1 | /* glpipm.c */ |
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2 | |
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3 | /*********************************************************************** |
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4 | * This code is part of GLPK (GNU Linear Programming Kit). |
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5 | * |
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6 | * Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, |
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7 | * 2009, 2010 Andrew Makhorin, Department for Applied Informatics, |
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8 | * Moscow Aviation Institute, Moscow, Russia. All rights reserved. |
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9 | * E-mail: <mao@gnu.org>. |
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10 | * |
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11 | * GLPK is free software: you can redistribute it and/or modify it |
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12 | * under the terms of the GNU General Public License as published by |
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13 | * the Free Software Foundation, either version 3 of the License, or |
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14 | * (at your option) any later version. |
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15 | * |
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16 | * GLPK is distributed in the hope that it will be useful, but WITHOUT |
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17 | * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
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18 | * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public |
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19 | * License for more details. |
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20 | * |
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21 | * You should have received a copy of the GNU General Public License |
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22 | * along with GLPK. If not, see <http://www.gnu.org/licenses/>. |
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23 | ***********************************************************************/ |
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24 | |
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25 | #include "glpipm.h" |
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26 | #include "glpmat.h" |
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27 | |
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28 | #define ITER_MAX 100 |
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29 | /* maximal number of iterations */ |
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30 | |
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31 | struct csa |
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32 | { /* common storage area */ |
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33 | /*--------------------------------------------------------------*/ |
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34 | /* LP data */ |
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35 | int m; |
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36 | /* number of rows (equality constraints) */ |
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37 | int n; |
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38 | /* number of columns (structural variables) */ |
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39 | int *A_ptr; /* int A_ptr[1+m+1]; */ |
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40 | int *A_ind; /* int A_ind[A_ptr[m+1]]; */ |
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41 | double *A_val; /* double A_val[A_ptr[m+1]]; */ |
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42 | /* mxn-matrix A in storage-by-rows format */ |
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43 | double *b; /* double b[1+m]; */ |
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44 | /* m-vector b of right-hand sides */ |
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45 | double *c; /* double c[1+n]; */ |
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46 | /* n-vector c of objective coefficients; c[0] is constant term of |
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47 | the objective function */ |
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48 | /*--------------------------------------------------------------*/ |
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49 | /* LP solution */ |
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50 | double *x; /* double x[1+n]; */ |
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51 | double *y; /* double y[1+m]; */ |
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52 | double *z; /* double z[1+n]; */ |
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53 | /* current point in primal-dual space; the best point on exit */ |
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54 | /*--------------------------------------------------------------*/ |
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55 | /* control parameters */ |
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56 | const glp_iptcp *parm; |
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57 | /*--------------------------------------------------------------*/ |
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58 | /* working arrays and variables */ |
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59 | double *D; /* double D[1+n]; */ |
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60 | /* diagonal nxn-matrix D = X*inv(Z), where X = diag(x[j]) and |
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61 | Z = diag(z[j]) */ |
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62 | int *P; /* int P[1+m+m]; */ |
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63 | /* permutation mxm-matrix P used to minimize fill-in in Cholesky |
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64 | factorization */ |
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65 | int *S_ptr; /* int S_ptr[1+m+1]; */ |
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66 | int *S_ind; /* int S_ind[S_ptr[m+1]]; */ |
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67 | double *S_val; /* double S_val[S_ptr[m+1]]; */ |
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68 | double *S_diag; /* double S_diag[1+m]; */ |
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69 | /* symmetric mxm-matrix S = P*A*D*A'*P' whose upper triangular |
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70 | part without diagonal elements is stored in S_ptr, S_ind, and |
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71 | S_val in storage-by-rows format, diagonal elements are stored |
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72 | in S_diag */ |
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73 | int *U_ptr; /* int U_ptr[1+m+1]; */ |
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74 | int *U_ind; /* int U_ind[U_ptr[m+1]]; */ |
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75 | double *U_val; /* double U_val[U_ptr[m+1]]; */ |
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76 | double *U_diag; /* double U_diag[1+m]; */ |
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77 | /* upper triangular mxm-matrix U defining Cholesky factorization |
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78 | S = U'*U; its non-diagonal elements are stored in U_ptr, U_ind, |
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79 | U_val in storage-by-rows format, diagonal elements are stored |
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80 | in U_diag */ |
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81 | int iter; |
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82 | /* iteration number (0, 1, 2, ...); iter = 0 corresponds to the |
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83 | initial point */ |
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84 | double obj; |
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85 | /* current value of the objective function */ |
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86 | double rpi; |
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87 | /* relative primal infeasibility rpi = ||A*x-b||/(1+||b||) */ |
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88 | double rdi; |
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89 | /* relative dual infeasibility rdi = ||A'*y+z-c||/(1+||c||) */ |
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90 | double gap; |
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91 | /* primal-dual gap = |c'*x-b'*y|/(1+|c'*x|) which is a relative |
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92 | difference between primal and dual objective functions */ |
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93 | double phi; |
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94 | /* merit function phi = ||A*x-b||/max(1,||b||) + |
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95 | + ||A'*y+z-c||/max(1,||c||) + |
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96 | + |c'*x-b'*y|/max(1,||b||,||c||) */ |
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97 | double mu; |
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98 | /* duality measure mu = x'*z/n (used as barrier parameter) */ |
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99 | double rmu; |
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100 | /* rmu = max(||A*x-b||,||A'*y+z-c||)/mu */ |
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101 | double rmu0; |
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102 | /* the initial value of rmu on iteration 0 */ |
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103 | double *phi_min; /* double phi_min[1+ITER_MAX]; */ |
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104 | /* phi_min[k] = min(phi[k]), where phi[k] is the value of phi on |
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105 | k-th iteration, 0 <= k <= iter */ |
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106 | int best_iter; |
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107 | /* iteration number, on which the value of phi reached its best |
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108 | (minimal) value */ |
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109 | double *best_x; /* double best_x[1+n]; */ |
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110 | double *best_y; /* double best_y[1+m]; */ |
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111 | double *best_z; /* double best_z[1+n]; */ |
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112 | /* best point (in the sense of the merit function phi) which has |
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113 | been reached on iteration iter_best */ |
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114 | double best_obj; |
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115 | /* objective value at the best point */ |
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116 | double *dx_aff; /* double dx_aff[1+n]; */ |
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117 | double *dy_aff; /* double dy_aff[1+m]; */ |
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118 | double *dz_aff; /* double dz_aff[1+n]; */ |
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119 | /* affine scaling direction */ |
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120 | double alfa_aff_p, alfa_aff_d; |
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121 | /* maximal primal and dual stepsizes in affine scaling direction, |
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122 | on which x and z are still non-negative */ |
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123 | double mu_aff; |
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124 | /* duality measure mu_aff = x_aff'*z_aff/n in the boundary point |
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125 | x_aff' = x+alfa_aff_p*dx_aff, z_aff' = z+alfa_aff_d*dz_aff */ |
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126 | double sigma; |
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127 | /* Mehrotra's heuristic parameter (0 <= sigma <= 1) */ |
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128 | double *dx_cc; /* double dx_cc[1+n]; */ |
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129 | double *dy_cc; /* double dy_cc[1+m]; */ |
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130 | double *dz_cc; /* double dz_cc[1+n]; */ |
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131 | /* centering corrector direction */ |
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132 | double *dx; /* double dx[1+n]; */ |
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133 | double *dy; /* double dy[1+m]; */ |
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134 | double *dz; /* double dz[1+n]; */ |
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135 | /* final combined direction dx = dx_aff+dx_cc, dy = dy_aff+dy_cc, |
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136 | dz = dz_aff+dz_cc */ |
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137 | double alfa_max_p; |
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138 | double alfa_max_d; |
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139 | /* maximal primal and dual stepsizes in combined direction, on |
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140 | which x and z are still non-negative */ |
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141 | }; |
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142 | |
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143 | /*********************************************************************** |
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144 | * initialize - allocate and initialize common storage area |
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145 | * |
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146 | * This routine allocates and initializes the common storage area (CSA) |
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147 | * used by interior-point method routines. */ |
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148 | |
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149 | static void initialize(struct csa *csa) |
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150 | { int m = csa->m; |
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151 | int n = csa->n; |
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152 | int i; |
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153 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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154 | xprintf("Matrix A has %d non-zeros\n", csa->A_ptr[m+1]-1); |
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155 | csa->D = xcalloc(1+n, sizeof(double)); |
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156 | /* P := I */ |
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157 | csa->P = xcalloc(1+m+m, sizeof(int)); |
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158 | for (i = 1; i <= m; i++) csa->P[i] = csa->P[m+i] = i; |
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159 | /* S := A*A', symbolically */ |
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160 | csa->S_ptr = xcalloc(1+m+1, sizeof(int)); |
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161 | csa->S_ind = adat_symbolic(m, n, csa->P, csa->A_ptr, csa->A_ind, |
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162 | csa->S_ptr); |
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163 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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164 | xprintf("Matrix S = A*A' has %d non-zeros (upper triangle)\n", |
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165 | csa->S_ptr[m+1]-1 + m); |
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166 | /* determine P using specified ordering algorithm */ |
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167 | if (csa->parm->ord_alg == GLP_ORD_NONE) |
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168 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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169 | xprintf("Original ordering is being used\n"); |
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170 | for (i = 1; i <= m; i++) |
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171 | csa->P[i] = csa->P[m+i] = i; |
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172 | } |
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173 | else if (csa->parm->ord_alg == GLP_ORD_QMD) |
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174 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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175 | xprintf("Minimum degree ordering (QMD)...\n"); |
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176 | min_degree(m, csa->S_ptr, csa->S_ind, csa->P); |
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177 | } |
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178 | else if (csa->parm->ord_alg == GLP_ORD_AMD) |
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179 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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180 | xprintf("Approximate minimum degree ordering (AMD)...\n"); |
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181 | amd_order1(m, csa->S_ptr, csa->S_ind, csa->P); |
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182 | } |
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183 | else if (csa->parm->ord_alg == GLP_ORD_SYMAMD) |
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184 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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185 | xprintf("Approximate minimum degree ordering (SYMAMD)...\n") |
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186 | ; |
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187 | symamd_ord(m, csa->S_ptr, csa->S_ind, csa->P); |
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188 | } |
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189 | else |
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190 | xassert(csa != csa); |
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191 | /* S := P*A*A'*P', symbolically */ |
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192 | xfree(csa->S_ind); |
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193 | csa->S_ind = adat_symbolic(m, n, csa->P, csa->A_ptr, csa->A_ind, |
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194 | csa->S_ptr); |
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195 | csa->S_val = xcalloc(csa->S_ptr[m+1], sizeof(double)); |
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196 | csa->S_diag = xcalloc(1+m, sizeof(double)); |
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197 | /* compute Cholesky factorization S = U'*U, symbolically */ |
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198 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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199 | xprintf("Computing Cholesky factorization S = L*L'...\n"); |
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200 | csa->U_ptr = xcalloc(1+m+1, sizeof(int)); |
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201 | csa->U_ind = chol_symbolic(m, csa->S_ptr, csa->S_ind, csa->U_ptr); |
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202 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
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203 | xprintf("Matrix L has %d non-zeros\n", csa->U_ptr[m+1]-1 + m); |
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204 | csa->U_val = xcalloc(csa->U_ptr[m+1], sizeof(double)); |
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205 | csa->U_diag = xcalloc(1+m, sizeof(double)); |
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206 | csa->iter = 0; |
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207 | csa->obj = 0.0; |
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208 | csa->rpi = 0.0; |
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209 | csa->rdi = 0.0; |
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210 | csa->gap = 0.0; |
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211 | csa->phi = 0.0; |
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212 | csa->mu = 0.0; |
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213 | csa->rmu = 0.0; |
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214 | csa->rmu0 = 0.0; |
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215 | csa->phi_min = xcalloc(1+ITER_MAX, sizeof(double)); |
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216 | csa->best_iter = 0; |
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217 | csa->best_x = xcalloc(1+n, sizeof(double)); |
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218 | csa->best_y = xcalloc(1+m, sizeof(double)); |
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219 | csa->best_z = xcalloc(1+n, sizeof(double)); |
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220 | csa->best_obj = 0.0; |
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221 | csa->dx_aff = xcalloc(1+n, sizeof(double)); |
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222 | csa->dy_aff = xcalloc(1+m, sizeof(double)); |
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223 | csa->dz_aff = xcalloc(1+n, sizeof(double)); |
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224 | csa->alfa_aff_p = 0.0; |
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225 | csa->alfa_aff_d = 0.0; |
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226 | csa->mu_aff = 0.0; |
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227 | csa->sigma = 0.0; |
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228 | csa->dx_cc = xcalloc(1+n, sizeof(double)); |
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229 | csa->dy_cc = xcalloc(1+m, sizeof(double)); |
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230 | csa->dz_cc = xcalloc(1+n, sizeof(double)); |
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231 | csa->dx = csa->dx_aff; |
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232 | csa->dy = csa->dy_aff; |
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233 | csa->dz = csa->dz_aff; |
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234 | csa->alfa_max_p = 0.0; |
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235 | csa->alfa_max_d = 0.0; |
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236 | return; |
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237 | } |
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238 | |
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239 | /*********************************************************************** |
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240 | * A_by_vec - compute y = A*x |
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241 | * |
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242 | * This routine computes matrix-vector product y = A*x, where A is the |
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243 | * constraint matrix. */ |
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244 | |
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245 | static void A_by_vec(struct csa *csa, double x[], double y[]) |
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246 | { /* compute y = A*x */ |
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247 | int m = csa->m; |
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248 | int *A_ptr = csa->A_ptr; |
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249 | int *A_ind = csa->A_ind; |
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250 | double *A_val = csa->A_val; |
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251 | int i, t, beg, end; |
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252 | double temp; |
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253 | for (i = 1; i <= m; i++) |
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254 | { temp = 0.0; |
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255 | beg = A_ptr[i], end = A_ptr[i+1]; |
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256 | for (t = beg; t < end; t++) temp += A_val[t] * x[A_ind[t]]; |
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257 | y[i] = temp; |
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258 | } |
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259 | return; |
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260 | } |
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261 | |
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262 | /*********************************************************************** |
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263 | * AT_by_vec - compute y = A'*x |
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264 | * |
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265 | * This routine computes matrix-vector product y = A'*x, where A' is a |
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266 | * matrix transposed to the constraint matrix A. */ |
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267 | |
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268 | static void AT_by_vec(struct csa *csa, double x[], double y[]) |
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269 | { /* compute y = A'*x, where A' is transposed to A */ |
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270 | int m = csa->m; |
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271 | int n = csa->n; |
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272 | int *A_ptr = csa->A_ptr; |
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273 | int *A_ind = csa->A_ind; |
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274 | double *A_val = csa->A_val; |
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275 | int i, j, t, beg, end; |
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276 | double temp; |
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277 | for (j = 1; j <= n; j++) y[j] = 0.0; |
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278 | for (i = 1; i <= m; i++) |
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279 | { temp = x[i]; |
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280 | if (temp == 0.0) continue; |
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281 | beg = A_ptr[i], end = A_ptr[i+1]; |
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282 | for (t = beg; t < end; t++) y[A_ind[t]] += A_val[t] * temp; |
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283 | } |
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284 | return; |
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285 | } |
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286 | |
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287 | /*********************************************************************** |
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288 | * decomp_NE - numeric factorization of matrix S = P*A*D*A'*P' |
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289 | * |
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290 | * This routine implements numeric phase of Cholesky factorization of |
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291 | * the matrix S = P*A*D*A'*P', which is a permuted matrix of the normal |
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292 | * equation system. Matrix D is assumed to be already computed. */ |
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293 | |
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294 | static void decomp_NE(struct csa *csa) |
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295 | { adat_numeric(csa->m, csa->n, csa->P, csa->A_ptr, csa->A_ind, |
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296 | csa->A_val, csa->D, csa->S_ptr, csa->S_ind, csa->S_val, |
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297 | csa->S_diag); |
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298 | chol_numeric(csa->m, csa->S_ptr, csa->S_ind, csa->S_val, |
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299 | csa->S_diag, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag); |
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300 | return; |
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301 | } |
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302 | |
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303 | /*********************************************************************** |
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304 | * solve_NE - solve normal equation system |
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305 | * |
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306 | * This routine solves the normal equation system: |
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307 | * |
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308 | * A*D*A'*y = h. |
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309 | * |
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310 | * It is assumed that the matrix A*D*A' has been previously factorized |
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311 | * by the routine decomp_NE. |
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312 | * |
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313 | * On entry the array y contains the vector of right-hand sides h. On |
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314 | * exit this array contains the computed vector of unknowns y. |
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315 | * |
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316 | * Once the vector y has been computed the routine checks for numeric |
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317 | * stability. If the residual vector: |
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318 | * |
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319 | * r = A*D*A'*y - h |
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320 | * |
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321 | * is relatively small, the routine returns zero, otherwise non-zero is |
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322 | * returned. */ |
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323 | |
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324 | static int solve_NE(struct csa *csa, double y[]) |
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325 | { int m = csa->m; |
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326 | int n = csa->n; |
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327 | int *P = csa->P; |
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328 | int i, j, ret = 0; |
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329 | double *h, *r, *w; |
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330 | /* save vector of right-hand sides h */ |
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331 | h = xcalloc(1+m, sizeof(double)); |
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332 | for (i = 1; i <= m; i++) h[i] = y[i]; |
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333 | /* solve normal equation system (A*D*A')*y = h */ |
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334 | /* since S = P*A*D*A'*P' = U'*U, then A*D*A' = P'*U'*U*P, so we |
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335 | have inv(A*D*A') = P'*inv(U)*inv(U')*P */ |
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336 | /* w := P*h */ |
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337 | w = xcalloc(1+m, sizeof(double)); |
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338 | for (i = 1; i <= m; i++) w[i] = y[P[i]]; |
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339 | /* w := inv(U')*w */ |
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340 | ut_solve(m, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag, w); |
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341 | /* w := inv(U)*w */ |
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342 | u_solve(m, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag, w); |
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343 | /* y := P'*w */ |
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344 | for (i = 1; i <= m; i++) y[i] = w[P[m+i]]; |
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345 | xfree(w); |
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346 | /* compute residual vector r = A*D*A'*y - h */ |
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347 | r = xcalloc(1+m, sizeof(double)); |
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348 | /* w := A'*y */ |
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349 | w = xcalloc(1+n, sizeof(double)); |
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350 | AT_by_vec(csa, y, w); |
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351 | /* w := D*w */ |
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352 | for (j = 1; j <= n; j++) w[j] *= csa->D[j]; |
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353 | /* r := A*w */ |
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354 | A_by_vec(csa, w, r); |
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355 | xfree(w); |
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356 | /* r := r - h */ |
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357 | for (i = 1; i <= m; i++) r[i] -= h[i]; |
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358 | /* check for numeric stability */ |
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359 | for (i = 1; i <= m; i++) |
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360 | { if (fabs(r[i]) / (1.0 + fabs(h[i])) > 1e-4) |
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361 | { ret = 1; |
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362 | break; |
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363 | } |
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364 | } |
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365 | xfree(h); |
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366 | xfree(r); |
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367 | return ret; |
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368 | } |
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369 | |
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370 | /*********************************************************************** |
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371 | * solve_NS - solve Newtonian system |
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372 | * |
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373 | * This routine solves the Newtonian system: |
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374 | * |
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375 | * A*dx = p |
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376 | * |
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377 | * A'*dy + dz = q |
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378 | * |
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379 | * Z*dx + X*dz = r |
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380 | * |
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381 | * where X = diag(x[j]), Z = diag(z[j]), by reducing it to the normal |
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382 | * equation system: |
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383 | * |
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384 | * (A*inv(Z)*X*A')*dy = A*inv(Z)*(X*q-r)+p |
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385 | * |
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386 | * (it is assumed that the matrix A*inv(Z)*X*A' has been factorized by |
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387 | * the routine decomp_NE). |
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388 | * |
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389 | * Once vector dy has been computed the routine computes vectors dx and |
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390 | * dz as follows: |
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391 | * |
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392 | * dx = inv(Z)*(X*(A'*dy-q)+r) |
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393 | * |
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394 | * dz = inv(X)*(r-Z*dx) |
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395 | * |
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396 | * The routine solve_NS returns the same code which was reported by the |
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397 | * routine solve_NE (see above). */ |
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398 | |
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399 | static int solve_NS(struct csa *csa, double p[], double q[], double r[], |
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400 | double dx[], double dy[], double dz[]) |
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401 | { int m = csa->m; |
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402 | int n = csa->n; |
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403 | double *x = csa->x; |
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404 | double *z = csa->z; |
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405 | int i, j, ret; |
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406 | double *w = dx; |
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407 | /* compute the vector of right-hand sides A*inv(Z)*(X*q-r)+p for |
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408 | the normal equation system */ |
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409 | for (j = 1; j <= n; j++) |
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410 | w[j] = (x[j] * q[j] - r[j]) / z[j]; |
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411 | A_by_vec(csa, w, dy); |
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412 | for (i = 1; i <= m; i++) dy[i] += p[i]; |
---|
413 | /* solve the normal equation system to compute vector dy */ |
---|
414 | ret = solve_NE(csa, dy); |
---|
415 | /* compute vectors dx and dz */ |
---|
416 | AT_by_vec(csa, dy, dx); |
---|
417 | for (j = 1; j <= n; j++) |
---|
418 | { dx[j] = (x[j] * (dx[j] - q[j]) + r[j]) / z[j]; |
---|
419 | dz[j] = (r[j] - z[j] * dx[j]) / x[j]; |
---|
420 | } |
---|
421 | return ret; |
---|
422 | } |
---|
423 | |
---|
424 | /*********************************************************************** |
---|
425 | * initial_point - choose initial point using Mehrotra's heuristic |
---|
426 | * |
---|
427 | * This routine chooses a starting point using a heuristic proposed in |
---|
428 | * the paper: |
---|
429 | * |
---|
430 | * S. Mehrotra. On the implementation of a primal-dual interior point |
---|
431 | * method. SIAM J. on Optim., 2(4), pp. 575-601, 1992. |
---|
432 | * |
---|
433 | * The starting point x in the primal space is chosen as a solution of |
---|
434 | * the following least squares problem: |
---|
435 | * |
---|
436 | * minimize ||x|| |
---|
437 | * |
---|
438 | * subject to A*x = b |
---|
439 | * |
---|
440 | * which can be computed explicitly as follows: |
---|
441 | * |
---|
442 | * x = A'*inv(A*A')*b |
---|
443 | * |
---|
444 | * Similarly, the starting point (y, z) in the dual space is chosen as |
---|
445 | * a solution of the following least squares problem: |
---|
446 | * |
---|
447 | * minimize ||z|| |
---|
448 | * |
---|
449 | * subject to A'*y + z = c |
---|
450 | * |
---|
451 | * which can be computed explicitly as follows: |
---|
452 | * |
---|
453 | * y = inv(A*A')*A*c |
---|
454 | * |
---|
455 | * z = c - A'*y |
---|
456 | * |
---|
457 | * However, some components of the vectors x and z may be non-positive |
---|
458 | * or close to zero, so the routine uses a Mehrotra's heuristic to find |
---|
459 | * a more appropriate starting point. */ |
---|
460 | |
---|
461 | static void initial_point(struct csa *csa) |
---|
462 | { int m = csa->m; |
---|
463 | int n = csa->n; |
---|
464 | double *b = csa->b; |
---|
465 | double *c = csa->c; |
---|
466 | double *x = csa->x; |
---|
467 | double *y = csa->y; |
---|
468 | double *z = csa->z; |
---|
469 | double *D = csa->D; |
---|
470 | int i, j; |
---|
471 | double dp, dd, ex, ez, xz; |
---|
472 | /* factorize A*A' */ |
---|
473 | for (j = 1; j <= n; j++) D[j] = 1.0; |
---|
474 | decomp_NE(csa); |
---|
475 | /* x~ = A'*inv(A*A')*b */ |
---|
476 | for (i = 1; i <= m; i++) y[i] = b[i]; |
---|
477 | solve_NE(csa, y); |
---|
478 | AT_by_vec(csa, y, x); |
---|
479 | /* y~ = inv(A*A')*A*c */ |
---|
480 | A_by_vec(csa, c, y); |
---|
481 | solve_NE(csa, y); |
---|
482 | /* z~ = c - A'*y~ */ |
---|
483 | AT_by_vec(csa, y,z); |
---|
484 | for (j = 1; j <= n; j++) z[j] = c[j] - z[j]; |
---|
485 | /* use Mehrotra's heuristic in order to choose more appropriate |
---|
486 | starting point with positive components of vectors x and z */ |
---|
487 | dp = dd = 0.0; |
---|
488 | for (j = 1; j <= n; j++) |
---|
489 | { if (dp < -1.5 * x[j]) dp = -1.5 * x[j]; |
---|
490 | if (dd < -1.5 * z[j]) dd = -1.5 * z[j]; |
---|
491 | } |
---|
492 | /* note that b = 0 involves x = 0, and c = 0 involves y = 0 and |
---|
493 | z = 0, so we need to be careful */ |
---|
494 | if (dp == 0.0) dp = 1.5; |
---|
495 | if (dd == 0.0) dd = 1.5; |
---|
496 | ex = ez = xz = 0.0; |
---|
497 | for (j = 1; j <= n; j++) |
---|
498 | { ex += (x[j] + dp); |
---|
499 | ez += (z[j] + dd); |
---|
500 | xz += (x[j] + dp) * (z[j] + dd); |
---|
501 | } |
---|
502 | dp += 0.5 * (xz / ez); |
---|
503 | dd += 0.5 * (xz / ex); |
---|
504 | for (j = 1; j <= n; j++) |
---|
505 | { x[j] += dp; |
---|
506 | z[j] += dd; |
---|
507 | xassert(x[j] > 0.0 && z[j] > 0.0); |
---|
508 | } |
---|
509 | return; |
---|
510 | } |
---|
511 | |
---|
512 | /*********************************************************************** |
---|
513 | * basic_info - perform basic computations at the current point |
---|
514 | * |
---|
515 | * This routine computes the following quantities at the current point: |
---|
516 | * |
---|
517 | * 1) value of the objective function: |
---|
518 | * |
---|
519 | * F = c'*x + c[0] |
---|
520 | * |
---|
521 | * 2) relative primal infeasibility: |
---|
522 | * |
---|
523 | * rpi = ||A*x-b|| / (1+||b||) |
---|
524 | * |
---|
525 | * 3) relative dual infeasibility: |
---|
526 | * |
---|
527 | * rdi = ||A'*y+z-c|| / (1+||c||) |
---|
528 | * |
---|
529 | * 4) primal-dual gap (relative difference between the primal and the |
---|
530 | * dual objective function values): |
---|
531 | * |
---|
532 | * gap = |c'*x-b'*y| / (1+|c'*x|) |
---|
533 | * |
---|
534 | * 5) merit function: |
---|
535 | * |
---|
536 | * phi = ||A*x-b|| / max(1,||b||) + ||A'*y+z-c|| / max(1,||c||) + |
---|
537 | * |
---|
538 | * + |c'*x-b'*y| / max(1,||b||,||c||) |
---|
539 | * |
---|
540 | * 6) duality measure: |
---|
541 | * |
---|
542 | * mu = x'*z / n |
---|
543 | * |
---|
544 | * 7) the ratio of infeasibility to mu: |
---|
545 | * |
---|
546 | * rmu = max(||A*x-b||,||A'*y+z-c||) / mu |
---|
547 | * |
---|
548 | * where ||*|| denotes euclidian norm, *' denotes transposition. */ |
---|
549 | |
---|
550 | static void basic_info(struct csa *csa) |
---|
551 | { int m = csa->m; |
---|
552 | int n = csa->n; |
---|
553 | double *b = csa->b; |
---|
554 | double *c = csa->c; |
---|
555 | double *x = csa->x; |
---|
556 | double *y = csa->y; |
---|
557 | double *z = csa->z; |
---|
558 | int i, j; |
---|
559 | double norm1, bnorm, norm2, cnorm, cx, by, *work, temp; |
---|
560 | /* compute value of the objective function */ |
---|
561 | temp = c[0]; |
---|
562 | for (j = 1; j <= n; j++) temp += c[j] * x[j]; |
---|
563 | csa->obj = temp; |
---|
564 | /* norm1 = ||A*x-b|| */ |
---|
565 | work = xcalloc(1+m, sizeof(double)); |
---|
566 | A_by_vec(csa, x, work); |
---|
567 | norm1 = 0.0; |
---|
568 | for (i = 1; i <= m; i++) |
---|
569 | norm1 += (work[i] - b[i]) * (work[i] - b[i]); |
---|
570 | norm1 = sqrt(norm1); |
---|
571 | xfree(work); |
---|
572 | /* bnorm = ||b|| */ |
---|
573 | bnorm = 0.0; |
---|
574 | for (i = 1; i <= m; i++) bnorm += b[i] * b[i]; |
---|
575 | bnorm = sqrt(bnorm); |
---|
576 | /* compute relative primal infeasibility */ |
---|
577 | csa->rpi = norm1 / (1.0 + bnorm); |
---|
578 | /* norm2 = ||A'*y+z-c|| */ |
---|
579 | work = xcalloc(1+n, sizeof(double)); |
---|
580 | AT_by_vec(csa, y, work); |
---|
581 | norm2 = 0.0; |
---|
582 | for (j = 1; j <= n; j++) |
---|
583 | norm2 += (work[j] + z[j] - c[j]) * (work[j] + z[j] - c[j]); |
---|
584 | norm2 = sqrt(norm2); |
---|
585 | xfree(work); |
---|
586 | /* cnorm = ||c|| */ |
---|
587 | cnorm = 0.0; |
---|
588 | for (j = 1; j <= n; j++) cnorm += c[j] * c[j]; |
---|
589 | cnorm = sqrt(cnorm); |
---|
590 | /* compute relative dual infeasibility */ |
---|
591 | csa->rdi = norm2 / (1.0 + cnorm); |
---|
592 | /* by = b'*y */ |
---|
593 | by = 0.0; |
---|
594 | for (i = 1; i <= m; i++) by += b[i] * y[i]; |
---|
595 | /* cx = c'*x */ |
---|
596 | cx = 0.0; |
---|
597 | for (j = 1; j <= n; j++) cx += c[j] * x[j]; |
---|
598 | /* compute primal-dual gap */ |
---|
599 | csa->gap = fabs(cx - by) / (1.0 + fabs(cx)); |
---|
600 | /* compute merit function */ |
---|
601 | csa->phi = 0.0; |
---|
602 | csa->phi += norm1 / (bnorm > 1.0 ? bnorm : 1.0); |
---|
603 | csa->phi += norm2 / (cnorm > 1.0 ? cnorm : 1.0); |
---|
604 | temp = 1.0; |
---|
605 | if (temp < bnorm) temp = bnorm; |
---|
606 | if (temp < cnorm) temp = cnorm; |
---|
607 | csa->phi += fabs(cx - by) / temp; |
---|
608 | /* compute duality measure */ |
---|
609 | temp = 0.0; |
---|
610 | for (j = 1; j <= n; j++) temp += x[j] * z[j]; |
---|
611 | csa->mu = temp / (double)n; |
---|
612 | /* compute the ratio of infeasibility to mu */ |
---|
613 | csa->rmu = (norm1 > norm2 ? norm1 : norm2) / csa->mu; |
---|
614 | return; |
---|
615 | } |
---|
616 | |
---|
617 | /*********************************************************************** |
---|
618 | * make_step - compute next point using Mehrotra's technique |
---|
619 | * |
---|
620 | * This routine computes the next point using the predictor-corrector |
---|
621 | * technique proposed in the paper: |
---|
622 | * |
---|
623 | * S. Mehrotra. On the implementation of a primal-dual interior point |
---|
624 | * method. SIAM J. on Optim., 2(4), pp. 575-601, 1992. |
---|
625 | * |
---|
626 | * At first, the routine computes so called affine scaling (predictor) |
---|
627 | * direction (dx_aff,dy_aff,dz_aff) which is a solution of the system: |
---|
628 | * |
---|
629 | * A*dx_aff = b - A*x |
---|
630 | * |
---|
631 | * A'*dy_aff + dz_aff = c - A'*y - z |
---|
632 | * |
---|
633 | * Z*dx_aff + X*dz_aff = - X*Z*e |
---|
634 | * |
---|
635 | * where (x,y,z) is the current point, X = diag(x[j]), Z = diag(z[j]), |
---|
636 | * e = (1,...,1)'. |
---|
637 | * |
---|
638 | * Then, the routine computes the centering parameter sigma, using the |
---|
639 | * following Mehrotra's heuristic: |
---|
640 | * |
---|
641 | * alfa_aff_p = inf{0 <= alfa <= 1 | x+alfa*dx_aff >= 0} |
---|
642 | * |
---|
643 | * alfa_aff_d = inf{0 <= alfa <= 1 | z+alfa*dz_aff >= 0} |
---|
644 | * |
---|
645 | * mu_aff = (x+alfa_aff_p*dx_aff)'*(z+alfa_aff_d*dz_aff)/n |
---|
646 | * |
---|
647 | * sigma = (mu_aff/mu)^3 |
---|
648 | * |
---|
649 | * where alfa_aff_p is the maximal stepsize along the affine scaling |
---|
650 | * direction in the primal space, alfa_aff_d is the maximal stepsize |
---|
651 | * along the same direction in the dual space. |
---|
652 | * |
---|
653 | * After determining sigma the routine computes so called centering |
---|
654 | * (corrector) direction (dx_cc,dy_cc,dz_cc) which is the solution of |
---|
655 | * the system: |
---|
656 | * |
---|
657 | * A*dx_cc = 0 |
---|
658 | * |
---|
659 | * A'*dy_cc + dz_cc = 0 |
---|
660 | * |
---|
661 | * Z*dx_cc + X*dz_cc = sigma*mu*e - X*Z*e |
---|
662 | * |
---|
663 | * Finally, the routine computes the combined direction |
---|
664 | * |
---|
665 | * (dx,dy,dz) = (dx_aff,dy_aff,dz_aff) + (dx_cc,dy_cc,dz_cc) |
---|
666 | * |
---|
667 | * and determines maximal primal and dual stepsizes along the combined |
---|
668 | * direction: |
---|
669 | * |
---|
670 | * alfa_max_p = inf{0 <= alfa <= 1 | x+alfa*dx >= 0} |
---|
671 | * |
---|
672 | * alfa_max_d = inf{0 <= alfa <= 1 | z+alfa*dz >= 0} |
---|
673 | * |
---|
674 | * In order to prevent the next point to be too close to the boundary |
---|
675 | * of the positive ortant, the routine decreases maximal stepsizes: |
---|
676 | * |
---|
677 | * alfa_p = gamma_p * alfa_max_p |
---|
678 | * |
---|
679 | * alfa_d = gamma_d * alfa_max_d |
---|
680 | * |
---|
681 | * where gamma_p and gamma_d are scaling factors, and computes the next |
---|
682 | * point: |
---|
683 | * |
---|
684 | * x_new = x + alfa_p * dx |
---|
685 | * |
---|
686 | * y_new = y + alfa_d * dy |
---|
687 | * |
---|
688 | * z_new = z + alfa_d * dz |
---|
689 | * |
---|
690 | * which becomes the current point on the next iteration. */ |
---|
691 | |
---|
692 | static int make_step(struct csa *csa) |
---|
693 | { int m = csa->m; |
---|
694 | int n = csa->n; |
---|
695 | double *b = csa->b; |
---|
696 | double *c = csa->c; |
---|
697 | double *x = csa->x; |
---|
698 | double *y = csa->y; |
---|
699 | double *z = csa->z; |
---|
700 | double *dx_aff = csa->dx_aff; |
---|
701 | double *dy_aff = csa->dy_aff; |
---|
702 | double *dz_aff = csa->dz_aff; |
---|
703 | double *dx_cc = csa->dx_cc; |
---|
704 | double *dy_cc = csa->dy_cc; |
---|
705 | double *dz_cc = csa->dz_cc; |
---|
706 | double *dx = csa->dx; |
---|
707 | double *dy = csa->dy; |
---|
708 | double *dz = csa->dz; |
---|
709 | int i, j, ret = 0; |
---|
710 | double temp, gamma_p, gamma_d, *p, *q, *r; |
---|
711 | /* allocate working arrays */ |
---|
712 | p = xcalloc(1+m, sizeof(double)); |
---|
713 | q = xcalloc(1+n, sizeof(double)); |
---|
714 | r = xcalloc(1+n, sizeof(double)); |
---|
715 | /* p = b - A*x */ |
---|
716 | A_by_vec(csa, x, p); |
---|
717 | for (i = 1; i <= m; i++) p[i] = b[i] - p[i]; |
---|
718 | /* q = c - A'*y - z */ |
---|
719 | AT_by_vec(csa, y,q); |
---|
720 | for (j = 1; j <= n; j++) q[j] = c[j] - q[j] - z[j]; |
---|
721 | /* r = - X * Z * e */ |
---|
722 | for (j = 1; j <= n; j++) r[j] = - x[j] * z[j]; |
---|
723 | /* solve the first Newtonian system */ |
---|
724 | if (solve_NS(csa, p, q, r, dx_aff, dy_aff, dz_aff)) |
---|
725 | { ret = 1; |
---|
726 | goto done; |
---|
727 | } |
---|
728 | /* alfa_aff_p = inf{0 <= alfa <= 1 | x + alfa*dx_aff >= 0} */ |
---|
729 | /* alfa_aff_d = inf{0 <= alfa <= 1 | z + alfa*dz_aff >= 0} */ |
---|
730 | csa->alfa_aff_p = csa->alfa_aff_d = 1.0; |
---|
731 | for (j = 1; j <= n; j++) |
---|
732 | { if (dx_aff[j] < 0.0) |
---|
733 | { temp = - x[j] / dx_aff[j]; |
---|
734 | if (csa->alfa_aff_p > temp) csa->alfa_aff_p = temp; |
---|
735 | } |
---|
736 | if (dz_aff[j] < 0.0) |
---|
737 | { temp = - z[j] / dz_aff[j]; |
---|
738 | if (csa->alfa_aff_d > temp) csa->alfa_aff_d = temp; |
---|
739 | } |
---|
740 | } |
---|
741 | /* mu_aff = (x+alfa_aff_p*dx_aff)' * (z+alfa_aff_d*dz_aff) / n */ |
---|
742 | temp = 0.0; |
---|
743 | for (j = 1; j <= n; j++) |
---|
744 | temp += (x[j] + csa->alfa_aff_p * dx_aff[j]) * |
---|
745 | (z[j] + csa->alfa_aff_d * dz_aff[j]); |
---|
746 | csa->mu_aff = temp / (double)n; |
---|
747 | /* sigma = (mu_aff/mu)^3 */ |
---|
748 | temp = csa->mu_aff / csa->mu; |
---|
749 | csa->sigma = temp * temp * temp; |
---|
750 | /* p = 0 */ |
---|
751 | for (i = 1; i <= m; i++) p[i] = 0.0; |
---|
752 | /* q = 0 */ |
---|
753 | for (j = 1; j <= n; j++) q[j] = 0.0; |
---|
754 | /* r = sigma * mu * e - X * Z * e */ |
---|
755 | for (j = 1; j <= n; j++) |
---|
756 | r[j] = csa->sigma * csa->mu - dx_aff[j] * dz_aff[j]; |
---|
757 | /* solve the second Newtonian system with the same coefficients |
---|
758 | but with altered right-hand sides */ |
---|
759 | if (solve_NS(csa, p, q, r, dx_cc, dy_cc, dz_cc)) |
---|
760 | { ret = 1; |
---|
761 | goto done; |
---|
762 | } |
---|
763 | /* (dx,dy,dz) = (dx_aff,dy_aff,dz_aff) + (dx_cc,dy_cc,dz_cc) */ |
---|
764 | for (j = 1; j <= n; j++) dx[j] = dx_aff[j] + dx_cc[j]; |
---|
765 | for (i = 1; i <= m; i++) dy[i] = dy_aff[i] + dy_cc[i]; |
---|
766 | for (j = 1; j <= n; j++) dz[j] = dz_aff[j] + dz_cc[j]; |
---|
767 | /* alfa_max_p = inf{0 <= alfa <= 1 | x + alfa*dx >= 0} */ |
---|
768 | /* alfa_max_d = inf{0 <= alfa <= 1 | z + alfa*dz >= 0} */ |
---|
769 | csa->alfa_max_p = csa->alfa_max_d = 1.0; |
---|
770 | for (j = 1; j <= n; j++) |
---|
771 | { if (dx[j] < 0.0) |
---|
772 | { temp = - x[j] / dx[j]; |
---|
773 | if (csa->alfa_max_p > temp) csa->alfa_max_p = temp; |
---|
774 | } |
---|
775 | if (dz[j] < 0.0) |
---|
776 | { temp = - z[j] / dz[j]; |
---|
777 | if (csa->alfa_max_d > temp) csa->alfa_max_d = temp; |
---|
778 | } |
---|
779 | } |
---|
780 | /* determine scale factors (not implemented yet) */ |
---|
781 | gamma_p = 0.90; |
---|
782 | gamma_d = 0.90; |
---|
783 | /* compute the next point */ |
---|
784 | for (j = 1; j <= n; j++) |
---|
785 | { x[j] += gamma_p * csa->alfa_max_p * dx[j]; |
---|
786 | xassert(x[j] > 0.0); |
---|
787 | } |
---|
788 | for (i = 1; i <= m; i++) |
---|
789 | y[i] += gamma_d * csa->alfa_max_d * dy[i]; |
---|
790 | for (j = 1; j <= n; j++) |
---|
791 | { z[j] += gamma_d * csa->alfa_max_d * dz[j]; |
---|
792 | xassert(z[j] > 0.0); |
---|
793 | } |
---|
794 | done: /* free working arrays */ |
---|
795 | xfree(p); |
---|
796 | xfree(q); |
---|
797 | xfree(r); |
---|
798 | return ret; |
---|
799 | } |
---|
800 | |
---|
801 | /*********************************************************************** |
---|
802 | * terminate - deallocate common storage area |
---|
803 | * |
---|
804 | * This routine frees all memory allocated to the common storage area |
---|
805 | * used by interior-point method routines. */ |
---|
806 | |
---|
807 | static void terminate(struct csa *csa) |
---|
808 | { xfree(csa->D); |
---|
809 | xfree(csa->P); |
---|
810 | xfree(csa->S_ptr); |
---|
811 | xfree(csa->S_ind); |
---|
812 | xfree(csa->S_val); |
---|
813 | xfree(csa->S_diag); |
---|
814 | xfree(csa->U_ptr); |
---|
815 | xfree(csa->U_ind); |
---|
816 | xfree(csa->U_val); |
---|
817 | xfree(csa->U_diag); |
---|
818 | xfree(csa->phi_min); |
---|
819 | xfree(csa->best_x); |
---|
820 | xfree(csa->best_y); |
---|
821 | xfree(csa->best_z); |
---|
822 | xfree(csa->dx_aff); |
---|
823 | xfree(csa->dy_aff); |
---|
824 | xfree(csa->dz_aff); |
---|
825 | xfree(csa->dx_cc); |
---|
826 | xfree(csa->dy_cc); |
---|
827 | xfree(csa->dz_cc); |
---|
828 | return; |
---|
829 | } |
---|
830 | |
---|
831 | /*********************************************************************** |
---|
832 | * ipm_main - main interior-point method routine |
---|
833 | * |
---|
834 | * This is a main routine of the primal-dual interior-point method. |
---|
835 | * |
---|
836 | * The routine ipm_main returns one of the following codes: |
---|
837 | * |
---|
838 | * 0 - optimal solution found; |
---|
839 | * 1 - problem has no feasible (primal or dual) solution; |
---|
840 | * 2 - no convergence; |
---|
841 | * 3 - iteration limit exceeded; |
---|
842 | * 4 - numeric instability on solving Newtonian system. |
---|
843 | * |
---|
844 | * In case of non-zero return code the routine returns the best point, |
---|
845 | * which has been reached during optimization. */ |
---|
846 | |
---|
847 | static int ipm_main(struct csa *csa) |
---|
848 | { int m = csa->m; |
---|
849 | int n = csa->n; |
---|
850 | int i, j, status; |
---|
851 | double temp; |
---|
852 | /* choose initial point using Mehrotra's heuristic */ |
---|
853 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
854 | xprintf("Guessing initial point...\n"); |
---|
855 | initial_point(csa); |
---|
856 | /* main loop starts here */ |
---|
857 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
858 | xprintf("Optimization begins...\n"); |
---|
859 | for (;;) |
---|
860 | { /* perform basic computations at the current point */ |
---|
861 | basic_info(csa); |
---|
862 | /* save initial value of rmu */ |
---|
863 | if (csa->iter == 0) csa->rmu0 = csa->rmu; |
---|
864 | /* accumulate values of min(phi[k]) and save the best point */ |
---|
865 | xassert(csa->iter <= ITER_MAX); |
---|
866 | if (csa->iter == 0 || csa->phi_min[csa->iter-1] > csa->phi) |
---|
867 | { csa->phi_min[csa->iter] = csa->phi; |
---|
868 | csa->best_iter = csa->iter; |
---|
869 | for (j = 1; j <= n; j++) csa->best_x[j] = csa->x[j]; |
---|
870 | for (i = 1; i <= m; i++) csa->best_y[i] = csa->y[i]; |
---|
871 | for (j = 1; j <= n; j++) csa->best_z[j] = csa->z[j]; |
---|
872 | csa->best_obj = csa->obj; |
---|
873 | } |
---|
874 | else |
---|
875 | csa->phi_min[csa->iter] = csa->phi_min[csa->iter-1]; |
---|
876 | /* display information at the current point */ |
---|
877 | if (csa->parm->msg_lev >= GLP_MSG_ON) |
---|
878 | xprintf("%3d: obj = %17.9e; rpi = %8.1e; rdi = %8.1e; gap =" |
---|
879 | " %8.1e\n", csa->iter, csa->obj, csa->rpi, csa->rdi, |
---|
880 | csa->gap); |
---|
881 | /* check if the current point is optimal */ |
---|
882 | if (csa->rpi < 1e-8 && csa->rdi < 1e-8 && csa->gap < 1e-8) |
---|
883 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
884 | xprintf("OPTIMAL SOLUTION FOUND\n"); |
---|
885 | status = 0; |
---|
886 | break; |
---|
887 | } |
---|
888 | /* check if the problem has no feasible solution */ |
---|
889 | temp = 1e5 * csa->phi_min[csa->iter]; |
---|
890 | if (temp < 1e-8) temp = 1e-8; |
---|
891 | if (csa->phi >= temp) |
---|
892 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
893 | xprintf("PROBLEM HAS NO FEASIBLE PRIMAL/DUAL SOLUTION\n") |
---|
894 | ; |
---|
895 | status = 1; |
---|
896 | break; |
---|
897 | } |
---|
898 | /* check for very slow convergence or divergence */ |
---|
899 | if (((csa->rpi >= 1e-8 || csa->rdi >= 1e-8) && csa->rmu / |
---|
900 | csa->rmu0 >= 1e6) || |
---|
901 | (csa->iter >= 30 && csa->phi_min[csa->iter] >= 0.5 * |
---|
902 | csa->phi_min[csa->iter - 30])) |
---|
903 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
904 | xprintf("NO CONVERGENCE; SEARCH TERMINATED\n"); |
---|
905 | status = 2; |
---|
906 | break; |
---|
907 | } |
---|
908 | /* check for maximal number of iterations */ |
---|
909 | if (csa->iter == ITER_MAX) |
---|
910 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
911 | xprintf("ITERATION LIMIT EXCEEDED; SEARCH TERMINATED\n"); |
---|
912 | status = 3; |
---|
913 | break; |
---|
914 | } |
---|
915 | /* start the next iteration */ |
---|
916 | csa->iter++; |
---|
917 | /* factorize normal equation system */ |
---|
918 | for (j = 1; j <= n; j++) csa->D[j] = csa->x[j] / csa->z[j]; |
---|
919 | decomp_NE(csa); |
---|
920 | /* compute the next point using Mehrotra's predictor-corrector |
---|
921 | technique */ |
---|
922 | if (make_step(csa)) |
---|
923 | { if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
924 | xprintf("NUMERIC INSTABILITY; SEARCH TERMINATED\n"); |
---|
925 | status = 4; |
---|
926 | break; |
---|
927 | } |
---|
928 | } |
---|
929 | /* restore the best point */ |
---|
930 | if (status != 0) |
---|
931 | { for (j = 1; j <= n; j++) csa->x[j] = csa->best_x[j]; |
---|
932 | for (i = 1; i <= m; i++) csa->y[i] = csa->best_y[i]; |
---|
933 | for (j = 1; j <= n; j++) csa->z[j] = csa->best_z[j]; |
---|
934 | if (csa->parm->msg_lev >= GLP_MSG_ALL) |
---|
935 | xprintf("Best point %17.9e was reached on iteration %d\n", |
---|
936 | csa->best_obj, csa->best_iter); |
---|
937 | } |
---|
938 | /* return to the calling program */ |
---|
939 | return status; |
---|
940 | } |
---|
941 | |
---|
942 | /*********************************************************************** |
---|
943 | * NAME |
---|
944 | * |
---|
945 | * ipm_solve - core LP solver based on the interior-point method |
---|
946 | * |
---|
947 | * SYNOPSIS |
---|
948 | * |
---|
949 | * #include "glpipm.h" |
---|
950 | * int ipm_solve(glp_prob *P, const glp_iptcp *parm); |
---|
951 | * |
---|
952 | * DESCRIPTION |
---|
953 | * |
---|
954 | * The routine ipm_solve is a core LP solver based on the primal-dual |
---|
955 | * interior-point method. |
---|
956 | * |
---|
957 | * The routine assumes the following standard formulation of LP problem |
---|
958 | * to be solved: |
---|
959 | * |
---|
960 | * minimize |
---|
961 | * |
---|
962 | * F = c[0] + c[1]*x[1] + c[2]*x[2] + ... + c[n]*x[n] |
---|
963 | * |
---|
964 | * subject to linear constraints |
---|
965 | * |
---|
966 | * a[1,1]*x[1] + a[1,2]*x[2] + ... + a[1,n]*x[n] = b[1] |
---|
967 | * |
---|
968 | * a[2,1]*x[1] + a[2,2]*x[2] + ... + a[2,n]*x[n] = b[2] |
---|
969 | * |
---|
970 | * . . . . . . |
---|
971 | * |
---|
972 | * a[m,1]*x[1] + a[m,2]*x[2] + ... + a[m,n]*x[n] = b[m] |
---|
973 | * |
---|
974 | * and non-negative variables |
---|
975 | * |
---|
976 | * x[1] >= 0, x[2] >= 0, ..., x[n] >= 0 |
---|
977 | * |
---|
978 | * where: |
---|
979 | * F is the objective function; |
---|
980 | * x[1], ..., x[n] are (structural) variables; |
---|
981 | * c[0] is a constant term of the objective function; |
---|
982 | * c[1], ..., c[n] are objective coefficients; |
---|
983 | * a[1,1], ..., a[m,n] are constraint coefficients; |
---|
984 | * b[1], ..., b[n] are right-hand sides. |
---|
985 | * |
---|
986 | * The solution is three vectors x, y, and z, which are stored by the |
---|
987 | * routine in the arrays x, y, and z, respectively. These vectors |
---|
988 | * correspond to the best primal-dual point found during optimization. |
---|
989 | * They are approximate solution of the following system (which is the |
---|
990 | * Karush-Kuhn-Tucker optimality conditions): |
---|
991 | * |
---|
992 | * A*x = b (primal feasibility condition) |
---|
993 | * |
---|
994 | * A'*y + z = c (dual feasibility condition) |
---|
995 | * |
---|
996 | * x'*z = 0 (primal-dual complementarity condition) |
---|
997 | * |
---|
998 | * x >= 0, z >= 0 (non-negativity condition) |
---|
999 | * |
---|
1000 | * where: |
---|
1001 | * x[1], ..., x[n] are primal (structural) variables; |
---|
1002 | * y[1], ..., y[m] are dual variables (Lagrange multipliers) for |
---|
1003 | * equality constraints; |
---|
1004 | * z[1], ..., z[n] are dual variables (Lagrange multipliers) for |
---|
1005 | * non-negativity constraints. |
---|
1006 | * |
---|
1007 | * RETURNS |
---|
1008 | * |
---|
1009 | * 0 LP has been successfully solved. |
---|
1010 | * |
---|
1011 | * GLP_ENOCVG |
---|
1012 | * No convergence. |
---|
1013 | * |
---|
1014 | * GLP_EITLIM |
---|
1015 | * Iteration limit exceeded. |
---|
1016 | * |
---|
1017 | * GLP_EINSTAB |
---|
1018 | * Numeric instability on solving Newtonian system. |
---|
1019 | * |
---|
1020 | * In case of non-zero return code the routine returns the best point, |
---|
1021 | * which has been reached during optimization. */ |
---|
1022 | |
---|
1023 | int ipm_solve(glp_prob *P, const glp_iptcp *parm) |
---|
1024 | { struct csa _dsa, *csa = &_dsa; |
---|
1025 | int m = P->m; |
---|
1026 | int n = P->n; |
---|
1027 | int nnz = P->nnz; |
---|
1028 | GLPROW *row; |
---|
1029 | GLPCOL *col; |
---|
1030 | GLPAIJ *aij; |
---|
1031 | int i, j, loc, ret, *A_ind, *A_ptr; |
---|
1032 | double dir, *A_val, *b, *c, *x, *y, *z; |
---|
1033 | xassert(m > 0); |
---|
1034 | xassert(n > 0); |
---|
1035 | /* allocate working arrays */ |
---|
1036 | A_ptr = xcalloc(1+m+1, sizeof(int)); |
---|
1037 | A_ind = xcalloc(1+nnz, sizeof(int)); |
---|
1038 | A_val = xcalloc(1+nnz, sizeof(double)); |
---|
1039 | b = xcalloc(1+m, sizeof(double)); |
---|
1040 | c = xcalloc(1+n, sizeof(double)); |
---|
1041 | x = xcalloc(1+n, sizeof(double)); |
---|
1042 | y = xcalloc(1+m, sizeof(double)); |
---|
1043 | z = xcalloc(1+n, sizeof(double)); |
---|
1044 | /* prepare rows and constraint coefficients */ |
---|
1045 | loc = 1; |
---|
1046 | for (i = 1; i <= m; i++) |
---|
1047 | { row = P->row[i]; |
---|
1048 | xassert(row->type == GLP_FX); |
---|
1049 | b[i] = row->lb * row->rii; |
---|
1050 | A_ptr[i] = loc; |
---|
1051 | for (aij = row->ptr; aij != NULL; aij = aij->r_next) |
---|
1052 | { A_ind[loc] = aij->col->j; |
---|
1053 | A_val[loc] = row->rii * aij->val * aij->col->sjj; |
---|
1054 | loc++; |
---|
1055 | } |
---|
1056 | } |
---|
1057 | A_ptr[m+1] = loc; |
---|
1058 | xassert(loc-1 == nnz); |
---|
1059 | /* prepare columns and objective coefficients */ |
---|
1060 | if (P->dir == GLP_MIN) |
---|
1061 | dir = +1.0; |
---|
1062 | else if (P->dir == GLP_MAX) |
---|
1063 | dir = -1.0; |
---|
1064 | else |
---|
1065 | xassert(P != P); |
---|
1066 | c[0] = dir * P->c0; |
---|
1067 | for (j = 1; j <= n; j++) |
---|
1068 | { col = P->col[j]; |
---|
1069 | xassert(col->type == GLP_LO && col->lb == 0.0); |
---|
1070 | c[j] = dir * col->coef * col->sjj; |
---|
1071 | } |
---|
1072 | /* allocate and initialize the common storage area */ |
---|
1073 | csa->m = m; |
---|
1074 | csa->n = n; |
---|
1075 | csa->A_ptr = A_ptr; |
---|
1076 | csa->A_ind = A_ind; |
---|
1077 | csa->A_val = A_val; |
---|
1078 | csa->b = b; |
---|
1079 | csa->c = c; |
---|
1080 | csa->x = x; |
---|
1081 | csa->y = y; |
---|
1082 | csa->z = z; |
---|
1083 | csa->parm = parm; |
---|
1084 | initialize(csa); |
---|
1085 | /* solve LP with the interior-point method */ |
---|
1086 | ret = ipm_main(csa); |
---|
1087 | /* deallocate the common storage area */ |
---|
1088 | terminate(csa); |
---|
1089 | /* determine solution status */ |
---|
1090 | if (ret == 0) |
---|
1091 | { /* optimal solution found */ |
---|
1092 | P->ipt_stat = GLP_OPT; |
---|
1093 | ret = 0; |
---|
1094 | } |
---|
1095 | else if (ret == 1) |
---|
1096 | { /* problem has no feasible (primal or dual) solution */ |
---|
1097 | P->ipt_stat = GLP_NOFEAS; |
---|
1098 | ret = 0; |
---|
1099 | } |
---|
1100 | else if (ret == 2) |
---|
1101 | { /* no convergence */ |
---|
1102 | P->ipt_stat = GLP_INFEAS; |
---|
1103 | ret = GLP_ENOCVG; |
---|
1104 | } |
---|
1105 | else if (ret == 3) |
---|
1106 | { /* iteration limit exceeded */ |
---|
1107 | P->ipt_stat = GLP_INFEAS; |
---|
1108 | ret = GLP_EITLIM; |
---|
1109 | } |
---|
1110 | else if (ret == 4) |
---|
1111 | { /* numeric instability on solving Newtonian system */ |
---|
1112 | P->ipt_stat = GLP_INFEAS; |
---|
1113 | ret = GLP_EINSTAB; |
---|
1114 | } |
---|
1115 | else |
---|
1116 | xassert(ret != ret); |
---|
1117 | /* store row solution components */ |
---|
1118 | for (i = 1; i <= m; i++) |
---|
1119 | { row = P->row[i]; |
---|
1120 | row->pval = row->lb; |
---|
1121 | row->dval = dir * y[i] * row->rii; |
---|
1122 | } |
---|
1123 | /* store column solution components */ |
---|
1124 | P->ipt_obj = P->c0; |
---|
1125 | for (j = 1; j <= n; j++) |
---|
1126 | { col = P->col[j]; |
---|
1127 | col->pval = x[j] * col->sjj; |
---|
1128 | col->dval = dir * z[j] / col->sjj; |
---|
1129 | P->ipt_obj += col->coef * col->pval; |
---|
1130 | } |
---|
1131 | /* free working arrays */ |
---|
1132 | xfree(A_ptr); |
---|
1133 | xfree(A_ind); |
---|
1134 | xfree(A_val); |
---|
1135 | xfree(b); |
---|
1136 | xfree(c); |
---|
1137 | xfree(x); |
---|
1138 | xfree(y); |
---|
1139 | xfree(z); |
---|
1140 | return ret; |
---|
1141 | } |
---|
1142 | |
---|
1143 | /* eof */ |
---|