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1 /* |
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2 |
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3 Copyright (C) 2000, 2001, 2004, 2005 Gabriele Pannocchia |
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4 |
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5 This file is part of Octave. |
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6 |
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7 Octave is free software; you can redistribute it and/or modify it |
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8 under the terms of the GNU General Public License as published by the |
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9 Free Software Foundation; either version 2, or (at your option) any |
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10 later version. |
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11 |
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12 Octave is distributed in the hope that it will be useful, but WITHOUT |
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13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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15 for more details. |
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16 |
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17 You should have received a copy of the GNU General Public License |
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18 along with Octave; see the file COPYING. If not, write to the Free |
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19 Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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20 02110-1301, USA. |
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21 |
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22 */ |
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23 |
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24 #ifdef HAVE_CONFIG_H |
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25 #include <config.h> |
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26 #endif |
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27 |
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28 #include <cfloat> |
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29 |
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30 #include "dbleCHOL.h" |
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31 #include "dbleSVD.h" |
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32 #include "mx-m-dm.h" |
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33 #include "EIG.h" |
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34 |
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35 #include "defun-dld.h" |
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36 #include "error.h" |
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37 #include "gripes.h" |
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38 #include "oct-obj.h" |
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39 #include "pr-output.h" |
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40 #include "utils.h" |
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41 |
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42 static Matrix |
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43 null (const Matrix& A, octave_idx_type& rank) |
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44 { |
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45 Matrix retval; |
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46 |
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47 rank = 0; |
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48 |
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49 if (! A.is_empty ()) |
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50 { |
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51 SVD A_svd (A); |
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52 |
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53 DiagMatrix S = A_svd.singular_values (); |
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54 |
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55 ColumnVector s = S.diag (); |
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56 |
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57 Matrix V = A_svd.right_singular_matrix (); |
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58 |
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59 octave_idx_type A_nr = A.rows (); |
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60 octave_idx_type A_nc = A.cols (); |
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61 |
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62 octave_idx_type tmp = A_nr > A_nc ? A_nr : A_nc; |
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63 |
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64 double tol = tmp * s(0) * DBL_EPSILON; |
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65 |
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66 octave_idx_type n = s.length (); |
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67 |
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68 for (octave_idx_type i = 0; i < n; i++) |
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69 { |
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70 if (s(i) > tol) |
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71 rank++; |
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72 } |
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73 |
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74 if (rank < A_nc) |
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75 retval = V.extract (0, rank, A_nc-1, A_nc-1); |
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76 else |
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77 retval.resize (A_nc, 0); |
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78 |
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79 for (octave_idx_type i = 0; i < retval.numel (); i++) |
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80 if (std::abs (retval(i)) < DBL_EPSILON) |
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81 retval(i) = 0; |
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82 } |
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83 |
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84 return retval; |
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85 } |
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86 |
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87 static int |
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88 qp (const Matrix& H, const ColumnVector& q, |
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89 const Matrix& Aeq, const ColumnVector& beq, |
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90 const Matrix& Ain, const ColumnVector& bin, |
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91 int maxit, |
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92 ColumnVector& x, ColumnVector& lambda, int& iter) |
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93 { |
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94 int info = 0; |
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95 |
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96 iter = 0; |
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97 |
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98 double rtol = sqrt (DBL_EPSILON); |
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99 |
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100 // Problem dimension. |
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101 octave_idx_type n = x.length (); |
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102 |
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103 // Dimension of constraints. |
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104 octave_idx_type n_eq = beq.length (); |
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105 octave_idx_type n_in = bin.length (); |
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106 |
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107 // Filling the current active set. |
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108 |
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109 octave_idx_type n_act = n_eq; |
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110 |
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111 octave_idx_type n_tot = n_eq + n_in; |
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112 |
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113 // Equality constraints come first. We won't check the sign of the |
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114 // Lagrange multiplier for those. |
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115 |
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116 Matrix Aact = Aeq; |
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117 ColumnVector bact = beq; |
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118 ColumnVector Wact; |
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119 |
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120 if (n_in > 0) |
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121 { |
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122 ColumnVector res = Ain*x - bin; |
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123 |
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124 for (octave_idx_type i = 0; i < n_in; i++) |
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125 { |
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126 res(i) /= (1.0 + std::abs (bin(i))); |
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127 |
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128 if (res(i) < rtol) |
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129 { |
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130 n_act++; |
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131 Aact = Aact.stack (Ain.row (i)); |
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132 bact.resize (n_act, bin(i)); |
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133 Wact.resize (n_act-n_eq, i); |
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134 } |
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135 } |
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136 } |
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137 |
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138 // Computing the ??? |
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139 |
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140 EIG eigH (H); |
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141 ColumnVector eigenvalH = real (eigH.eigenvalues ()); |
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142 Matrix eigenvecH = real (eigH.eigenvectors ()); |
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143 double minReal = eigenvalH.min (); |
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144 octave_idx_type indminR = 0; |
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145 for (octave_idx_type i = 0; i < n; i++) |
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146 { |
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147 if (minReal == eigenvalH(i)) |
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148 { |
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149 indminR = i; |
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150 break; |
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151 } |
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152 } |
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153 |
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154 bool done = false; |
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155 |
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156 double alpha = 0.0; |
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157 |
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158 Matrix R; |
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159 Matrix Y (n, 0, 0.0); |
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160 |
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161 ColumnVector g (n, 0.0); |
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162 ColumnVector p (n, 0.0); |
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163 |
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164 ColumnVector lambda_tmp (n_in, 0.0); |
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165 |
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166 while (! done) |
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167 { |
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168 iter++; |
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169 |
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170 // Current Gradient |
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171 // g = q + H * x; |
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172 |
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173 g = q + H * x; |
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174 |
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175 if (n_act == 0) |
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176 { |
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177 // There are no active constraints. |
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178 |
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179 if (minReal > 0.0) |
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180 { |
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181 // Inverting the Hessian. Using the Cholesky |
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182 // factorization since the Hessian is positive |
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183 // definite. |
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184 |
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185 CHOL cholH (H); |
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186 |
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187 R = cholH.chol_matrix (); |
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188 |
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189 Matrix Hinv = chol2inv (R); |
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190 |
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191 // Computing the unconstrained step. |
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192 // p = -Hinv * g; |
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193 |
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194 p = -Hinv * g; |
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195 |
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196 info = 0; |
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197 } |
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198 else |
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199 { |
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200 // Finding the negative curvature of H. |
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201 |
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202 p = eigenvecH.column (indminR); |
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203 |
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204 // Following the negative curvature of H. |
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205 |
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206 if (p.transpose () * g > DBL_EPSILON) |
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207 p = -p; |
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208 |
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209 info = 1; |
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210 } |
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211 |
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212 // Multipliers are zero. |
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213 lambda_tmp.fill (0.0); |
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214 } |
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215 else |
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216 { |
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217 // There are active constraints. |
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218 |
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219 // Computing the null space. |
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220 |
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221 octave_idx_type rank; |
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222 |
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223 Matrix Z = null (Aact, rank); |
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224 |
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225 octave_idx_type dimZ = n - rank; |
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226 |
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227 // FIXME -- still remain to handle the case of |
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228 // non-full rank active set matrix. |
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229 |
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230 // Reduced Hessian |
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231 Matrix Zt = Z.transpose (); |
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232 Matrix rH = Zt * H * Z; |
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233 |
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234 octave_idx_type pR = 0; |
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235 |
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236 if (dimZ > 0) |
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237 { |
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238 // Computing the Cholesky factorization (pR = 0 means |
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239 // that the reduced Hessian was positive definite). |
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240 |
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241 CHOL cholrH (rH, pR); |
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242 Matrix tR = cholrH.chol_matrix (); |
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243 if (pR == 0) |
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244 R = tR; |
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245 } |
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246 |
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247 if (pR == 0) |
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248 { |
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249 info = 0; |
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250 |
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251 // Computing the step pz. |
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252 if (dimZ > 0) |
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253 { |
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254 // Using the Cholesky factorization to invert rH |
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255 |
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256 Matrix rHinv = chol2inv (R); |
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257 |
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258 ColumnVector pz = -rHinv * Zt * g; |
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259 |
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260 // Global step. |
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261 p = Z * pz; |
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262 } |
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263 else |
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264 { |
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265 // Global step. |
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266 p.fill (0.0); |
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267 } |
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268 } |
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269 else |
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270 { |
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271 info = 1; |
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272 |
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273 // Searching for the most negative curvature. |
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274 |
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275 EIG eigrH (rH); |
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276 ColumnVector eigenvalrH = real (eigrH.eigenvalues ()); |
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277 Matrix eigenvecrH = real (eigrH.eigenvectors ()); |
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278 double mRrH = eigenvalrH.min (); |
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279 indminR = 0; |
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280 for (octave_idx_type i = 0; i < n; i++) |
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281 { |
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282 if (mRrH == eigenvalH(i)) |
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283 { |
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284 indminR = i; |
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285 break; |
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286 } |
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287 } |
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288 |
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289 ColumnVector eVrH = eigenvecrH.column (indminR); |
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290 |
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291 // Computing the step pz. |
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292 p = Z * eVrH; |
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293 |
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294 if (p.transpose () * g > DBL_EPSILON) |
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295 p = -p; |
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296 } |
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297 } |
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298 |
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299 // Checking the step-size. |
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300 ColumnVector abs_p (n); |
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301 for (octave_idx_type i = 0; i < n; i++) |
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302 abs_p(i) = std::abs (p(i)); |
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303 double max_p = abs_p.max (); |
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304 |
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305 if (max_p < rtol) |
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306 { |
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307 // The step is null. Checking constraints. |
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308 if (n_act - n_eq == 0) |
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309 // Solution is found because no inequality |
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310 // constraints are active. |
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311 done = true; |
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312 else |
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313 { |
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314 // Computing the multipliers only for the inequality |
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315 // constraints that are active. We do NOT compute |
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316 // multipliers for the equality constraints. |
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317 Y = Aact.pseudo_inverse (); |
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318 Matrix Yt = Y.transpose (); |
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319 Yt = Yt.extract_n (n_eq, 0, n_act-n_eq, n); |
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320 lambda_tmp = Yt * (g + H * p); |
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321 |
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322 // Checking the multipliers. We remove the most |
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323 // negative from the set (if any). |
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324 double min_lambda = lambda_tmp.min (); |
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325 if (min_lambda >= 0) |
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326 { |
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327 // Solution is found. |
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328 done = true; |
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329 } |
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330 else |
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331 { |
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332 octave_idx_type which_eig = 0; |
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333 for (octave_idx_type i = 0; i < n_act; i++) |
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334 { |
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335 if (lambda_tmp(i) == min_lambda) |
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336 { |
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337 which_eig = i; |
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338 break; |
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339 } |
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340 } |
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341 |
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342 // At least one multiplier is negative, we |
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343 // remove it from the set. |
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344 |
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345 for (octave_idx_type i = which_eig; i < n_act - n_eq; i++) |
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346 { |
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347 Wact(i) = Wact(i+1); |
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348 for (octave_idx_type j = 0; j < n; j++) |
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349 Aact(n_eq+i,j) = Aact(n_eq+i+1,j); |
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350 bact(n_eq+i) = bact(n_eq+i+1); |
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351 } |
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352 n_act--; |
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353 |
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354 // Resizing the active set. |
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355 Wact.resize (n_act-n_eq); |
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356 bact.resize (n_act); |
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357 Aact.resize (n_act, n); |
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358 } |
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359 } |
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360 } |
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361 else |
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362 { |
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363 // The step is not null. |
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364 if (n_act - n_eq == n_in) |
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365 { |
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366 // All inequality constraints were active. We can |
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367 // add the whole step. |
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368 x += p; |
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369 } |
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370 else |
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371 { |
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372 // Some constraints were not active. Checking if |
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373 // there is a blocking constraint. |
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374 alpha = 1.0; |
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375 octave_idx_type is_block = -1; |
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376 |
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377 for (octave_idx_type i = 0; i < n_in; i++) |
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378 { |
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379 bool found = false; |
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380 |
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381 for (octave_idx_type j = 0; j < n_act-n_eq; j++) |
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382 { |
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383 if (Wact(j) == i - n_eq) |
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384 { |
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385 found = true; |
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386 break; |
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387 } |
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388 } |
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389 |
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390 if (! found) |
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391 { |
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392 // The i-th constraint was not in the set. Is it a |
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393 // blocking constraint? |
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394 |
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395 RowVector tmp_row = Ain.row (i); |
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396 double tmp = tmp_row * p; |
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397 double res = tmp_row * x; |
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398 |
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399 if (tmp < 0.0) |
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400 { |
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401 double alpha_tmp = (bin(i) - res) / tmp; |
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402 |
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403 if (alpha_tmp < alpha) |
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404 { |
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405 alpha = alpha_tmp; |
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406 is_block = i; |
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407 } |
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408 } |
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409 } |
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410 } |
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411 |
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412 // In is_block there is the index of the blocking |
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413 // constraint (if any). |
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414 if (is_block >= 0) |
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415 { |
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416 // There is a blocking constraint (index in |
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417 // is_block) which is added to the active set. |
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418 n_act++; |
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419 Aact = Aact.stack (Ain.row (is_block)); |
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420 bact.resize (n_act, bin(is_block)); |
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421 Wact.resize (n_act-n_eq, is_block); |
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422 |
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423 // Adding the reduced step |
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424 x += alpha * p; |
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425 } |
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426 else |
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427 { |
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428 // There are no blocking constraints. Adding the |
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429 // whole step. |
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430 x += alpha * p; |
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431 } |
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432 } |
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433 } |
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434 |
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435 if (iter == maxit) |
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436 { |
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437 done = true; |
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438 // warning ("qp_main: maximum number of iteration reached"); |
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439 info = 3; |
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440 } |
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441 } |
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442 |
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443 lambda_tmp = Y.transpose () * (g + H * p); |
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444 |
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445 // Reordering the Lagrange multipliers. |
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446 |
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447 lambda.resize (n_tot); |
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448 lambda.fill (0.0); |
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449 for (octave_idx_type i = 0; i < n_eq; i++) |
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450 lambda(i) = lambda_tmp(i); |
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451 |
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452 for (octave_idx_type i = n_eq; i < n_tot; i++) |
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453 { |
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454 for (octave_idx_type j = 0; j < n_act-n_eq; j++) |
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455 { |
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456 if (Wact(j) == i) |
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457 { |
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458 lambda(i) = lambda_tmp(n_eq+j); |
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459 break; |
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460 } |
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461 } |
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462 } |
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463 |
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464 return info; |
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465 } |
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466 |
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467 DEFUN_DLD (__qp__, args, , |
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468 "-*- texinfo -*-\n\ |
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469 @deftypefn {Loadable Function} {[@var{x}, @var{lambda}, @var{info}, @var{iter}] =} __qp__ (@var{x0}, @var{H}, @var{q}, @var{Aeq}, @var{beq}, @var{Ain}, @var{bin}, @var{maxit})\n\ |
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470 Undocumented internal function.\n\ |
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471 @end deftypefn") |
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472 { |
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473 octave_value_list retval; |
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474 |
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475 if (args.length () == 8) |
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476 { |
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477 const ColumnVector x0 (args(0) . vector_value ()); |
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478 const Matrix H (args(1) . matrix_value ()); |
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479 const ColumnVector q (args(2) . vector_value ()); |
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480 const Matrix Aeq (args(3) . matrix_value ()); |
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481 const ColumnVector beq (args(4) . vector_value ()); |
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482 const Matrix Ain (args(5) . matrix_value ()); |
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483 const ColumnVector bin (args(6) . vector_value ()); |
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484 const int maxit (args(7) . int_value ()); |
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485 |
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486 if (! error_state) |
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487 { |
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488 int iter = 0; |
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489 |
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490 // Copying the initial guess in the working variable |
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491 ColumnVector x = x0; |
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492 |
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493 // Reordering the Lagrange multipliers |
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494 ColumnVector lambda; |
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495 |
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496 int info = qp (H, q, Aeq, beq, Ain, bin, maxit, x, lambda, iter); |
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497 |
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498 retval(3) = iter; |
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499 retval(2) = info; |
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500 retval(1) = lambda; |
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501 retval(0) = x; |
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502 } |
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503 else |
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504 error ("__qp__: invalid arguments"); |
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505 } |
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506 else |
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5823
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507 print_usage (); |
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5290
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508 |
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509 return retval; |
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510 } |
