan (ctl(6)), rst = ctl(6); end
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113 else
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114 if isfield (ctl, "crit") && ! isnan (ctl.crit ), crit = ctl.crit ; end
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115 if isfield (ctl, "tol") && ! isnan (ctl.tol ), tol = ctl.tol ; end
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116 if isfield (ctl, "ftol") && ! isnan (ctl.ftol ), ftol = ctl.ftol ; end
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117 if isfield (ctl, "rtol") && ! isnan (ctl.rtol ), rtol = ctl.rtol ; end
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118 if isfield (ctl, "vtol") && ! isnan (ctl.vtol ), vtol = ctl.vtol ; end
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119 if isfield (ctl, "narg") && ! isnan (ctl.narg ), narg = ctl.narg ; end
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120 if isfield (ctl,"maxev") && ! isnan (ctl.maxev), maxev = ctl.maxev; end
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121 if isfield (ctl, "isz") && ! isnan (ctl.isz ), isz = ctl.isz ; end
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122 if isfield (ctl, "rst") && ! isnan (ctl.rst ), rst = ctl.rst ; end
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123 if isfield(ctl,"verbose")&& !isnan(ctl.verbose),verbose=ctl.verbose;end
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124 end
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125 end
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126
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127
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128 if crit == 1, ftol = tol;
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129 elseif crit == 2, rtol = tol;
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130 elseif crit == 3, vtol = tol;
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131 elseif crit, error ("crit is %i. Should be 1,2 or 3.\n");
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132 end
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133
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134 if iscell (args)
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135 x = args{1};
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136 else # Single argument
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137 x = args;
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138 args = {args};
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139 endif
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140
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141 if narg > length (args) # Check
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142 error ("nelder_mead_min : narg==%i, length (args)==%i\n",
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143 narg, length (args));
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144 end
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145
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146 [R,C] = size (x);
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147 N = R*C; # Size of argument
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148 x = x(:);
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149 # Initial simplex
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150 u = isz * eye (N+1,N) + ones(N+1,1)*x';
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151
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152 y = zeros (N+1,1);
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153 for i = 1:N+1,
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154 y(i) = feval (f, args{1:narg-1},reshape(u(i,:),R,C),args{narg+1:end});
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155 end ;
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156 nev = N+1;
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157
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158 [ymin,imin] = min(y);
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159 ymin0 = ymin;
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160 ## y
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161 nextprint = 0 ;
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162 v = nan;
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163 while nev <= maxev,
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164
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165 ## ymin, ymax, ymx2 : lowest, highest and 2nd highest function values
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166 ## imin, imax, imx2 : indices of vertices with these values
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167 [ymin,imin] = min(y); [ymax,imax] = max(y) ;
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168 y(imax) = ymin ;
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169 [ymx2,imx2] = max(y) ;
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170 y(imax) = ymax ;
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171
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172 ## ymin may be > ymin0 after restarting
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173 ## if ymin > ymin0 ,
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174 ## "nelder-mead : Whoa 'downsimplex' Should be renamed 'upsimplex'"
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175 ## keyboard
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176 ## end
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177
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178 # Compute stopping criterion
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179 done = 0;
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180 if ! isnan (ftol),
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181 done |= ((max(y)-min(y)) / max(1,max(abs(y))) < ftol);
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182 end
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183 if ! isnan (rtol),
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184 done |= (2*max (max (u) - min (u)) < rtol);
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185 end
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186 if ! isnan (vtol)
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187 done |= (abs (det (u(1:N,:)-ones(N,1)*u(N+1,:)))/factorial(N) < vtol);
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188 end
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189 ## [ 2*max (max (u) - min (u)), abs (det (u(1:N,:)-ones(N,1)*u(N+1,:)))/factorial(N);\
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190 ## rtol, vtol]
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191
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192 # Eventually print some info
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193 if verbose && nev > nextprint && ! done
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194
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195 printf("nev=%-5d imin=%-3d ymin=%-8.3g done=%i\n",\
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196 nev,imin,ymin,done) ;
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197
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198 nextprint = nextprint + 100 ;
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199 end
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200
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201 if done # Termination test
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202 if (rst > 0) && (isnan (v) || v > ymin)
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203 rst--;
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204 if verbose
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205 if isnan (v),
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206 printf ("Restarting next to minimum %10.3e\n",ymin);
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207 else
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208 printf ("Restarting next to minimum %10.3e\n",ymin-v);
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209 end
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210 end
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211 # Keep best minimum
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212 x = reshape (u(imin,:), R, C) ;
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213 v = ymin ;
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214
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215 jumplen = 10 * max (max (u) - min (u));
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216
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217 u += jumplen * randn (size (u));
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218 for i = 1:N+1, y(i) = \
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219 feval (f, args{1:narg-1},reshape(u(i,:),R,C),args{narg+1:length(args)});
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220 end
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221 nev += N+1;
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222 [ymin,imin] = min(y); [ymax,imax] = max(y);
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223 y(imax) = ymin;
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224 [ymx2,imx2] = max(y);
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225 y(imax) = ymax ;
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226 else
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227 if isnan (v),
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228 x = reshape (u(imin,:), R, C) ;
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229 v = ymin ;
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230 end
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231 if verbose,
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232 printf("nev=%-5d imin=%-3d ymin=%-8.3g done=%i. Done\n",\
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233 nev,imin,ymin,done) ;
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234 end
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235 return
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236 end
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237
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238 end
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239 ## [ y' u ]
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240
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241 tra = 0 ; # 'trace' debug var contains flags
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242 if verbose > 1, str = sprintf (" %i : %10.3e --",done,ymin); end
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243
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244 # Look for a new point
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245 xsum = sum(u) ; # Consider reflection of worst vertice
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246 # around centroid.
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247 ## f1 = (1-(-1))/N = 2/N;
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248 ## f2 = f1 - (-1) = 2/N + 1 = (N+2)/N
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249 xnew = (2*xsum - (N+2)*u(imax,:)) / N;
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250 ## xnew = (2*xsum - N*u(imax,:)) / N;
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251 ynew = feval (f, args{1:narg-1},reshape(xnew,R,C),args{narg+1:length(args)});
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252 nev++;
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253
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254 if ynew <= ymin , # Reflection is good
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255
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256 tra += 1 ;
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257 if verbose > 1
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258 str = [str,sprintf(" %3i : %10.3e good refl >>",nev,ynew-ymin)];
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259 end
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260 y(imax) = ynew; u(imax,:) = xnew ;
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261 ## ymin = ynew;
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262 ## imin = imax;
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263 xsum = sum(u) ;
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264
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265 ## f1 = (1-2)/N = -1/N
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266 ## f2 = f1 - 2 = -1/N - 2 = -(2*N+1)/N
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267 xnew = ( -xsum + (2*N+1)*u(imax,:) ) / N;
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268 ynew = feval (f, args{1:narg-1},reshape(xnew,R,C),args{narg+1:length(args)});
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269 nev++;
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270
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271 if ynew <= ymin , # expansion improves
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272 tra += 2 ;
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273 ## 'expanded reflection'
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274 y(imax) = ynew ; u(imax,:) = xnew ;
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275 xsum = sum(u) ;
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276 if verbose > 1
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277 str = [str,sprintf(" %3i : %10.3e expd refl",nev,ynew-ymin)];
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278 end
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279 else
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280 tra += 4 ;
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281 ## 'plain reflection'
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282 ## Updating of y and u has already been done
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283 if verbose > 1
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284 str = [str,sprintf(" %3i : %10.3e plain ref",nev,ynew-ymin)];
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285 end
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286 end
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287 # Reflexion is really bad
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288 elseif ynew >= ymax ,
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289
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290 tra += 8 ;
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291 if verbose > 1
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292 str = [str,sprintf(" %3i : %10.3e intermedt >>",nev,ynew-ymin)];
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293 end
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294 ## look for intermediate point
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295 # Bring worst point closer to centroid
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296 ## f1 = (1-0.5)/N = 0.5/N
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297 ## f2 = f1 - 0.5 = 0.5*(1 - N)/N
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298 xnew = 0.5*(xsum + (N-1)*u(imax,:)) / N;
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299 ynew = feval (f, args{1:narg-1},reshape(xnew,R,C),args{narg+1:length(args)});
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300 nev++;
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301
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302 if ynew >= ymax , # New point is even worse. Contract whole
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303 # simplex
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304
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305 nev += N + 1 ;
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306 ## u0 = u;
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307 u = (u + ones(N+1,1)*u(imin,:)) / 2;
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308 ## keyboard
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309
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310 ## Code that doesn't care about value of empty_list_elements_ok
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311 if imin == 1 , ii = 2:N+1;
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312 elseif imin == N+1, ii = 1:N;
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313 else ii = [1:imin-1,imin+1:N+1]; end
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314 for i = ii
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315 y(i) = \
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316 ynew = feval (f, args{1:narg-1},reshape(u(i,:),R,C),args{narg+1:length(args)});
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317 end
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318 ## 'contraction'
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319 tra += 16 ;
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320 if verbose > 1
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321 str = [str,sprintf(" %3i contractn",nev)];
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322 end
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323 else # Replace highest point
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324 y(imax) = ynew ; u(imax,:) = xnew ;
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325 xsum = sum(u) ;
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326 ## 'intermediate'
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327 tra += 32 ;
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328 if verbose > 1
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329 str = [str,sprintf(" %3i : %10.3e intermedt",nev,ynew-ymin)];
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330 end
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331 end
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332
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333 else # Reflexion is neither good nor bad
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334 y(imax) = ynew ; u(imax,:) = xnew ;
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335 xsum = sum(u) ;
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336 ## 'plain reflection (2)'
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337 tra += 64 ;
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338 if verbose > 1
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339 str = [str,sprintf(" %3i : %10.3e keep refl",nev,ynew-ymin)];
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340 end
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341 end
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342 if verbose > 1, printf ("%s\n",str); end
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343 end
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344
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345 if verbose >= 0
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346 printf ("nelder_mead : Too many iterations. Returning\n");
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347 end
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348
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349 if isnan (v) || v > ymin,
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350 x = reshape (u(imin,:), R, C) ;
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351 v = ymin ;
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352 end