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properly compress sparse matrices after assembly
author Jaroslav Hajek <highegg@gmail.com>
date Mon, 12 Apr 2010 12:57:44 +0200
parents 5c66978f3fdf
children 95c3e38098bf
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1 ## Copyright (C) 2008, 2009 VZLU Prague, a.s.
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18 ##
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19 ## Author: Jaroslav Hajek <highegg@gmail.com>
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20
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21 ## -*- texinfo -*-
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22 ## @deftypefn{Function File} {} fminunc (@var{fcn}, @var{x0}, @var{options})
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23 ## @deftypefnx{Function File} {[@var{x}, @var{fvec}, @var{info}, @var{output}, @var{grad}, @var{hess}]} = fminunc (@var{fcn}, @dots{})
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24 ## Solve a unconstrained optimization problem defined by the function @var{fcn}.
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25 ## @var{fcn} should accepts a vector (array) defining the unknown variables,
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26 ## and return the objective function value, optionally with gradient.
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27 ## In other words, this function attempts to determine a vector @var{x} such
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28 ## that @code{@var{fcn} (@var{x})} is a local minimum.
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29 ## @var{x0} determines a starting guess. The shape of @var{x0} is preserved
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30 ## in all calls to @var{fcn}, but otherwise it is treated as a column vector.
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31 ## @var{options} is a structure specifying additional options.
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32 ## Currently, @code{fminunc} recognizes these options:
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33 ## @code{"FunValCheck"}, @code{"OutputFcn"}, @code{"TolX"},
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34 ## @code{"TolFun"}, @code{"MaxIter"}, @code{"MaxFunEvals"},
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35 ## @code{"GradObj"}, @code{"FinDiffType"},
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36 ## @code{"TypicalX"}, @code{"AutoScaling"}.
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37 ##
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38 ## If @code{"GradObj"} is @code{"on"}, it specifies that @var{fcn},
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39 ## called with 2 output arguments, also returns the Jacobian matrix
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40 ## of right-hand sides at the requested point. @code{"TolX"} specifies
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41 ## the termination tolerance in the unknown variables, while
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42 ## @code{"TolFun"} is a tolerance for equations. Default is @code{1e-7}
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43 ## for both @code{"TolX"} and @code{"TolFun"}.
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44 ##
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45 ## For description of the other options, see @code{optimset}.
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46 ##
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47 ## On return, @var{fval} contains the value of the function @var{fcn}
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48 ## evaluated at @var{x}, and @var{info} may be one of the following values:
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49 ##
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50 ## @table @asis
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51 ## @item 1
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52 ## Converged to a solution point. Relative gradient error is less than specified
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53 ## by TolFun.
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54 ## @item 2
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55 ## Last relative step size was less that TolX.
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56 ## @item 3
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57 ## Last relative decrease in func value was less than TolF.
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58 ## @item 0
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59 ## Iteration limit exceeded.
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60 ## @item -3
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61 ## The trust region radius became excessively small.
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62 ## @end table
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63 ##
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64 ## Optionally, fminunc can also yield a structure with convergence statistics
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65 ## (@var{output}), the output gradient (@var{grad}) and approximate hessian
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66 ## (@var{hess}).
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67 ##
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68 ## Note: If you only have a single nonlinear equation of one variable, using
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69 ## @code{fminbnd} is usually a much better idea.
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70 ## @seealso{fminbnd, optimset}
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71 ## @end deftypefn
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72
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73 ## PKG_ADD: __all_opts__ ("fminunc");
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74
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75 function [x, fval, info, output, grad, hess] = fminunc (fcn, x0, options = struct ())
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76
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77 ## Get default options if requested.
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78 if (nargin == 1 && ischar (fcn) && strcmp (fcn, 'defaults'))
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79 x = optimset ("MaxIter", 400, "MaxFunEvals", Inf, \
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80 "GradObj", "off", "TolX", 1e-7, "TolFun", 1e-7,
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81 "OutputFcn", [], "FunValCheck", "off",
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82 "FinDiffType", "central",
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83 "TypicalX", [], "AutoScaling", "off");
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84 return;
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85 endif
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86
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87 if (nargin < 2 || nargin > 3 || ! ismatrix (x0))
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88 print_usage ();
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89 endif
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90
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91 if (ischar (fcn))
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92 fcn = str2func (fcn, "global");
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93 endif
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94
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95 xsiz = size (x0);
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96 n = numel (x0);
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97
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98 has_grad = strcmpi (optimget (options, "GradObj", "off"), "on");
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99 cdif = strcmpi (optimget (options, "FinDiffType", "central"), "central");
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100 maxiter = optimget (options, "MaxIter", 400);
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101 maxfev = optimget (options, "MaxFunEvals", Inf);
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102 outfcn = optimget (options, "OutputFcn");
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103
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104 ## Get scaling matrix using the TypicalX option. If set to "auto", the
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105 ## scaling matrix is estimated using the jacobian.
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106 typicalx = optimget (options, "TypicalX");
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107 if (isempty (typicalx))
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108 typicalx = ones (n, 1);
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109 endif
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110 autoscale = strcmpi (optimget (options, "AutoScaling", "off"), "on");
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111 if (! autoscale)
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112 dg = 1 ./ typicalx;
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113 endif
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114
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115 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on");
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116
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117 if (funvalchk)
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118 ## Replace fcn with a guarded version.
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119 fcn = @(x) guarded_eval (fcn, x);
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120 endif
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121
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122 ## These defaults are rather stringent. I think that normally, user
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123 ## prefers accuracy to performance.
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124
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125 macheps = eps (class (x0));
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126
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127 tolx = optimget (options, "TolX", 1e-7);
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128 tolf = optimget (options, "TolFun", 1e-7);
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129
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130 factor = 0.1;
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131 ## FIXME: TypicalX corresponds to user scaling (???)
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132 autodg = true;
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133
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134 niter = 1;
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135 nfev = 0;
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136
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137 x = x0(:);
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138 info = 0;
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139
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140 ## Initial evaluation.
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141 fval = fcn (reshape (x, xsiz));
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142 n = length (x);
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143
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144 if (! isempty (outfcn))
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145 optimvalues.iter = niter;
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146 optimvalues.funccount = nfev;
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147 optimvalues.fval = fval;
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148 optimvalues.searchdirection = zeros (n, 1);
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149 state = 'init';
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150 stop = outfcn (x, optimvalues, state);
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151 if (stop)
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152 info = -1;
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153 break;
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154 endif
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155 endif
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156
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157 nsuciter = 0;
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158 lastratio = 0;
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159
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160 grad = [];
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161
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162 ## Outer loop.
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163 while (niter < maxiter && nfev < maxfev && ! info)
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164
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165 grad0 = grad;
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166
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167 ## Calculate function value and gradient (possibly via FD).
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168 if (has_grad)
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169 [fval, grad] = fcn (reshape (x, xsiz));
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170 grad = grad(:);
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171 nfev ++;
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172 else
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173 grad = __fdjac__ (fcn, reshape (x, xsiz), fval, typicalx, cdif)(:);
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174 nfev += (1 + cdif) * length (x);
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175 endif
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176
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177 if (niter == 1)
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178 ## Initialize by identity matrix.
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179 hesr = eye (n);
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180 else
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181 ## Use the damped BFGS formula.
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182 y = grad - grad0;
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183 sBs = sumsq (w);
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184 Bs = hesr'*w;
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185 sy = y'*s;
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186 theta = 0.8 / max (1 - sy / sBs, 0.8);
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187 r = theta * y + (1-theta) * Bs;
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188 hesr = cholupdate (hesr, r / sqrt (s'*r), "+");
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189 [hesr, info] = cholupdate (hesr, Bs / sqrt (sBs), "-");
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190 if (info)
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191 hesr = eye (n);
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192 endif
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193 endif
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194
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195 if (autoscale)
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196 ## Second derivatives approximate the hessian.
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197 d2f = norm (hesr, 'columns').';
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198 if (niter == 1)
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199 dg = d2f;
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200 else
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201 ## FIXME: maybe fixed lower and upper bounds?
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202 dg = max (0.1*dg, d2f);
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203 endif
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204 endif
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205
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206 if (niter == 1)
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207 xn = norm (dg .* x);
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208 ## FIXME: something better?
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209 delta = factor * max (xn, 1);
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210 endif
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211
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212 ## FIXME -- why tolf*n*xn? If abs (e) ~ abs(x) * eps is a vector
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213 ## of perturbations of x, then norm (hesr*e) <= eps*xn, i.e. by
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214 ## tolf ~ eps we demand as much accuracy as we can expect.
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215 if (norm (grad) <= tolf*n*xn)
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216 info = 1;
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217 break;
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218 endif
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219
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220 suc = false;
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221 decfac = 0.5;
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222
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223 ## Inner loop.
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224 while (! suc && niter <= maxiter && nfev < maxfev && ! info)
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225
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226 s = - __doglegm__ (hesr, grad, dg, delta);
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227
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228 sn = norm (dg .* s);
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229 if (niter == 1)
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230 delta = min (delta, sn);
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231 endif
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232
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233 fval1 = fcn (reshape (x + s, xsiz)) (:);
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234 nfev ++;
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235
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236 if (fval1 < fval)
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237 ## Scaled actual reduction.
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238 actred = (fval - fval1) / (abs (fval1) + abs (fval));
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239 else
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240 actred = -1;
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241 endif
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242
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243 w = hesr*s;
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244 ## Scaled predicted reduction, and ratio.
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245 t = 1/2 * sumsq (w) + grad'*s;
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246 if (t < 0)
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247 prered = -t/(abs (fval) + abs (fval + t));
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248 ratio = actred / prered;
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249 else
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250 prered = 0;
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251 ratio = 0;
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252 endif
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253
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254 ## Update delta.
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255 if (ratio < min(max(0.1, 0.8*lastratio), 0.9))
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256 delta *= decfac;
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257 decfac ^= 1.4142;
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258 if (delta <= 1e1*macheps*xn)
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259 ## Trust region became uselessly small.
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260 info = -3;
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261 break;
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262 endif
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263 else
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264 lastratio = ratio;
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265 decfac = 0.5;
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266 if (abs (1-ratio) <= 0.1)
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267 delta = 1.4142*sn;
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268 elseif (ratio >= 0.5)
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269 delta = max (delta, 1.4142*sn);
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270 endif
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271 endif
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272
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273 if (ratio >= 1e-4)
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274 ## Successful iteration.
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275 x += s;
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276 xn = norm (dg .* x);
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277 fval = fval1;
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278 nsuciter ++;
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279 suc = true;
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280 endif
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281
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282 niter ++;
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283
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284 ## FIXME: should outputfcn be only called after a successful iteration?
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285 if (! isempty (outfcn))
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286 optimvalues.iter = niter;
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287 optimvalues.funccount = nfev;
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288 optimvalues.fval = fval;
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289 optimvalues.searchdirection = s;
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290 state = 'iter';
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291 stop = outfcn (x, optimvalues, state);
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292 if (stop)
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293 info = -1;
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294 break;
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295 endif
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296 endif
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297
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298 ## Tests for termination conditions. A mysterious place, anything
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299 ## can happen if you change something here...
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300
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301 ## The rule of thumb (which I'm not sure M*b is quite following)
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302 ## is that for a tolerance that depends on scaling, only 0 makes
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303 ## sense as a default value. But 0 usually means uselessly long
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304 ## iterations, so we need scaling-independent tolerances wherever
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305 ## possible.
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306
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307 ## The following tests done only after successful step.
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308 if (ratio >= 1e-4)
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309 ## This one is classic. Note that we use scaled variables again,
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310 ## but compare to scaled step, so nothing bad.
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311 if (sn <= tolx*xn)
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312 info = 2;
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313 ## Again a classic one.
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314 elseif (actred < tolf)
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315 info = 3;
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316 endif
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317 endif
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318
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319 endwhile
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320 endwhile
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321
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322 ## Restore original shapes.
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323 x = reshape (x, xsiz);
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324
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325 output.iterations = niter;
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326 output.successful = nsuciter;
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327 output.funcCount = nfev;
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328
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329 if (nargout > 5)
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330 hess = hesr'*hesr;
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331 endif
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332
9084
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333 endfunction
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334
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335 ## An assistant function that evaluates a function handle and checks for
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336 ## bad results.
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337 function [fx, gx] = guarded_eval (fun, x)
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338 if (nargout > 1)
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339 [fx, gx] = fun (x);
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340 else
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341 fx = fun (x);
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342 gx = [];
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343 endif
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344
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345 if (! (isreal (fx) && isreal (jx)))
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346 error ("fminunc:notreal", "fminunc: non-real value encountered");
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347 elseif (complexeqn && ! (isnumeric (fx) && isnumeric(jx)))
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348 error ("fminunc:notnum", "fminunc: non-numeric value encountered");
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349 elseif (any (isnan (fx(:))))
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350 error ("fminunc:isnan", "fminunc: NaN value encountered");
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351 endif
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352 endfunction
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353
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354 %!function f = rosenb (x)
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355 %! n = length (x);
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356 %! f = sumsq (1 - x(1:n-1)) + 100 * sumsq (x(2:n) - x(1:n-1).^2);
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357 %!test
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358 %! [x, fval, info, out] = fminunc (@rosenb, [5, -5]);
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359 %! tol = 2e-5;
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360 %! assert (info > 0);
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361 %! assert (x, ones (1, 2), tol);
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362 %! assert (fval, 0, tol);
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363 %!test
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364 %! [x, fval, info, out] = fminunc (@rosenb, zeros (1, 4));
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365 %! tol = 2e-5;
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366 %! assert (info > 0);
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367 %! assert (x, ones (1, 4), tol);
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368 %! assert (fval, 0, tol);
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369
9899
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370 ## Solve the double dogleg trust-region minimization problem:
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371 ## Minimize 1/2*norm(r*x)^2 subject to the constraint norm(d.*x) <= delta,
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372 ## x being a convex combination of the gauss-newton and scaled gradient.
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373
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374 ## TODO: error checks
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375 ## TODO: handle singularity, or leave it up to mldivide?
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376
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377 function x = __doglegm__ (r, g, d, delta)
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378 ## Get Gauss-Newton direction.
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379 b = r' \ g;
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380 x = r \ b;
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381 xn = norm (d .* x);
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382 if (xn > delta)
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383 ## GN is too big, get scaled gradient.
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384 s = g ./ d;
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385 sn = norm (s);
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386 if (sn > 0)
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387 ## Normalize and rescale.
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388 s = (s / sn) ./ d;
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389 ## Get the line minimizer in s direction.
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390 tn = norm (r*s);
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391 snm = (sn / tn) / tn;
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392 if (snm < delta)
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393 ## Get the dogleg path minimizer.
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394 bn = norm (b);
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395 dxn = delta/xn; snmd = snm/delta;
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396 t = (bn/sn) * (bn/xn) * snmd;
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397 t -= dxn * snmd^2 - sqrt ((t-dxn)^2 + (1-dxn^2)*(1-snmd^2));
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398 alpha = dxn*(1-snmd^2) / t;
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399 else
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400 alpha = 0;
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401 endif
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402 else
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John W. Eaton <jwe@octave.org>
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403 alpha = delta / xn;
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404 snm = 0;
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405 endif
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John W. Eaton <jwe@octave.org>
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406 ## Form the appropriate convex combination.
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407 x = alpha * x + ((1-alpha) * min (snm, delta)) * s;
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408 endif
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409 endfunction