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1 ## Copyright (C) 2005 John W. Eaton |
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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 2, or (at your option) |
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8 ## 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, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{iter}, @var{nf}, @var{lambda}] =} sqp (@var{x}, @var{phi}, @var{g}, @var{h}) |
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22 ## Solve the nonlinear program |
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23 ## @ifinfo |
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24 ## |
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25 ## @example |
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26 ## min phi (x) |
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27 ## x |
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28 ## @end example |
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29 ## |
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30 ## @end ifinfo |
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31 ## @iftex |
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32 ## @tex |
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33 ## @end tex |
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34 ## @end iftex |
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35 ## subject to |
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36 ## @ifinfo |
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37 ## |
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38 ## @example |
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39 ## g(x) = 0 |
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40 ## h(x) >= 0 |
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41 ## @end example |
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42 ## @end ifinfo |
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43 ## @iftex |
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44 ## @tex |
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45 ## @end tex |
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46 ## @end iftex |
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47 ## |
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48 ## @noindent |
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49 ## using a successive quadratic programming method. |
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50 ## |
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51 ## The first argument is the initial guess for the vector @var{x}. |
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52 ## |
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53 ## The second argument is a function handle pointing to the ojective |
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54 ## function. The objective function must be of the form |
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55 ## |
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56 ## @example |
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57 ## y = phi (x) |
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58 ## @end example |
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59 ## |
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60 ## @noindent |
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61 ## in which @var{x} is a vector and @var{y} is a scalar. |
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62 ## |
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63 ## The second argument may also be a 2- or 3-element cell array of |
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64 ## function handles. The first element should point to the objective |
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65 ## function, the second should point to a function that computes the |
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66 ## gradient of the objective function, and the third should point to a |
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67 ## function to compute the hessian of the objective function. If the |
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68 ## gradient function is not supplied, the gradient is computed by finite |
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69 ## differences. If the hessian function is not supplied, a BFGS update |
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70 ## formula is used to approximate the hessian. |
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71 ## |
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72 ## If supplied, the gradient function must be of the form |
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73 ## |
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74 ## @example |
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75 ## g = gradient (x) |
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76 ## @end example |
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77 ## |
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78 ## @noindent |
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79 ## in which @var{x} is a vector and @var{g} is a vector. |
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80 ## |
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81 ## If supplied, the hessian function must be of the form |
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82 ## |
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83 ## @example |
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84 ## h = hessian (x) |
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85 ## @end example |
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86 ## |
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87 ## @noindent |
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88 ## in which @var{x} is a vector and @var{h} is a matrix. |
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89 ## |
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90 ## The third and fourth arguments are function handles pointing to |
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91 ## functions that compute the equality constraints and the inequality |
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92 ## constraints, respectively. |
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93 ## |
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94 ## If your problem does not have equality (or inequality) constraints, |
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95 ## you may pass an empty matrix for @var{cef} (or @var{cif}). |
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96 ## |
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97 ## If supplied, the equality and inequality constraint functions must be |
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98 ## of the form |
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99 ## |
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100 ## @example |
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101 ## r = f (x) |
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102 ## @end example |
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103 ## |
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104 ## @noindent |
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105 ## in which @var{x} is a vector and @var{r} is a vector. |
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106 ## |
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107 ## The third and fourth arguments may also be 2-element cell arrays of |
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108 ## function handles. The first element should point to the constraint |
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109 ## function and the second should point to a function that computes the |
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110 ## gradient of the constraint function: |
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111 ## |
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112 ## @example |
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113 ## [ d f(x) d f(x) d f(x) ] |
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114 ## transpose ( [ ------ ----- ... ------ ] ) |
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115 ## [ dx_1 dx_2 dx_N ] |
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116 ## @end example |
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117 ## |
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118 ## Here is an example of calling @code{sqp}: |
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119 ## |
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120 ## @example |
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121 ## function r = g (x) |
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122 ## r = [ sumsq(x)-10; x(2)*x(3)-5*x(4)*x(5); x(1)^3+x(2)^3+1]; |
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123 ## endfunction |
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124 ## |
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125 ## function obj = phi (x) |
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126 ## obj = exp(prod(x)) - 0.5*(x(1)^3+x(2)^3+1)^2; |
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127 ## endfunction |
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128 ## |
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129 ## x0 = [-1.8; 1.7; 1.9; -0.8; -0.8]; |
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130 ## |
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131 ## [x, obj, info, iter, nf, lambda] = sqp (x0, @@phi, @@g, []) |
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132 ## |
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133 ## x = |
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134 ## |
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135 ## -1.71714 |
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136 ## 1.59571 |
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137 ## 1.82725 |
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138 ## -0.76364 |
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139 ## -0.76364 |
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140 ## |
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141 ## obj = 0.053950 |
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142 ## info = 101 |
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143 ## iter = 8 |
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144 ## nf = 10 |
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145 ## lambda = |
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146 ## |
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147 ## -0.0401627 |
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148 ## 0.0379578 |
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149 ## -0.0052227 |
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150 ## @end example |
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151 ## |
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152 ## The value returned in @var{info} may be one of the following: |
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153 ## @table @asis |
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154 ## @item 101 |
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155 ## The algorithm terminated because the norm of the last step was less |
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156 ## than @code{tol * norm (x))} (the value of tol is currently fixed at |
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157 ## @code{sqrt (eps)}---edit @file{sqp.m} to modify this value. |
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158 ## @item 102 |
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159 ## The BFGS update failed. |
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160 ## @item 103 |
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161 ## The maximum number of iterations was reached (the maximum number of |
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162 ## allowed iterations is currently fixed at 100---edit @file{sqp.m} to |
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163 ## increase this value). |
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164 ## @end table |
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165 ## @end deftypefn |
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166 ## @seealso{qp} |
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167 |
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168 function [x, obj, info, iter, nf, lambda] = sqp (x, objf, cef, cif) |
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169 |
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170 global nfun; |
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171 global __sqp_obj_fun__; |
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172 global __sqp_ce_fun__; |
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173 global __sqp_ci_fun__; |
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174 |
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175 if (nargin >= 2 && nargin <= 4) |
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176 |
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177 ## Choose an initial NxN symmetric positive definite Hessan |
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178 ## approximation B. |
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179 |
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180 n = length (x); |
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181 |
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182 B = eye (n, n); |
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183 |
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184 ## Evaluate objective function, constraints, and gradients at initial |
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185 ## value of x. |
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186 ## |
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187 ## obj_fun |
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188 ## obj_grad |
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189 ## ce_fun -- equality constraint functions |
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190 ## ci_fun -- inequality constraint functions |
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191 ## A == [grad_{x_1} cx_fun, grad_{x_2} cx_fun, ..., grad_{x_n} cx_fun]^T |
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192 |
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193 obj_grd = @fd_obj_grd; |
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194 have_hess = 0; |
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195 if (iscell (objf)) |
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196 if (length (objf) > 0) |
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197 __sqp_obj_fun__ = obj_fun = objf{1}; |
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198 if (length (objf) > 1) |
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199 obj_grd = objf{2}; |
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200 if (length (objf) > 2) |
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201 obj_hess = objf{3}; |
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202 have_hess = 1; |
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203 B = feval (obj_hess, x); |
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204 endif |
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205 endif |
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206 else |
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207 error ("sqp: invalid objective function"); |
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208 endif |
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209 else |
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210 __sqp_obj_fun__ = obj_fun = objf; |
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211 endif |
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212 |
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213 ce_fun = @empty_cf; |
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214 ce_grd = @empty_jac; |
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215 if (nargin > 2) |
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216 ce_grd = @fd_ce_jac; |
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217 if (iscell (cef)) |
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218 if (length (cef) > 0) |
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219 __sqp_ce_fun__ = ce_fun = cef{1}; |
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220 if (length (cef) > 1) |
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221 ce_grd = cef{2}; |
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222 endif |
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223 else |
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224 error ("sqp: invalid equality constraint function"); |
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225 endif |
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226 elseif (! isempty (cef)) |
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227 ce_fun = cef; |
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228 endif |
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229 endif |
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230 __sqp_ce_fun__ = ce_fun; |
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231 |
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232 ci_fun = @empty_cf; |
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233 ci_grd = @empty_jac; |
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234 if (nargin > 3) |
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235 ci_grd = @fd_ci_jac; |
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236 if (iscell (cif)) |
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237 if (length (cif) > 0) |
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238 __sqp_ci_fun__ = ci_fun = cif{1}; |
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239 if (length (cif) > 1) |
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240 ci_grd = cif{2}; |
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241 endif |
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242 else |
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243 error ("sqp: invalid equality constraint function"); |
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244 endif |
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245 elseif (! isempty (cif)) |
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246 ci_fun = cif; |
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247 endif |
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248 endif |
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249 __sqp_ci_fun__ = ci_fun; |
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250 |
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251 iter_max = 100; |
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252 |
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253 iter = 0; |
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254 |
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255 obj = feval (obj_fun, x); |
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256 nfun = 1; |
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257 |
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258 c = feval (obj_grd, x); |
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259 |
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260 ce = feval (ce_fun, x); |
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261 F = feval (ce_grd, x); |
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262 |
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263 ci = feval (ci_fun, x); |
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264 C = feval (ci_grd, x); |
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265 |
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266 A = [F; C]; |
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267 |
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268 ## Choose an initial lambda (x is provided by the caller). |
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269 |
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270 lambda = 100 * ones (rows (A), 1); |
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271 |
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272 qp_iter = 1; |
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273 alpha = 1; |
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274 |
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275 ## report (); |
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276 |
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277 ## report (iter, qp_iter, alpha, nfun, obj); |
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278 |
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279 while (++iter < iter_max) |
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280 |
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281 ## Check convergence. This is just a simple check on the first |
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282 ## order necessary conditions. |
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283 |
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284 ## IDX is the indices of the active inequality constraints. |
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285 |
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286 nr_f = rows (F); |
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287 |
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288 lambda_e = lambda((1:nr_f)'); |
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289 lambda_i = lambda((nr_f+1:end)'); |
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290 |
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291 con = [ce; ci]; |
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292 |
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293 t0 = norm (c - A' * lambda); |
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294 t1 = norm (ce); |
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295 t2 = all (ci >= 0); |
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296 t3 = all (lambda_i >= 0); |
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297 t4 = norm (lambda .* con); |
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298 |
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299 tol = sqrt (eps); |
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300 |
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301 if (t2 && t3 && max ([t0; t1; t4]) < tol) |
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302 break; |
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303 endif |
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304 |
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305 ## Compute search direction p by solving QP. |
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306 |
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307 g = -ce; |
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308 d = -ci; |
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309 |
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310 ## Discard inequality constraints that have -Inf bounds since those |
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311 ## will never be active. |
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312 idx = isinf (d) & d < 0; |
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313 d(idx) = []; |
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314 C(idx,:) = []; |
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315 |
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316 [p, obj_qp, INFO, lambda] = qp (x, B, c, F, g, [], [], d, C, |
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317 Inf * ones (size (d))); |
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318 |
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319 info = INFO.info; |
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320 |
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321 ## Check QP solution and attempt to recover if it has failed. |
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322 |
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323 ## Choose mu such that p is a descent direction for the chosen |
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324 ## merit function phi. |
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325 |
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326 [x_new, alpha, obj_new] = linesearch_L1 (x, p, obj_fun, obj_grd, |
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327 ce_fun, ci_fun, lambda, obj); |
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328 |
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329 ## Evaluate objective function, constraints, and gradients at |
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330 ## x_new. |
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331 |
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332 c_new = feval (obj_grd, x_new); |
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333 |
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334 ce_new = feval (ce_fun, x_new); |
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335 F_new = feval (ce_grd, x_new); |
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336 |
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337 ci_new = feval (ci_fun, x_new); |
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338 C_new = feval (ci_grd, x_new); |
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339 |
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340 A_new = [F_new; C_new]; |
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341 |
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342 ## Set |
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343 ## |
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344 ## s = alpha * p |
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345 ## y = grad_x L (x_new, lambda) - grad_x L (x, lambda}) |
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346 |
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347 y = c_new - c; |
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348 |
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349 if (! isempty (A)) |
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350 t = ((A_new - A)'*lambda); |
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351 y -= t; |
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352 endif |
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353 |
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354 delx = x_new - x; |
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355 |
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356 if (norm (delx) < tol * norm (x)) |
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357 info = 101; |
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358 break; |
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359 endif |
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360 |
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361 if (have_hess) |
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362 |
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363 B = feval (obj_hess, x); |
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364 |
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365 else |
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366 |
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367 ## Update B using a quasi-Newton formula. |
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368 |
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369 delxt = delx'; |
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370 |
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371 ## Damped BFGS. Or maybe we would actually want to use the Hessian |
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372 ## of the Lagrangian, computed directly. |
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373 |
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374 d1 = delxt*B*delx; |
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375 |
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376 t1 = 0.2 * d1; |
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377 t2 = delxt*y; |
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378 |
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379 if (t2 < t1) |
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380 theta = 0.8*d1/(d1 - t2); |
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381 else |
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382 theta = 1; |
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383 endif |
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384 |
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385 r = theta*y + (1-theta)*B*delx; |
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386 |
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387 d2 = delxt*r; |
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388 |
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389 if (d1 == 0 || d2 == 0) |
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390 info = 102; |
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391 break; |
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392 endif |
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393 |
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394 B = B - B*delx*delxt*B/d1 + r*r'/d2; |
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395 |
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396 endif |
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397 |
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398 x = x_new; |
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399 |
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400 obj = obj_new; |
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401 |
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402 c = c_new; |
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403 |
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404 ce = ce_new; |
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405 F = F_new; |
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406 |
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407 ci = ci_new; |
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408 C = C_new; |
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409 |
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410 A = A_new; |
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411 |
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412 ## report (iter, qp_iter, alpha, nfun, obj); |
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413 |
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414 endwhile |
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415 |
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416 if (iter >= iter_max) |
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417 info = 103; |
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418 endif |
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419 |
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420 nf = nfun; |
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421 |
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422 else |
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423 |
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424 usage ("[x, obj, info, iter, nf, lambda] = sqp (x, objf, cef, cif)"); |
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425 |
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426 endif |
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427 |
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428 ### endfunction |
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429 |
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430 |
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431 function [merit, obj] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, mu) |
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432 |
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433 global nfun; |
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434 |
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435 ce = feval (ce_fun, x); |
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436 ci = feval (ci_fun, x); |
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437 |
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438 idx = ci < 0; |
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439 |
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440 con = [ce; ci(idx)]; |
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441 |
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442 if (isempty (obj)) |
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443 obj = feval (obj_fun, x); |
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444 nfun++; |
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445 endif |
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446 |
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447 merit = obj; |
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448 t = norm (con, 1) / mu; |
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449 |
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450 if (! isempty (t)) |
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451 merit += t; |
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452 endif |
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453 |
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454 ### endfunction |
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455 |
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456 |
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457 function [x_new, alpha, obj] = linesearch_L1 (x, p, obj_fun, obj_grd, |
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458 ce_fun, ci_fun, lambda, obj) |
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459 |
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460 ## Choose parameters |
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461 ## |
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462 ## eta in the range (0, 0.5) |
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463 ## tau in the range (0, 1) |
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464 |
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465 eta = 0.25; |
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466 tau = 0.5; |
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467 |
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468 delta_bar = sqrt (eps); |
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469 |
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470 if (isempty (lambda)) |
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471 mu = 1 / delta_bar; |
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472 else |
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473 mu = 1 / (norm (lambda, Inf) + delta_bar); |
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474 endif |
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475 |
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476 alpha = 1; |
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477 |
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478 c = feval (obj_grd, x); |
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479 ce = feval (ce_fun, x); |
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480 |
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481 [phi_x_mu, obj] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, mu); |
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482 |
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483 D_phi_x_mu = c' * p; |
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484 d = feval (ci_fun, x); |
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485 ## only those elements of d corresponding |
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486 ## to violated constraints should be included. |
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487 idx = d < 0; |
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488 t = - norm ([ce; d(idx)], 1) / mu; |
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489 if (! isempty (t)) |
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490 D_phi_x_mu += t; |
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491 endif |
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492 |
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493 while (1) |
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494 [p1, obj] = phi_L1 ([], obj_fun, ce_fun, ci_fun, x+alpha*p, mu); |
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495 p2 = phi_x_mu+eta*alpha*D_phi_x_mu; |
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496 if (p1 > p2) |
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497 ## Reset alpha = tau_alpha * alpha for some tau_alpha in the |
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498 ## range (0, tau). |
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499 tau_alpha = 0.9 * tau; ## ?? |
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500 alpha = tau_alpha * alpha; |
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501 else |
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502 break; |
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503 endif |
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504 endwhile |
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505 |
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506 ## Set x_new = x + alpha * p; |
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507 |
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508 x_new = x + alpha * p; |
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509 |
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510 ### endfunction |
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511 |
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512 |
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513 function report (iter, qp_iter, alpha, nfun, obj) |
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|
514 |
|
|
515 if (nargin == 0) |
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516 printf (" Itn ItQP Step Nfun Objective\n"); |
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|
517 else |
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518 printf ("%5d %4d %8.1g %5d %13.6e\n", iter, qp_iter, alpha, nfun, obj); |
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|
519 endif |
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|
520 |
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|
521 ### endfunction |
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|
522 |
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523 |
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|
524 function grd = fdgrd (f, x) |
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|
525 |
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|
526 if (! isempty (f)) |
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|
527 y0 = feval (f, x); |
|
|
528 nx = length (x); |
|
|
529 grd = zeros (nx, 1); |
|
|
530 deltax = sqrt (eps); |
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531 for i = 1:nx |
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532 t = x(i); |
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|
533 x(i) += deltax; |
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534 grd(i) = (feval (f, x) - y0) / deltax; |
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535 x(i) = t; |
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536 endfor |
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537 else |
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|
538 grd = zeros (0, 1); |
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|
539 endif |
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|
540 |
|
|
541 ### endfunction |
|
|
542 |
|
|
543 |
|
|
544 function jac = fdjac (f, x) |
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|
545 |
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|
546 if (! isempty (f)) |
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|
547 y0 = feval (f, x); |
|
|
548 nf = length (y0); |
|
|
549 nx = length (x); |
|
|
550 jac = zeros (nf, nx); |
|
|
551 deltax = sqrt (eps); |
|
|
552 for i = 1:nx |
|
|
553 t = x(i); |
|
|
554 x(i) += deltax; |
|
|
555 jac(:,i) = (feval (f, x) - y0) / deltax; |
|
|
556 x(i) = t; |
|
|
557 endfor |
|
|
558 else |
|
|
559 jac = zeros (0, nx); |
|
|
560 endif |
|
|
561 |
|
|
562 ### endfunction |
|
|
563 |
|
|
564 |
|
|
565 function grd = fd_obj_grd (x) |
|
|
566 |
|
|
567 global __sqp_obj_fun__; |
|
|
568 |
|
|
569 grd = fdgrd (__sqp_obj_fun__, x); |
|
|
570 |
|
|
571 ### endfunction |
|
|
572 |
|
|
573 |
|
|
574 function res = empty_cf (x) |
|
|
575 |
|
|
576 res = zeros (0, 1); |
|
|
577 |
|
|
578 ### endfunction |
|
|
579 |
|
|
580 |
|
|
581 function res = empty_jac (x) |
|
|
582 |
|
|
583 res = zeros (0, length (x)); |
|
|
584 |
|
|
585 ### endfunction |
|
|
586 |
|
|
587 |
|
|
588 function jac = fd_ce_jac (x) |
|
|
589 |
|
|
590 global __sqp_ce_fun__; |
|
|
591 |
|
|
592 jac = fdjac (__sqp_ce_fun__, x); |
|
|
593 |
|
|
594 ### endfunction |
|
|
595 |
|
|
596 |
|
|
597 function jac = fd_ci_jac (x) |
|
|
598 |
|
|
599 global __sqp_ci_fun__; |
|
|
600 |
|
|
601 jac = fdjac (__sqp_ci_fun__, x); |
|
|
602 |
|
|
603 ### endfunction |
