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1 ## Copyright (C) 2005, 2006, 2007 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 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 ## -*- texinfo -*- |
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20 ## @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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21 ## Solve the nonlinear program |
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22 ## @iftex |
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23 ## @tex |
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24 ## $$ |
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25 ## \min_x \phi (x) |
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26 ## $$ |
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27 ## @end tex |
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28 ## @end iftex |
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29 ## @ifnottex |
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30 ## |
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31 ## @example |
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32 ## min phi (x) |
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33 ## x |
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34 ## @end example |
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35 ## |
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36 ## @end ifnottex |
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37 ## subject to |
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38 ## @iftex |
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39 ## @tex |
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40 ## $$ |
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41 ## g(x) = 0 \qquad h(x) \geq 0 |
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42 ## $$ |
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43 ## @end tex |
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44 ## @end iftex |
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45 ## @ifnottex |
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46 ## |
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47 ## @example |
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48 ## g(x) = 0 |
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49 ## h(x) >= 0 |
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50 ## @end example |
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51 ## @end ifnottex |
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52 ## |
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53 ## @noindent |
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54 ## using a successive quadratic programming method. |
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55 ## |
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56 ## The first argument is the initial guess for the vector @var{x}. |
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57 ## |
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58 ## The second argument is a function handle pointing to the objective |
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59 ## function. The objective function must be of the form |
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60 ## |
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61 ## @example |
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62 ## y = phi (x) |
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63 ## @end example |
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64 ## |
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65 ## @noindent |
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66 ## in which @var{x} is a vector and @var{y} is a scalar. |
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67 ## |
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68 ## The second argument may also be a 2- or 3-element cell array of |
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69 ## function handles. The first element should point to the objective |
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70 ## function, the second should point to a function that computes the |
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71 ## gradient of the objective function, and the third should point to a |
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72 ## function to compute the hessian of the objective function. If the |
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73 ## gradient function is not supplied, the gradient is computed by finite |
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74 ## differences. If the hessian function is not supplied, a BFGS update |
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75 ## formula is used to approximate the hessian. |
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76 ## |
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77 ## If supplied, the gradient function must be of the form |
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78 ## |
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79 ## @example |
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80 ## g = gradient (x) |
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81 ## @end example |
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82 ## |
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83 ## @noindent |
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84 ## in which @var{x} is a vector and @var{g} is a vector. |
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85 ## |
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86 ## If supplied, the hessian function must be of the form |
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87 ## |
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88 ## @example |
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89 ## h = hessian (x) |
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90 ## @end example |
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91 ## |
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92 ## @noindent |
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93 ## in which @var{x} is a vector and @var{h} is a matrix. |
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94 ## |
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95 ## The third and fourth arguments are function handles pointing to |
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96 ## functions that compute the equality constraints and the inequality |
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97 ## constraints, respectively. |
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98 ## |
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99 ## If your problem does not have equality (or inequality) constraints, |
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100 ## you may pass an empty matrix for @var{cef} (or @var{cif}). |
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101 ## |
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102 ## If supplied, the equality and inequality constraint functions must be |
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103 ## of the form |
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104 ## |
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105 ## @example |
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106 ## r = f (x) |
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107 ## @end example |
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108 ## |
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109 ## @noindent |
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110 ## in which @var{x} is a vector and @var{r} is a vector. |
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111 ## |
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112 ## The third and fourth arguments may also be 2-element cell arrays of |
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113 ## function handles. The first element should point to the constraint |
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114 ## function and the second should point to a function that computes the |
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115 ## gradient of the constraint function: |
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116 ## |
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117 ## @iftex |
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118 ## @tex |
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119 ## $$ |
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120 ## \Bigg( {\partial f(x) \over \partial x_1}, |
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121 ## {\partial f(x) \over \partial x_2}, \ldots, |
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122 ## {\partial f(x) \over \partial x_N} \Bigg)^T |
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123 ## $$ |
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124 ## @end tex |
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125 ## @end iftex |
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126 ## @ifnottex |
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127 ## @example |
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128 ## [ d f(x) d f(x) d f(x) ] |
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129 ## transpose ( [ ------ ----- ... ------ ] ) |
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130 ## [ dx_1 dx_2 dx_N ] |
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131 ## @end example |
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132 ## @end ifnottex |
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133 ## |
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134 ## Here is an example of calling @code{sqp}: |
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135 ## |
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136 ## @example |
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137 ## function r = g (x) |
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138 ## r = [ sumsq(x)-10; x(2)*x(3)-5*x(4)*x(5); x(1)^3+x(2)^3+1]; |
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139 ## endfunction |
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140 ## |
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141 ## function obj = phi (x) |
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142 ## obj = exp(prod(x)) - 0.5*(x(1)^3+x(2)^3+1)^2; |
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143 ## endfunction |
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144 ## |
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145 ## x0 = [-1.8; 1.7; 1.9; -0.8; -0.8]; |
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146 ## |
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147 ## [x, obj, info, iter, nf, lambda] = sqp (x0, @@phi, @@g, []) |
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148 ## |
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149 ## x = |
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150 ## |
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151 ## -1.71714 |
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152 ## 1.59571 |
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153 ## 1.82725 |
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154 ## -0.76364 |
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155 ## -0.76364 |
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156 ## |
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157 ## obj = 0.053950 |
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158 ## info = 101 |
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159 ## iter = 8 |
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160 ## nf = 10 |
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161 ## lambda = |
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162 ## |
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163 ## -0.0401627 |
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164 ## 0.0379578 |
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165 ## -0.0052227 |
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166 ## @end example |
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167 ## |
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168 ## The value returned in @var{info} may be one of the following: |
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169 ## @table @asis |
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170 ## @item 101 |
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171 ## The algorithm terminated because the norm of the last step was less |
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172 ## than @code{tol * norm (x))} (the value of tol is currently fixed at |
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173 ## @code{sqrt (eps)}---edit @file{sqp.m} to modify this value. |
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174 ## @item 102 |
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175 ## The BFGS update failed. |
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176 ## @item 103 |
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177 ## The maximum number of iterations was reached (the maximum number of |
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178 ## allowed iterations is currently fixed at 100---edit @file{sqp.m} to |
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179 ## increase this value). |
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180 ## @end table |
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181 ## @seealso{qp} |
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182 ## @end deftypefn |
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183 |
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184 function [x, obj, info, iter, nf, lambda] = sqp (x, objf, cef, cif, lb, ub, maxiter, tolerance) |
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185 |
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186 global __sqp_nfun__; |
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187 global __sqp_obj_fun__; |
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188 global __sqp_ce_fun__; |
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189 global __sqp_ci_fun__; |
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190 global __sqp_cif__; |
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191 global __sqp_cifcn__; |
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192 |
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193 if (nargin >= 2 && nargin <= 8 && nargin != 5) |
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194 |
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195 ## Choose an initial NxN symmetric positive definite Hessan |
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196 ## approximation B. |
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197 |
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198 n = length (x); |
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199 |
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200 ## Evaluate objective function, constraints, and gradients at initial |
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201 ## value of x. |
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202 ## |
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203 ## obj_fun |
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204 ## obj_grad |
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205 ## ce_fun -- equality constraint functions |
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206 ## ci_fun -- inequality constraint functions |
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207 ## A == [grad_{x_1} cx_fun, grad_{x_2} cx_fun, ..., grad_{x_n} cx_fun]^T |
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208 |
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209 obj_grd = @fd_obj_grd; |
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210 have_hess = 0; |
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211 if (iscell (objf)) |
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212 if (length (objf) > 0) |
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213 __sqp_obj_fun__ = obj_fun = objf{1}; |
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214 if (length (objf) > 1) |
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215 obj_grd = objf{2}; |
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216 if (length (objf) > 2) |
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217 obj_hess = objf{3}; |
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218 have_hess = 1; |
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219 endif |
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220 endif |
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221 else |
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222 error ("sqp: invalid objective function"); |
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223 endif |
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224 else |
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225 __sqp_obj_fun__ = obj_fun = objf; |
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226 endif |
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227 |
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228 ce_fun = @empty_cf; |
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229 ce_grd = @empty_jac; |
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230 if (nargin > 2) |
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231 ce_grd = @fd_ce_jac; |
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232 if (iscell (cef)) |
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233 if (length (cef) > 0) |
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234 __sqp_ce_fun__ = ce_fun = cef{1}; |
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235 if (length (cef) > 1) |
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236 ce_grd = cef{2}; |
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237 endif |
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238 else |
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239 error ("sqp: invalid equality constraint function"); |
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240 endif |
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241 elseif (! isempty (cef)) |
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242 ce_fun = cef; |
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243 endif |
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244 endif |
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245 __sqp_ce_fun__ = ce_fun; |
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246 |
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247 ci_fun = @empty_cf; |
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248 ci_grd = @empty_jac; |
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249 |
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250 if (nargin > 3) |
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251 ## constraint function given by user with possibly gradient |
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252 __sqp_cif__ = cif; |
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253 ## constraint function given by user without gradient |
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254 __sqp_cifcn__ = @empty_cf; |
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255 if (iscell (__sqp_cif__)) |
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256 if (length (__sqp_cif__) > 0) |
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257 __sqp_cifcn__ = __sqp_cif__{1}; |
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258 endif |
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259 elseif (! isempty (__sqp_cif__)) |
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260 __sqp_cifcn__ = __sqp_cif__; |
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261 endif |
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262 |
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263 if (nargin < 5) |
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264 ci_grd = @fd_ci_jac; |
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265 if (iscell (cif)) |
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266 if (length (cif) > 0) |
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267 __sqp_ci_fun__ = ci_fun = cif{1}; |
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268 if (length (cif) > 1) |
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269 ci_grd = cif{2}; |
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270 endif |
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271 else |
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272 error ("sqp: invalid equality constraint function"); |
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273 endif |
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274 elseif (! isempty (cif)) |
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275 ci_fun = cif; |
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276 endif |
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277 else |
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278 global __sqp_lb__; |
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279 if (isvector (lb)) |
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280 __sqp_lb__ = lb; |
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281 elseif (isempty (lb)) |
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282 __sqp_lb__ = -realmax; |
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283 else |
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284 error ("sqp: invalid lower bound"); |
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285 endif |
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286 |
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287 global __sqp_ub__; |
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288 if (isvector (ub)) |
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289 __sqp_ub__ = ub; |
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290 elseif (isempty (lb)) |
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291 __sqp_ub__ = realmax; |
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292 else |
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293 error ("sqp: invalid upper bound"); |
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294 endif |
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295 |
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296 if (lb > ub) |
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297 error ("sqp: upper bound smaller than lower bound"); |
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298 endif |
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299 __sqp_ci_fun__ = ci_fun = @cf_ub_lb; |
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300 ci_grd = @cigrad_ub_lb; |
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301 endif |
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302 __sqp_ci_fun__ = ci_fun; |
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303 endif |
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304 |
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305 iter_max = 100; |
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306 if (nargin > 6 && ! isempty (maxiter)) |
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307 if (isscalar (maxiter) && maxiter > 0 && round (maxiter) == maxiter) |
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308 iter_max = maxiter; |
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309 else |
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310 error ("sqp: invalid number of maximum iterations"); |
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311 endif |
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312 endif |
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313 |
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314 tol = sqrt (eps); |
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315 if (nargin > 7 && ! isempty (tolerance)) |
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316 if (isscalar (tolerance) && tolerance > 0) |
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317 tol = tolerance; |
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318 else |
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319 error ("sqp: invalid value for tolerance"); |
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320 endif |
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321 endif |
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322 |
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323 iter = 0; |
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324 |
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325 obj = feval (obj_fun, x); |
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326 __sqp_nfun__ = 1; |
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327 |
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328 c = feval (obj_grd, x); |
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329 |
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330 if (have_hess) |
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331 B = feval (obj_hess, x); |
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332 else |
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333 B = eye (n, n); |
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334 endif |
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335 |
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336 ce = feval (ce_fun, x); |
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337 F = feval (ce_grd, x); |
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338 |
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339 ci = feval (ci_fun, x); |
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340 C = feval (ci_grd, x); |
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341 |
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342 A = [F; C]; |
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343 |
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344 ## Choose an initial lambda (x is provided by the caller). |
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345 |
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346 lambda = 100 * ones (rows (A), 1); |
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347 |
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348 qp_iter = 1; |
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349 alpha = 1; |
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350 |
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351 ## report (); |
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352 |
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353 ## report (iter, qp_iter, alpha, __sqp_nfun__, obj); |
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354 |
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355 info = 0; |
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356 |
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357 while (++iter < iter_max) |
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358 |
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359 ## Check convergence. This is just a simple check on the first |
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360 ## order necessary conditions. |
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361 |
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362 ## IDX is the indices of the active inequality constraints. |
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363 |
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364 nr_f = rows (F); |
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365 |
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366 lambda_e = lambda((1:nr_f)'); |
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367 lambda_i = lambda((nr_f+1:end)'); |
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368 |
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369 con = [ce; ci]; |
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370 |
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371 t0 = norm (c - A' * lambda); |
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372 t1 = norm (ce); |
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373 t2 = all (ci >= 0); |
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374 t3 = all (lambda_i >= 0); |
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375 t4 = norm (lambda .* con); |
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376 |
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377 if (t2 && t3 && max ([t0; t1; t4]) < tol) |
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378 break; |
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379 endif |
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380 |
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381 ## Compute search direction p by solving QP. |
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382 |
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383 g = -ce; |
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384 d = -ci; |
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385 |
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386 ## Discard inequality constraints that have -Inf bounds since those |
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387 ## will never be active. |
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388 idx = isinf (d) & d < 0; |
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389 d(idx) = []; |
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390 C(idx,:) = []; |
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391 |
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392 [p, obj_qp, INFO, lambda] = qp (x, B, c, F, g, [], [], d, C, |
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393 Inf * ones (size (d))); |
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394 |
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395 info = INFO.info; |
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396 |
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397 ## Check QP solution and attempt to recover if it has failed. |
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398 |
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399 ## Choose mu such that p is a descent direction for the chosen |
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400 ## merit function phi. |
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401 |
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402 [x_new, alpha, obj_new] = linesearch_L1 (x, p, obj_fun, obj_grd, |
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403 ce_fun, ci_fun, lambda, obj); |
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404 |
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405 ## Evaluate objective function, constraints, and gradients at |
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406 ## x_new. |
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407 |
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408 c_new = feval (obj_grd, x_new); |
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409 |
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410 ce_new = feval (ce_fun, x_new); |
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411 F_new = feval (ce_grd, x_new); |
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412 |
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413 ci_new = feval (ci_fun, x_new); |
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414 C_new = feval (ci_grd, x_new); |
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415 |
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416 A_new = [F_new; C_new]; |
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417 |
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418 ## Set |
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419 ## |
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420 ## s = alpha * p |
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421 ## y = grad_x L (x_new, lambda) - grad_x L (x, lambda}) |
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422 |
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423 y = c_new - c; |
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424 |
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425 if (! isempty (A)) |
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426 t = ((A_new - A)'*lambda); |
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427 y -= t; |
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428 endif |
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429 |
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430 delx = x_new - x; |
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431 |
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432 if (norm (delx) < tol * norm (x)) |
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433 info = 101; |
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434 break; |
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435 endif |
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436 |
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437 if (have_hess) |
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438 |
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439 B = feval (obj_hess, x); |
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440 |
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441 else |
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442 |
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443 ## Update B using a quasi-Newton formula. |
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444 |
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445 delxt = delx'; |
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446 |
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447 ## Damped BFGS. Or maybe we would actually want to use the Hessian |
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448 ## of the Lagrangian, computed directly. |
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449 |
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450 d1 = delxt*B*delx; |
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451 |
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452 t1 = 0.2 * d1; |
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453 t2 = delxt*y; |
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454 |
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455 if (t2 < t1) |
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456 theta = 0.8*d1/(d1 - t2); |
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457 else |
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458 theta = 1; |
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459 endif |
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460 |
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461 r = theta*y + (1-theta)*B*delx; |
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462 |
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463 d2 = delxt*r; |
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464 |
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465 if (d1 == 0 || d2 == 0) |
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466 info = 102; |
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467 break; |
|
|
468 endif |
|
|
469 |
|
|
470 B = B - B*delx*delxt*B/d1 + r*r'/d2; |
|
|
471 |
|
|
472 endif |
|
|
473 |
|
|
474 x = x_new; |
|
|
475 |
|
|
476 obj = obj_new; |
|
|
477 |
|
|
478 c = c_new; |
|
|
479 |
|
|
480 ce = ce_new; |
|
|
481 F = F_new; |
|
|
482 |
|
|
483 ci = ci_new; |
|
|
484 C = C_new; |
|
|
485 |
|
|
486 A = A_new; |
|
|
487 |
|
6768
|
488 ## report (iter, qp_iter, alpha, __sqp_nfun__, obj); |
|
5289
|
489 |
|
|
490 endwhile |
|
|
491 |
|
|
492 if (iter >= iter_max) |
|
|
493 info = 103; |
|
|
494 endif |
|
|
495 |
|
6768
|
496 nf = __sqp_nfun__; |
|
5289
|
497 |
|
|
498 else |
|
|
499 |
|
6046
|
500 print_usage (); |
|
5289
|
501 |
|
|
502 endif |
|
|
503 |
|
|
504 ### endfunction |
|
|
505 |
|
|
506 |
|
|
507 function [merit, obj] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, mu) |
|
|
508 |
|
6768
|
509 global __sqp_nfun__; |
|
5289
|
510 |
|
|
511 ce = feval (ce_fun, x); |
|
|
512 ci = feval (ci_fun, x); |
|
|
513 |
|
|
514 idx = ci < 0; |
|
|
515 |
|
|
516 con = [ce; ci(idx)]; |
|
|
517 |
|
|
518 if (isempty (obj)) |
|
|
519 obj = feval (obj_fun, x); |
|
6768
|
520 __sqp_nfun__++; |
|
5289
|
521 endif |
|
|
522 |
|
|
523 merit = obj; |
|
|
524 t = norm (con, 1) / mu; |
|
|
525 |
|
|
526 if (! isempty (t)) |
|
|
527 merit += t; |
|
|
528 endif |
|
|
529 |
|
|
530 ### endfunction |
|
|
531 |
|
|
532 |
|
|
533 function [x_new, alpha, obj] = linesearch_L1 (x, p, obj_fun, obj_grd, |
|
|
534 ce_fun, ci_fun, lambda, obj) |
|
|
535 |
|
|
536 ## Choose parameters |
|
|
537 ## |
|
|
538 ## eta in the range (0, 0.5) |
|
|
539 ## tau in the range (0, 1) |
|
|
540 |
|
|
541 eta = 0.25; |
|
|
542 tau = 0.5; |
|
|
543 |
|
|
544 delta_bar = sqrt (eps); |
|
|
545 |
|
|
546 if (isempty (lambda)) |
|
|
547 mu = 1 / delta_bar; |
|
|
548 else |
|
|
549 mu = 1 / (norm (lambda, Inf) + delta_bar); |
|
|
550 endif |
|
|
551 |
|
|
552 alpha = 1; |
|
|
553 |
|
|
554 c = feval (obj_grd, x); |
|
|
555 ce = feval (ce_fun, x); |
|
|
556 |
|
|
557 [phi_x_mu, obj] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, mu); |
|
|
558 |
|
|
559 D_phi_x_mu = c' * p; |
|
|
560 d = feval (ci_fun, x); |
|
|
561 ## only those elements of d corresponding |
|
|
562 ## to violated constraints should be included. |
|
|
563 idx = d < 0; |
|
|
564 t = - norm ([ce; d(idx)], 1) / mu; |
|
|
565 if (! isempty (t)) |
|
|
566 D_phi_x_mu += t; |
|
|
567 endif |
|
|
568 |
|
|
569 while (1) |
|
|
570 [p1, obj] = phi_L1 ([], obj_fun, ce_fun, ci_fun, x+alpha*p, mu); |
|
|
571 p2 = phi_x_mu+eta*alpha*D_phi_x_mu; |
|
|
572 if (p1 > p2) |
|
|
573 ## Reset alpha = tau_alpha * alpha for some tau_alpha in the |
|
|
574 ## range (0, tau). |
|
|
575 tau_alpha = 0.9 * tau; ## ?? |
|
|
576 alpha = tau_alpha * alpha; |
|
|
577 else |
|
|
578 break; |
|
|
579 endif |
|
|
580 endwhile |
|
|
581 |
|
|
582 ## Set x_new = x + alpha * p; |
|
|
583 |
|
|
584 x_new = x + alpha * p; |
|
|
585 |
|
|
586 ### endfunction |
|
|
587 |
|
|
588 |
|
|
589 function report (iter, qp_iter, alpha, nfun, obj) |
|
|
590 |
|
|
591 if (nargin == 0) |
|
|
592 printf (" Itn ItQP Step Nfun Objective\n"); |
|
|
593 else |
|
|
594 printf ("%5d %4d %8.1g %5d %13.6e\n", iter, qp_iter, alpha, nfun, obj); |
|
|
595 endif |
|
|
596 |
|
|
597 ### endfunction |
|
|
598 |
|
|
599 |
|
|
600 function grd = fdgrd (f, x) |
|
|
601 |
|
|
602 if (! isempty (f)) |
|
|
603 y0 = feval (f, x); |
|
|
604 nx = length (x); |
|
|
605 grd = zeros (nx, 1); |
|
|
606 deltax = sqrt (eps); |
|
|
607 for i = 1:nx |
|
|
608 t = x(i); |
|
|
609 x(i) += deltax; |
|
|
610 grd(i) = (feval (f, x) - y0) / deltax; |
|
|
611 x(i) = t; |
|
|
612 endfor |
|
|
613 else |
|
|
614 grd = zeros (0, 1); |
|
|
615 endif |
|
|
616 |
|
|
617 ### endfunction |
|
|
618 |
|
|
619 |
|
|
620 function jac = fdjac (f, x) |
|
6768
|
621 nx = length (x); |
|
5289
|
622 if (! isempty (f)) |
|
|
623 y0 = feval (f, x); |
|
|
624 nf = length (y0); |
|
|
625 nx = length (x); |
|
|
626 jac = zeros (nf, nx); |
|
|
627 deltax = sqrt (eps); |
|
|
628 for i = 1:nx |
|
|
629 t = x(i); |
|
|
630 x(i) += deltax; |
|
|
631 jac(:,i) = (feval (f, x) - y0) / deltax; |
|
|
632 x(i) = t; |
|
|
633 endfor |
|
|
634 else |
|
|
635 jac = zeros (0, nx); |
|
|
636 endif |
|
|
637 |
|
|
638 ### endfunction |
|
|
639 |
|
|
640 |
|
|
641 function grd = fd_obj_grd (x) |
|
|
642 |
|
|
643 global __sqp_obj_fun__; |
|
|
644 |
|
|
645 grd = fdgrd (__sqp_obj_fun__, x); |
|
|
646 |
|
|
647 ### endfunction |
|
|
648 |
|
|
649 |
|
|
650 function res = empty_cf (x) |
|
|
651 |
|
|
652 res = zeros (0, 1); |
|
|
653 |
|
|
654 ### endfunction |
|
|
655 |
|
|
656 |
|
|
657 function res = empty_jac (x) |
|
|
658 |
|
|
659 res = zeros (0, length (x)); |
|
|
660 |
|
|
661 ### endfunction |
|
|
662 |
|
|
663 |
|
|
664 function jac = fd_ce_jac (x) |
|
|
665 |
|
|
666 global __sqp_ce_fun__; |
|
|
667 |
|
|
668 jac = fdjac (__sqp_ce_fun__, x); |
|
|
669 |
|
|
670 ### endfunction |
|
|
671 |
|
|
672 |
|
|
673 function jac = fd_ci_jac (x) |
|
|
674 |
|
6768
|
675 global __sqp_cifcn__; |
|
|
676 ## __sqp_cifcn__ = constraint function without gradients and lb or ub |
|
|
677 jac = fdjac (__sqp_cifcn__, x); |
|
|
678 |
|
|
679 ### endfunction |
|
|
680 |
|
7017
|
681 |
|
6768
|
682 function res = cf_ub_lb (x) |
|
5289
|
683 |
|
6768
|
684 ## combine constraint function with ub and lb |
|
|
685 global __sqp_cifcn__ __sqp_lb__ __sqp_ub__ |
|
|
686 |
|
|
687 res = [x-__sqp_lb__; __sqp_ub__-x]; |
|
|
688 |
|
|
689 if (! isempty (__sqp_cifcn__)) |
|
|
690 res = [feval(__sqp_cifcn__,x); x-__sqp_lb__; __sqp_ub__-x]; |
|
|
691 endif |
|
5289
|
692 |
|
|
693 ### endfunction |
|
6768
|
694 |
|
7017
|
695 |
|
6768
|
696 function res = cigrad_ub_lb (x) |
|
|
697 |
|
|
698 global __sqp_cif__ |
|
|
699 |
|
|
700 res = [eye(numel(x)); -eye(numel(x))]; |
|
|
701 |
|
|
702 cigradfcn = @fd_ci_jac; |
|
|
703 |
|
|
704 if (iscell (__sqp_cif__) && length (__sqp_cif__) > 1) |
|
|
705 cigradfcn = __sqp_cif__{2}; |
|
|
706 endif |
|
|
707 |
|
|
708 if (! isempty (cigradfcn)) |
|
|
709 res = [feval(cigradfcn,x); eye(numel(x)); -eye(numel(x))]; |
|
|
710 endif |
|
|
711 |
|
7001
|
712 ### endfunction |
