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1 ## Copyright (C) 2000, 2001, 2004, 2005 Gabriele Pannocchia. |
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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{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, @var{A_in}, @var{A_ub}) |
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21 ## Solve the quadratic program |
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22 ## @iftex |
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23 ## @tex |
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24 ## $$ |
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25 ## \min_x {1 \over 2} x^T H x + x^T q |
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26 ## $$ |
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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 0.5 x'*H*x + x'*q |
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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 ## $$ |
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41 ## Ax = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} \leq A_{ub} |
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42 ## $$ |
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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 ## A*x = b |
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49 ## lb <= x <= ub |
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50 ## A_lb <= A_in*x <= A_ub |
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51 ## @end example |
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52 ## @end ifnottex |
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53 ## |
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54 ## @noindent |
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55 ## using a null-space active-set method. |
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56 ## |
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57 ## Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, |
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58 ## @var{A_ub}) may be set to the empty matrix (@code{[]}) if not |
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59 ## present. If the initial guess is feasible the algorithm is faster. |
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60 ## |
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61 ## The value @var{info} is a structure with the following fields: |
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62 ## @table @code |
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63 ## @item solveiter |
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64 ## The number of iterations required to find the solution. |
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65 ## @item info |
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66 ## An integer indicating the status of the solution, as follows: |
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67 ## @table @asis |
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68 ## @item 0 |
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69 ## The problem is feasible and convex. Global solution found. |
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70 ## @item 1 |
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71 ## The problem is not convex. Local solution found. |
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72 ## @item 2 |
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73 ## The problem is not convex and unbounded. |
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74 ## @item 3 |
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75 ## Maximum number of iterations reached. |
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76 ## @item 6 |
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77 ## The problem is infeasible. |
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78 ## @end table |
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79 ## @end table |
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80 ## @end deftypefn |
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81 |
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82 function [x, obj, INFO, lambda] = qp (x0, H, q, A, b, lb, ub, A_lb, A_in, A_ub) |
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83 |
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84 if (nargin == 5 || nargin == 7 || nargin == 10) |
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85 |
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86 ## Checking the quadratic penalty |
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87 n = issquare (H); |
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88 if (n == 0) |
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89 error ("qp: quadratic penalty matrix not square"); |
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90 endif |
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91 |
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92 n1 = issymmetric (H); |
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93 if (n1 == 0) |
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94 ## warning ("qp: quadratic penalty matrix not symmetric"); |
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95 H = (H + H')/2; |
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96 endif |
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97 |
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98 ## Checking the initial guess (if empty it is resized to the |
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99 ## right dimension and filled with 0) |
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100 if (isempty (x0)) |
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101 x0 = zeros (n, 1); |
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102 elseif (length (x0) != n) |
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103 error ("qp: the initial guess has incorrect length"); |
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104 endif |
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105 |
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106 ## Linear penalty. |
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107 if (length (q) != n) |
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108 error ("qp: the linear term has incorrect length"); |
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109 endif |
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110 |
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111 ## Equality constraint matrices |
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112 if (isempty (A) || isempty(b)) |
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113 n_eq = 0; |
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114 A = zeros (n_eq, n); |
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115 b = zeros (n_eq, 1); |
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116 else |
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117 [n_eq, n1] = size (A); |
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118 if (n1 != n) |
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119 error ("qp: equality constraint matrix has incorrect column dimension"); |
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120 endif |
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121 if (length (b) != n_eq) |
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122 error ("qp: equality constraint matrix and vector have inconsistent dimension"); |
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123 endif |
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124 endif |
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125 |
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126 ## Bound constraints |
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127 Ain = zeros (0, n); |
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128 bin = zeros (0, 1); |
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129 n_in = 0; |
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130 if (nargin > 5) |
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131 if (! isempty (lb)) |
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132 if (length(lb) != n) |
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133 error ("qp: lower bound has incorrect length"); |
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134 else |
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135 Ain = [Ain; eye(n)]; |
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136 bin = [bin; lb]; |
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137 endif |
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138 endif |
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139 |
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140 if (! isempty (ub)) |
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141 if (length (ub) != n) |
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142 error ("qp: upper bound has incorrect length"); |
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143 else |
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144 Ain = [Ain; -eye(n)]; |
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145 bin = [bin; -ub]; |
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146 endif |
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147 endif |
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148 endif |
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149 |
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150 ## Inequality constraints |
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151 if (nargin > 7) |
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152 [dimA_in, n1] = size (A_in); |
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153 if (n1 != n) |
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154 error ("qp: inequality constraint matrix has incorrect column dimension"); |
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155 else |
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156 if (! isempty (A_lb)) |
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157 if (length (A_lb) != dimA_in) |
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158 error ("qp: inequality constraint matrix and lower bound vector inconsistent"); |
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159 else |
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160 Ain = [Ain; A_in]; |
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161 bin = [bin; A_lb]; |
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162 endif |
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163 endif |
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164 if (! isempty (A_ub)) |
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165 if (length (A_ub) != dimA_in) |
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166 error ("qp: inequality constraint matrix and upper bound vector inconsistent"); |
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167 else |
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168 Ain = [Ain; -A_in]; |
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169 bin = [bin; -A_ub]; |
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170 endif |
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171 endif |
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172 endif |
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173 endif |
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174 |
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175 ## Now we should have the following QP: |
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176 ## |
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177 ## min_x 0.5*x'*H*x + x'*q |
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178 ## s.t. A*x = b |
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179 ## Ain*x >= bin |
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180 |
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181 ## Discard inequality constraints that have -Inf bounds since those |
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182 ## will never be active. |
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183 idx = isinf (bin) & bin < 0; |
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184 |
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185 bin(idx) = []; |
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186 Ain(idx,:) = []; |
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187 |
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188 n_in = length (bin); |
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189 |
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190 ## Check if the initial guess is feasible. |
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191 rtol = sqrt (eps); |
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192 |
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193 eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+norm (b))); |
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194 in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+norm (bin)))); |
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195 |
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196 info = 0; |
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197 if (eq_infeasible || in_infeasible) |
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198 ## The initial guess is not feasible. |
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199 ## First define xbar that is feasible with respect to the equality |
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200 ## constraints. |
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201 if (eq_infeasible) |
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202 if (rank (A) < n_eq) |
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203 error ("qp: equality constraint matrix must be full row rank") |
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204 endif |
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205 xbar = pinv (A) * b; |
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206 else |
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207 xbar = x0; |
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208 endif |
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209 |
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210 ## Check if xbar is feasible with respect to the inequality |
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211 ## constraints also. |
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212 if (n_in > 0) |
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213 res = Ain * xbar - bin; |
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214 if (any (res < -rtol * (1 + norm (bin)))) |
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215 ## xbar is not feasible with respect to the inequality |
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216 ## constraints. Compute a step in the null space of the |
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217 ## equality constraints, by solving a QP. If the slack is |
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218 ## small, we have a feasible initial guess. Otherwise, the |
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219 ## problem is infeasible. |
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220 if (n_eq > 0) |
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221 Z = null (A); |
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222 if (isempty (Z)) |
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223 ## The problem is infeasible because A is square and full |
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224 ## rank, but xbar is not feasible. |
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225 info = 6; |
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226 endif |
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227 endif |
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228 |
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229 if (info != 6) |
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230 ## Solve an LP with additional slack variables to find |
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231 ## a feasible starting point. |
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232 gamma = eye (n_in); |
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233 if (n_eq > 0) |
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234 Atmp = [Ain*Z, gamma]; |
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235 btmp = -res; |
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236 else |
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237 Atmp = [Ain, gamma]; |
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238 btmp = bin; |
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239 endif |
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240 ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; |
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241 lb = [-Inf*ones(n-n_eq,1); zeros(n_in,1)]; |
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242 ub = []; |
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243 ctype = repmat ("L", n_in, 1); |
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244 [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); |
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245 if ((status == 180 || status == 181 || status == 151) |
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246 && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) |
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247 ## We found a feasible starting point |
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248 if (n_eq > 0) |
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249 x0 = xbar + Z*P(1:n-n_eq); |
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250 else |
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251 x0 = P(1:n); |
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252 endif |
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253 else |
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254 ## The problem is infeasible |
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255 info = 6; |
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256 endif |
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257 endif |
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258 else |
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259 ## xbar is feasible. We use it a starting point. |
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260 x0 = xbar; |
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261 endif |
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262 else |
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263 ## xbar is feasible. We use it a starting point. |
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264 x0 = xbar; |
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265 endif |
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266 endif |
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267 |
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268 if (info == 0) |
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269 ## FIXME -- make maxit a user option. |
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270 ## The initial (or computed) guess is feasible. |
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271 ## We call the solver. |
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272 maxit = 200; |
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273 [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); |
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274 else |
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275 iter = 0; |
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276 x = x0; |
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277 lambda = []; |
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278 endif |
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279 obj = 0.5 * x' * H * x + q' * x; |
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280 INFO.solveiter = iter; |
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281 INFO.info = info; |
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282 |
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283 else |
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284 print_usage (); |
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285 endif |
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286 |
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287 endfunction |
