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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 inline double |
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43 ABS (double x) |
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44 { |
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45 return x < 0 ? -x : x; |
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46 } |
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47 |
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48 static Matrix |
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49 null (const Matrix& A, octave_idx_type& rank) |
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50 { |
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51 Matrix retval; |
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52 |
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53 rank = 0; |
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54 |
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55 if (! A.is_empty ()) |
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56 { |
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57 SVD A_svd (A); |
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58 |
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59 DiagMatrix S = A_svd.singular_values (); |
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60 |
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61 ColumnVector s = S.diag (); |
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62 |
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63 Matrix V = A_svd.right_singular_matrix (); |
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64 |
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65 octave_idx_type A_nr = A.rows (); |
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66 octave_idx_type A_nc = A.cols (); |
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67 |
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68 octave_idx_type tmp = A_nr > A_nc ? A_nr : A_nc; |
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69 |
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70 double tol = tmp * s(0) * DBL_EPSILON; |
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71 |
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72 octave_idx_type n = s.length (); |
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73 |
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74 for (octave_idx_type i = 0; i < n; i++) |
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75 { |
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76 if (s(i) > tol) |
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77 rank++; |
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78 } |
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79 |
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80 if (rank < A_nc) |
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81 retval = V.extract (0, rank, A_nc-1, A_nc-1); |
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82 else |
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83 retval.resize (A_nc, 0); |
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84 } |
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85 |
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86 return retval; |
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87 } |
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88 |
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89 static int |
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90 qp (const Matrix& H, const ColumnVector& q, |
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91 const Matrix& Aeq, const ColumnVector& beq, |
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92 const Matrix& Ain, const ColumnVector& bin, |
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93 int maxit, |
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94 ColumnVector& x, ColumnVector& lambda, int& iter) |
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95 { |
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96 int info = 0; |
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97 |
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98 iter = 0; |
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99 |
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100 double rtol = sqrt (DBL_EPSILON); |
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101 |
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102 // Problem dimension. |
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103 octave_idx_type n = x.length (); |
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104 |
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105 // Dimension of constraints. |
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106 octave_idx_type n_eq = beq.length (); |
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107 octave_idx_type n_in = bin.length (); |
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108 |
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109 // Filling the current active set. |
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110 |
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111 octave_idx_type n_act = n_eq; |
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112 |
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113 octave_idx_type n_tot = n_eq + n_in; |
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114 |
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115 // Equality constraints come first. We won't check the sign of the |
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116 // Lagrange multiplier for those. |
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117 |
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118 Matrix Aact = Aeq; |
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119 ColumnVector bact = beq; |
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120 ColumnVector Wact; |
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121 |
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122 if (n_in > 0) |
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123 { |
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124 ColumnVector res = Ain*x - bin; |
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125 |
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126 for (octave_idx_type i = 0; i < n_in; i++) |
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127 { |
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128 res(i) /= (1.0 + ABS (bin(i))); |
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129 |
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130 if (res(i) < rtol) |
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131 { |
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132 n_act++; |
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133 Aact = Aact.stack (Ain.row (i)); |
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134 bact.resize (n_act, bin(i)); |
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135 Wact.resize (n_act, i); |
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136 } |
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137 } |
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138 } |
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139 |
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140 // Computing the ??? |
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141 |
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142 EIG eigH (H); |
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143 ColumnVector eigenvalH = real (eigH.eigenvalues ()); |
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144 Matrix eigenvecH = real (eigH.eigenvectors ()); |
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145 double minReal = eigenvalH.min (); |
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146 octave_idx_type indminR = 0; |
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147 for (octave_idx_type i = 0; i < n; i++) |
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148 { |
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149 if (minReal == eigenvalH(i)) |
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150 { |
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151 indminR = i; |
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152 break; |
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153 } |
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154 } |
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155 |
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156 bool done = false; |
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157 |
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158 double alpha = 0.0; |
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159 |
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160 Matrix R; |
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161 Matrix Y (n, 0, 0.0); |
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162 |
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163 ColumnVector g (n, 0.0); |
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164 ColumnVector p (n, 0.0); |
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165 |
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166 ColumnVector lambda_tmp (n_in, 0.0); |
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167 |
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168 while (! done) |
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169 { |
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170 iter++; |
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171 |
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172 // Current Gradient |
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173 // g = q + H * x; |
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174 |
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175 g = q + H * x; |
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176 |
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177 if (n_act == 0) |
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178 { |
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179 // There are no active constraints. |
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180 |
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181 if (minReal > 0.0) |
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182 { |
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183 // Inverting the Hessian. Using the Cholesky |
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184 // factorization since the Hessian is positive |
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185 // definite. |
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186 |
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187 CHOL cholH (H); |
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188 |
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189 R = cholH.chol_matrix (); |
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190 |
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191 Matrix Hinv = chol2inv (R); |
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192 |
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193 // Computing the unconstrained step. |
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194 // p = -Hinv * g; |
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195 |
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196 p = -Hinv * g; |
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197 |
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198 info = 0; |
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199 } |
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200 else |
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201 { |
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202 // Finding the negative curvature of H. |
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203 |
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204 p = eigenvecH.column (indminR); |
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205 |
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206 // Following the negative curvature of H. |
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207 |
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208 if (p.transpose () * g > DBL_EPSILON) |
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209 p = -p; |
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210 |
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211 info = 1; |
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212 } |
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213 |
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214 // Multipliers are zero. |
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215 lambda_tmp.fill (0.0); |
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216 } |
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217 else |
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218 { |
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219 // There are active constraints. |
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220 |
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221 // Computing the null space. |
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222 |
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223 octave_idx_type rank; |
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224 |
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225 Matrix Z = null (Aact, rank); |
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226 |
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227 octave_idx_type dimZ = n - rank; |
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228 |
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229 // FIXME -- still remain to handle the case of |
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230 // non-full rank active set matrix. |
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231 |
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232 // Computing the Y matrix (orthogonal to Z) |
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233 Y = Aact.pseudo_inverse (); |
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234 |
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235 // Reduced Hessian |
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236 Matrix Zt = Z.transpose (); |
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237 Matrix rH = Zt * H * Z; |
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238 |
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239 octave_idx_type pR = 0; |
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240 |
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241 if (dimZ > 0) |
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242 { |
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243 // Computing the Cholesky factorization (pR = 0 means |
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244 // that the reduced Hessian was positive definite). |
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245 |
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246 CHOL cholrH (rH, pR); |
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247 Matrix tR = cholrH.chol_matrix (); |
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248 if (pR == 0) |
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249 R = tR; |
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250 } |
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251 |
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252 if (pR == 0) |
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253 { |
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254 info = 0; |
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255 |
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256 // Computing the step pz. |
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257 if (dimZ > 0) |
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258 { |
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259 // Using the Cholesky factorization to invert rH |
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260 |
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261 Matrix rHinv = chol2inv (R); |
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262 |
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263 ColumnVector pz = -rHinv * Zt * g; |
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264 |
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265 // Global step. |
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266 p = Z * pz; |
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267 } |
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268 else |
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269 { |
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270 // Global step. |
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271 p.fill (0.0); |
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272 } |
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273 } |
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274 else |
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275 { |
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276 info = 1; |
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277 |
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278 // Searching for the most negative curvature. |
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279 |
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280 EIG eigrH (rH); |
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281 ColumnVector eigenvalrH = real (eigrH.eigenvalues ()); |
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282 Matrix eigenvecrH = real (eigrH.eigenvectors ()); |
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283 double mRrH = eigenvalrH.min (); |
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284 indminR = 0; |
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285 for (octave_idx_type i = 0; i < n; i++) |
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286 { |
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287 if (mRrH == eigenvalH(i)) |
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288 { |
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289 indminR = i; |
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290 break; |
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291 } |
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292 } |
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293 |
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294 ColumnVector eVrH = eigenvecrH.column (indminR); |
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295 |
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296 // Computing the step pz. |
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297 p = Z * eVrH; |
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298 |
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299 if (p.transpose () * g > DBL_EPSILON) |
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300 p = -p; |
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301 } |
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302 } |
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303 |
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304 // Checking the step-size. |
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305 ColumnVector abs_p (n); |
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306 for (octave_idx_type i = 0; i < n; i++) |
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307 abs_p(i) = ABS (p(i)); |
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308 double max_p = abs_p.max (); |
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309 |
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310 if (max_p < rtol) |
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311 { |
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312 // The step is null. Checking constraints. |
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313 if (n_act - n_eq == 0) |
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314 // Solution is found because no inequality |
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315 // constraints are active. |
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316 done = true; |
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317 else |
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318 { |
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319 // Computing the multipliers only for the inequality |
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320 // constraints that are active. We do NOT compute |
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321 // multipliers for the equality constraints. |
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322 Matrix Yt = Y.transpose (); |
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323 Yt = Yt.extract_n (n_eq, 0, n_act-n_eq, n); |
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324 lambda_tmp = Yt * (g + H * p); |
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325 if (n_act - n_eq < n_in) |
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326 { |
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327 lambda_tmp.resize (n_in, 0.0); |
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328 |
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329 for (octave_idx_type i = n_act-n_eq; i < n_in; i++) |
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330 lambda_tmp(i) = 0; |
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331 } |
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332 |
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333 // Checking the multipliers. We remove the most |
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334 // negative from the set (if any). |
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335 double min_lambda = lambda_tmp.min (); |
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336 if (min_lambda >= 0) |
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337 { |
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338 // Solution is found. |
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339 done = true; |
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340 } |
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341 else |
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342 { |
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343 octave_idx_type which_eig = 0; |
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344 for (octave_idx_type i = 0; i < n_act; i++) |
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345 { |
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346 if (lambda_tmp(i) == min_lambda) |
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347 { |
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348 which_eig = i; |
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349 break; |
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350 } |
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351 } |
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352 |
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353 // At least one multiplier is negative, we |
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354 // remove it from the set. |
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355 |
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356 for (octave_idx_type i = which_eig; i < n_act - n_eq; i++) |
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357 { |
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358 Wact(i) = Wact(i+1); |
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359 for (octave_idx_type j = 0; j < n; j++) |
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360 Aact(n_eq+i,j) = Aact(n_eq+i+1,j); |
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361 bact(n_eq+i) = bact(n_eq+i+1); |
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362 } |
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363 n_act--; |
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364 |
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365 // Resizing the active set. |
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366 Wact.resize (n_act-n_eq); |
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367 bact.resize (n_act); |
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368 Aact.resize (n_act, n); |
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369 } |
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370 } |
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371 } |
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372 else |
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373 { |
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374 // The step is not null. |
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375 if (n_act - n_eq == n_in) |
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376 { |
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377 // All inequality constraints were active. We can |
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378 // add the whole step. |
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379 x += p; |
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380 } |
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381 else |
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382 { |
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383 // Some constraints were not active. Checking if |
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384 // there is a blocking constraint. |
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385 alpha = 1.0; |
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386 octave_idx_type is_block = -1; |
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387 |
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388 for (octave_idx_type i = 0; i < n_in; i++) |
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389 { |
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390 bool found = false; |
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391 |
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392 for (octave_idx_type j = 0; j < n_act-n_eq; j++) |
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393 { |
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394 if (Wact(j) == i) |
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395 { |
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396 found = true; |
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397 break; |
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398 } |
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399 } |
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400 |
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401 if (! found) |
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402 { |
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403 // The i-th constraint was not in the set. Is it a |
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404 // blocking constraint? |
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405 |
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406 RowVector tmp_row = Ain.row (i); |
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407 double tmp = tmp_row * p; |
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408 double res = tmp_row * x; |
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409 |
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410 if (tmp < 0.0) |
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411 { |
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412 double alpha_tmp = (bin(i) - res) / tmp; |
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413 |
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414 if (alpha_tmp < alpha) |
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415 { |
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416 alpha = alpha_tmp; |
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417 is_block = i; |
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418 } |
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419 } |
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420 } |
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421 } |
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422 |
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423 // In is_block there is the index of the blocking |
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424 // constraint (if any). |
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425 if (is_block >= 0) |
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426 { |
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427 // There is a blocking constraint (index in |
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428 // is_block) which is added to the active set. |
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429 n_act++; |
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430 Aact = Aact.stack (Ain.row (is_block)); |
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431 bact.resize (n_act, bin(is_block)); |
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432 Wact.resize (n_act, is_block); |
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433 |
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434 // Adding the reduced step |
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435 x += alpha * p; |
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436 } |
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437 else |
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438 { |
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439 // There are no blocking constraints. Adding the |
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440 // whole step. |
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441 x += alpha * p; |
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442 } |
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443 } |
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444 } |
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445 |
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446 if (iter == maxit) |
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447 { |
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448 done = true; |
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449 // warning ("qp_main: maximum number of iteration reached"); |
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450 info = 3; |
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451 } |
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452 } |
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453 |
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454 lambda_tmp = Y.transpose () * (g + H * p); |
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455 |
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456 // Reordering the Lagrange multipliers. |
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457 |
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458 lambda.resize (n_tot); |
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459 lambda.fill (0.0); |
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460 for (octave_idx_type i = 0; i < n_eq; i++) |
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461 lambda(i) = lambda_tmp(i); |
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462 |
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463 for (octave_idx_type i = n_eq; i < n_tot; i++) |
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464 { |
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465 for (octave_idx_type j = 0; j < n_act-n_eq; j++) |
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466 { |
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467 if (Wact(j) == i) |
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468 { |
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469 lambda(i) = lambda_tmp(n_eq+j); |
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470 break; |
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471 } |
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472 } |
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473 } |
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474 |
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475 return info; |
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476 } |
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477 |
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478 DEFUN_DLD (__qp__, args, , |
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479 "[x, lambda, info, iter] = __qp__ (x0, H, q, Aeq, beq, Ain, bin, maxit)") |
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480 { |
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481 octave_value_list retval; |
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482 |
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483 if (args.length () == 8) |
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484 { |
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485 const ColumnVector x0 (args(0) . vector_value ()); |
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486 const Matrix H (args(1) . matrix_value ()); |
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487 const ColumnVector q (args(2) . vector_value ()); |
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488 const Matrix Aeq (args(3) . matrix_value ()); |
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489 const ColumnVector beq (args(4) . vector_value ()); |
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490 const Matrix Ain (args(5) . matrix_value ()); |
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491 const ColumnVector bin (args(6) . vector_value ()); |
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492 const int maxit (args(7) . int_value ()); |
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493 |
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494 if (! error_state) |
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495 { |
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496 int iter = 0; |
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497 |
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498 // Copying the initial guess in the working variable |
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499 ColumnVector x = x0; |
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500 |
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501 // Reordering the Lagrange multipliers |
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502 ColumnVector lambda; |
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503 |
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504 int info = qp (H, q, Aeq, beq, Ain, bin, maxit, x, lambda, iter); |
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505 |
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506 retval(3) = iter; |
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507 retval(2) = info; |
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508 retval(1) = lambda; |
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509 retval(0) = x; |
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510 } |
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511 else |
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512 error ("__qp__: invalid arguments"); |
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513 } |
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514 else |
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5823
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515 print_usage (); |
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5290
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516 |
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517 return retval; |
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518 } |
