1 | /* glpscl.c */ |
---|
2 | |
---|
3 | /*********************************************************************** |
---|
4 | * This code is part of GLPK (GNU Linear Programming Kit). |
---|
5 | * |
---|
6 | * Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, |
---|
7 | * 2009, 2010 Andrew Makhorin, Department for Applied Informatics, |
---|
8 | * Moscow Aviation Institute, Moscow, Russia. All rights reserved. |
---|
9 | * E-mail: <mao@gnu.org>. |
---|
10 | * |
---|
11 | * GLPK is free software: you can redistribute it and/or modify it |
---|
12 | * under the terms of the GNU General Public License as published by |
---|
13 | * the Free Software Foundation, either version 3 of the License, or |
---|
14 | * (at your option) any later version. |
---|
15 | * |
---|
16 | * GLPK is distributed in the hope that it will be useful, but WITHOUT |
---|
17 | * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
---|
18 | * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public |
---|
19 | * License for more details. |
---|
20 | * |
---|
21 | * You should have received a copy of the GNU General Public License |
---|
22 | * along with GLPK. If not, see <http://www.gnu.org/licenses/>. |
---|
23 | ***********************************************************************/ |
---|
24 | |
---|
25 | #include "glpapi.h" |
---|
26 | |
---|
27 | /*********************************************************************** |
---|
28 | * min_row_aij - determine minimal |a[i,j]| in i-th row |
---|
29 | * |
---|
30 | * This routine returns minimal magnitude of (non-zero) constraint |
---|
31 | * coefficients in i-th row of the constraint matrix. |
---|
32 | * |
---|
33 | * If the parameter scaled is zero, the original constraint matrix A is |
---|
34 | * assumed. Otherwise, the scaled constraint matrix R*A*S is assumed. |
---|
35 | * |
---|
36 | * If i-th row of the matrix is empty, the routine returns 1. */ |
---|
37 | |
---|
38 | static double min_row_aij(glp_prob *lp, int i, int scaled) |
---|
39 | { GLPAIJ *aij; |
---|
40 | double min_aij, temp; |
---|
41 | xassert(1 <= i && i <= lp->m); |
---|
42 | min_aij = 1.0; |
---|
43 | for (aij = lp->row[i]->ptr; aij != NULL; aij = aij->r_next) |
---|
44 | { temp = fabs(aij->val); |
---|
45 | if (scaled) temp *= (aij->row->rii * aij->col->sjj); |
---|
46 | if (aij->r_prev == NULL || min_aij > temp) |
---|
47 | min_aij = temp; |
---|
48 | } |
---|
49 | return min_aij; |
---|
50 | } |
---|
51 | |
---|
52 | /*********************************************************************** |
---|
53 | * max_row_aij - determine maximal |a[i,j]| in i-th row |
---|
54 | * |
---|
55 | * This routine returns maximal magnitude of (non-zero) constraint |
---|
56 | * coefficients in i-th row of the constraint matrix. |
---|
57 | * |
---|
58 | * If the parameter scaled is zero, the original constraint matrix A is |
---|
59 | * assumed. Otherwise, the scaled constraint matrix R*A*S is assumed. |
---|
60 | * |
---|
61 | * If i-th row of the matrix is empty, the routine returns 1. */ |
---|
62 | |
---|
63 | static double max_row_aij(glp_prob *lp, int i, int scaled) |
---|
64 | { GLPAIJ *aij; |
---|
65 | double max_aij, temp; |
---|
66 | xassert(1 <= i && i <= lp->m); |
---|
67 | max_aij = 1.0; |
---|
68 | for (aij = lp->row[i]->ptr; aij != NULL; aij = aij->r_next) |
---|
69 | { temp = fabs(aij->val); |
---|
70 | if (scaled) temp *= (aij->row->rii * aij->col->sjj); |
---|
71 | if (aij->r_prev == NULL || max_aij < temp) |
---|
72 | max_aij = temp; |
---|
73 | } |
---|
74 | return max_aij; |
---|
75 | } |
---|
76 | |
---|
77 | /*********************************************************************** |
---|
78 | * min_col_aij - determine minimal |a[i,j]| in j-th column |
---|
79 | * |
---|
80 | * This routine returns minimal magnitude of (non-zero) constraint |
---|
81 | * coefficients in j-th column of the constraint matrix. |
---|
82 | * |
---|
83 | * If the parameter scaled is zero, the original constraint matrix A is |
---|
84 | * assumed. Otherwise, the scaled constraint matrix R*A*S is assumed. |
---|
85 | * |
---|
86 | * If j-th column of the matrix is empty, the routine returns 1. */ |
---|
87 | |
---|
88 | static double min_col_aij(glp_prob *lp, int j, int scaled) |
---|
89 | { GLPAIJ *aij; |
---|
90 | double min_aij, temp; |
---|
91 | xassert(1 <= j && j <= lp->n); |
---|
92 | min_aij = 1.0; |
---|
93 | for (aij = lp->col[j]->ptr; aij != NULL; aij = aij->c_next) |
---|
94 | { temp = fabs(aij->val); |
---|
95 | if (scaled) temp *= (aij->row->rii * aij->col->sjj); |
---|
96 | if (aij->c_prev == NULL || min_aij > temp) |
---|
97 | min_aij = temp; |
---|
98 | } |
---|
99 | return min_aij; |
---|
100 | } |
---|
101 | |
---|
102 | /*********************************************************************** |
---|
103 | * max_col_aij - determine maximal |a[i,j]| in j-th column |
---|
104 | * |
---|
105 | * This routine returns maximal magnitude of (non-zero) constraint |
---|
106 | * coefficients in j-th column of the constraint matrix. |
---|
107 | * |
---|
108 | * If the parameter scaled is zero, the original constraint matrix A is |
---|
109 | * assumed. Otherwise, the scaled constraint matrix R*A*S is assumed. |
---|
110 | * |
---|
111 | * If j-th column of the matrix is empty, the routine returns 1. */ |
---|
112 | |
---|
113 | static double max_col_aij(glp_prob *lp, int j, int scaled) |
---|
114 | { GLPAIJ *aij; |
---|
115 | double max_aij, temp; |
---|
116 | xassert(1 <= j && j <= lp->n); |
---|
117 | max_aij = 1.0; |
---|
118 | for (aij = lp->col[j]->ptr; aij != NULL; aij = aij->c_next) |
---|
119 | { temp = fabs(aij->val); |
---|
120 | if (scaled) temp *= (aij->row->rii * aij->col->sjj); |
---|
121 | if (aij->c_prev == NULL || max_aij < temp) |
---|
122 | max_aij = temp; |
---|
123 | } |
---|
124 | return max_aij; |
---|
125 | } |
---|
126 | |
---|
127 | /*********************************************************************** |
---|
128 | * min_mat_aij - determine minimal |a[i,j]| in constraint matrix |
---|
129 | * |
---|
130 | * This routine returns minimal magnitude of (non-zero) constraint |
---|
131 | * coefficients in the constraint matrix. |
---|
132 | * |
---|
133 | * If the parameter scaled is zero, the original constraint matrix A is |
---|
134 | * assumed. Otherwise, the scaled constraint matrix R*A*S is assumed. |
---|
135 | * |
---|
136 | * If the matrix is empty, the routine returns 1. */ |
---|
137 | |
---|
138 | static double min_mat_aij(glp_prob *lp, int scaled) |
---|
139 | { int i; |
---|
140 | double min_aij, temp; |
---|
141 | min_aij = 1.0; |
---|
142 | for (i = 1; i <= lp->m; i++) |
---|
143 | { temp = min_row_aij(lp, i, scaled); |
---|
144 | if (i == 1 || min_aij > temp) |
---|
145 | min_aij = temp; |
---|
146 | } |
---|
147 | return min_aij; |
---|
148 | } |
---|
149 | |
---|
150 | /*********************************************************************** |
---|
151 | * max_mat_aij - determine maximal |a[i,j]| in constraint matrix |
---|
152 | * |
---|
153 | * This routine returns maximal magnitude of (non-zero) constraint |
---|
154 | * coefficients in the constraint matrix. |
---|
155 | * |
---|
156 | * If the parameter scaled is zero, the original constraint matrix A is |
---|
157 | * assumed. Otherwise, the scaled constraint matrix R*A*S is assumed. |
---|
158 | * |
---|
159 | * If the matrix is empty, the routine returns 1. */ |
---|
160 | |
---|
161 | static double max_mat_aij(glp_prob *lp, int scaled) |
---|
162 | { int i; |
---|
163 | double max_aij, temp; |
---|
164 | max_aij = 1.0; |
---|
165 | for (i = 1; i <= lp->m; i++) |
---|
166 | { temp = max_row_aij(lp, i, scaled); |
---|
167 | if (i == 1 || max_aij < temp) |
---|
168 | max_aij = temp; |
---|
169 | } |
---|
170 | return max_aij; |
---|
171 | } |
---|
172 | |
---|
173 | /*********************************************************************** |
---|
174 | * eq_scaling - perform equilibration scaling |
---|
175 | * |
---|
176 | * This routine performs equilibration scaling of rows and columns of |
---|
177 | * the constraint matrix. |
---|
178 | * |
---|
179 | * If the parameter flag is zero, the routine scales rows at first and |
---|
180 | * then columns. Otherwise, the routine scales columns and then rows. |
---|
181 | * |
---|
182 | * Rows are scaled as follows: |
---|
183 | * |
---|
184 | * n |
---|
185 | * a'[i,j] = a[i,j] / max |a[i,j]|, i = 1,...,m. |
---|
186 | * j=1 |
---|
187 | * |
---|
188 | * This makes the infinity (maximum) norm of each row of the matrix |
---|
189 | * equal to 1. |
---|
190 | * |
---|
191 | * Columns are scaled as follows: |
---|
192 | * |
---|
193 | * n |
---|
194 | * a'[i,j] = a[i,j] / max |a[i,j]|, j = 1,...,n. |
---|
195 | * i=1 |
---|
196 | * |
---|
197 | * This makes the infinity (maximum) norm of each column of the matrix |
---|
198 | * equal to 1. */ |
---|
199 | |
---|
200 | static void eq_scaling(glp_prob *lp, int flag) |
---|
201 | { int i, j, pass; |
---|
202 | double temp; |
---|
203 | xassert(flag == 0 || flag == 1); |
---|
204 | for (pass = 0; pass <= 1; pass++) |
---|
205 | { if (pass == flag) |
---|
206 | { /* scale rows */ |
---|
207 | for (i = 1; i <= lp->m; i++) |
---|
208 | { temp = max_row_aij(lp, i, 1); |
---|
209 | glp_set_rii(lp, i, glp_get_rii(lp, i) / temp); |
---|
210 | } |
---|
211 | } |
---|
212 | else |
---|
213 | { /* scale columns */ |
---|
214 | for (j = 1; j <= lp->n; j++) |
---|
215 | { temp = max_col_aij(lp, j, 1); |
---|
216 | glp_set_sjj(lp, j, glp_get_sjj(lp, j) / temp); |
---|
217 | } |
---|
218 | } |
---|
219 | } |
---|
220 | return; |
---|
221 | } |
---|
222 | |
---|
223 | /*********************************************************************** |
---|
224 | * gm_scaling - perform geometric mean scaling |
---|
225 | * |
---|
226 | * This routine performs geometric mean scaling of rows and columns of |
---|
227 | * the constraint matrix. |
---|
228 | * |
---|
229 | * If the parameter flag is zero, the routine scales rows at first and |
---|
230 | * then columns. Otherwise, the routine scales columns and then rows. |
---|
231 | * |
---|
232 | * Rows are scaled as follows: |
---|
233 | * |
---|
234 | * a'[i,j] = a[i,j] / sqrt(alfa[i] * beta[i]), i = 1,...,m, |
---|
235 | * |
---|
236 | * where: |
---|
237 | * n n |
---|
238 | * alfa[i] = min |a[i,j]|, beta[i] = max |a[i,j]|. |
---|
239 | * j=1 j=1 |
---|
240 | * |
---|
241 | * This allows decreasing the ratio beta[i] / alfa[i] for each row of |
---|
242 | * the matrix. |
---|
243 | * |
---|
244 | * Columns are scaled as follows: |
---|
245 | * |
---|
246 | * a'[i,j] = a[i,j] / sqrt(alfa[j] * beta[j]), j = 1,...,n, |
---|
247 | * |
---|
248 | * where: |
---|
249 | * m m |
---|
250 | * alfa[j] = min |a[i,j]|, beta[j] = max |a[i,j]|. |
---|
251 | * i=1 i=1 |
---|
252 | * |
---|
253 | * This allows decreasing the ratio beta[j] / alfa[j] for each column |
---|
254 | * of the matrix. */ |
---|
255 | |
---|
256 | static void gm_scaling(glp_prob *lp, int flag) |
---|
257 | { int i, j, pass; |
---|
258 | double temp; |
---|
259 | xassert(flag == 0 || flag == 1); |
---|
260 | for (pass = 0; pass <= 1; pass++) |
---|
261 | { if (pass == flag) |
---|
262 | { /* scale rows */ |
---|
263 | for (i = 1; i <= lp->m; i++) |
---|
264 | { temp = min_row_aij(lp, i, 1) * max_row_aij(lp, i, 1); |
---|
265 | glp_set_rii(lp, i, glp_get_rii(lp, i) / sqrt(temp)); |
---|
266 | } |
---|
267 | } |
---|
268 | else |
---|
269 | { /* scale columns */ |
---|
270 | for (j = 1; j <= lp->n; j++) |
---|
271 | { temp = min_col_aij(lp, j, 1) * max_col_aij(lp, j, 1); |
---|
272 | glp_set_sjj(lp, j, glp_get_sjj(lp, j) / sqrt(temp)); |
---|
273 | } |
---|
274 | } |
---|
275 | } |
---|
276 | return; |
---|
277 | } |
---|
278 | |
---|
279 | /*********************************************************************** |
---|
280 | * max_row_ratio - determine worst scaling "quality" for rows |
---|
281 | * |
---|
282 | * This routine returns the worst scaling "quality" for rows of the |
---|
283 | * currently scaled constraint matrix: |
---|
284 | * |
---|
285 | * m |
---|
286 | * ratio = max ratio[i], |
---|
287 | * i=1 |
---|
288 | * where: |
---|
289 | * n n |
---|
290 | * ratio[i] = max |a[i,j]| / min |a[i,j]|, 1 <= i <= m, |
---|
291 | * j=1 j=1 |
---|
292 | * |
---|
293 | * is the scaling "quality" of i-th row. */ |
---|
294 | |
---|
295 | static double max_row_ratio(glp_prob *lp) |
---|
296 | { int i; |
---|
297 | double ratio, temp; |
---|
298 | ratio = 1.0; |
---|
299 | for (i = 1; i <= lp->m; i++) |
---|
300 | { temp = max_row_aij(lp, i, 1) / min_row_aij(lp, i, 1); |
---|
301 | if (i == 1 || ratio < temp) ratio = temp; |
---|
302 | } |
---|
303 | return ratio; |
---|
304 | } |
---|
305 | |
---|
306 | /*********************************************************************** |
---|
307 | * max_col_ratio - determine worst scaling "quality" for columns |
---|
308 | * |
---|
309 | * This routine returns the worst scaling "quality" for columns of the |
---|
310 | * currently scaled constraint matrix: |
---|
311 | * |
---|
312 | * n |
---|
313 | * ratio = max ratio[j], |
---|
314 | * j=1 |
---|
315 | * where: |
---|
316 | * m m |
---|
317 | * ratio[j] = max |a[i,j]| / min |a[i,j]|, 1 <= j <= n, |
---|
318 | * i=1 i=1 |
---|
319 | * |
---|
320 | * is the scaling "quality" of j-th column. */ |
---|
321 | |
---|
322 | static double max_col_ratio(glp_prob *lp) |
---|
323 | { int j; |
---|
324 | double ratio, temp; |
---|
325 | ratio = 1.0; |
---|
326 | for (j = 1; j <= lp->n; j++) |
---|
327 | { temp = max_col_aij(lp, j, 1) / min_col_aij(lp, j, 1); |
---|
328 | if (j == 1 || ratio < temp) ratio = temp; |
---|
329 | } |
---|
330 | return ratio; |
---|
331 | } |
---|
332 | |
---|
333 | /*********************************************************************** |
---|
334 | * gm_iterate - perform iterative geometric mean scaling |
---|
335 | * |
---|
336 | * This routine performs iterative geometric mean scaling of rows and |
---|
337 | * columns of the constraint matrix. |
---|
338 | * |
---|
339 | * The parameter it_max specifies the maximal number of iterations. |
---|
340 | * Recommended value of it_max is 15. |
---|
341 | * |
---|
342 | * The parameter tau specifies a minimal improvement of the scaling |
---|
343 | * "quality" on each iteration, 0 < tau < 1. It means than the scaling |
---|
344 | * process continues while the following condition is satisfied: |
---|
345 | * |
---|
346 | * ratio[k] <= tau * ratio[k-1], |
---|
347 | * |
---|
348 | * where ratio = max |a[i,j]| / min |a[i,j]| is the scaling "quality" |
---|
349 | * to be minimized, k is the iteration number. Recommended value of tau |
---|
350 | * is 0.90. */ |
---|
351 | |
---|
352 | static void gm_iterate(glp_prob *lp, int it_max, double tau) |
---|
353 | { int k, flag; |
---|
354 | double ratio = 0.0, r_old; |
---|
355 | /* if the scaling "quality" for rows is better than for columns, |
---|
356 | the rows are scaled first; otherwise, the columns are scaled |
---|
357 | first */ |
---|
358 | flag = (max_row_ratio(lp) > max_col_ratio(lp)); |
---|
359 | for (k = 1; k <= it_max; k++) |
---|
360 | { /* save the scaling "quality" from previous iteration */ |
---|
361 | r_old = ratio; |
---|
362 | /* determine the current scaling "quality" */ |
---|
363 | ratio = max_mat_aij(lp, 1) / min_mat_aij(lp, 1); |
---|
364 | #if 0 |
---|
365 | xprintf("k = %d; ratio = %g\n", k, ratio); |
---|
366 | #endif |
---|
367 | /* if improvement is not enough, terminate scaling */ |
---|
368 | if (k > 1 && ratio > tau * r_old) break; |
---|
369 | /* otherwise, perform another iteration */ |
---|
370 | gm_scaling(lp, flag); |
---|
371 | } |
---|
372 | return; |
---|
373 | } |
---|
374 | |
---|
375 | /*********************************************************************** |
---|
376 | * NAME |
---|
377 | * |
---|
378 | * scale_prob - scale problem data |
---|
379 | * |
---|
380 | * SYNOPSIS |
---|
381 | * |
---|
382 | * #include "glpscl.h" |
---|
383 | * void scale_prob(glp_prob *lp, int flags); |
---|
384 | * |
---|
385 | * DESCRIPTION |
---|
386 | * |
---|
387 | * The routine scale_prob performs automatic scaling of problem data |
---|
388 | * for the specified problem object. */ |
---|
389 | |
---|
390 | static void scale_prob(glp_prob *lp, int flags) |
---|
391 | { static const char *fmt = |
---|
392 | "%s: min|aij| = %10.3e max|aij| = %10.3e ratio = %10.3e\n"; |
---|
393 | double min_aij, max_aij, ratio; |
---|
394 | xprintf("Scaling...\n"); |
---|
395 | /* cancel the current scaling effect */ |
---|
396 | glp_unscale_prob(lp); |
---|
397 | /* report original scaling "quality" */ |
---|
398 | min_aij = min_mat_aij(lp, 1); |
---|
399 | max_aij = max_mat_aij(lp, 1); |
---|
400 | ratio = max_aij / min_aij; |
---|
401 | xprintf(fmt, " A", min_aij, max_aij, ratio); |
---|
402 | /* check if the problem is well scaled */ |
---|
403 | if (min_aij >= 0.10 && max_aij <= 10.0) |
---|
404 | { xprintf("Problem data seem to be well scaled\n"); |
---|
405 | /* skip scaling, if required */ |
---|
406 | if (flags & GLP_SF_SKIP) goto done; |
---|
407 | } |
---|
408 | /* perform iterative geometric mean scaling, if required */ |
---|
409 | if (flags & GLP_SF_GM) |
---|
410 | { gm_iterate(lp, 15, 0.90); |
---|
411 | min_aij = min_mat_aij(lp, 1); |
---|
412 | max_aij = max_mat_aij(lp, 1); |
---|
413 | ratio = max_aij / min_aij; |
---|
414 | xprintf(fmt, "GM", min_aij, max_aij, ratio); |
---|
415 | } |
---|
416 | /* perform equilibration scaling, if required */ |
---|
417 | if (flags & GLP_SF_EQ) |
---|
418 | { eq_scaling(lp, max_row_ratio(lp) > max_col_ratio(lp)); |
---|
419 | min_aij = min_mat_aij(lp, 1); |
---|
420 | max_aij = max_mat_aij(lp, 1); |
---|
421 | ratio = max_aij / min_aij; |
---|
422 | xprintf(fmt, "EQ", min_aij, max_aij, ratio); |
---|
423 | } |
---|
424 | /* round scale factors to nearest power of two, if required */ |
---|
425 | if (flags & GLP_SF_2N) |
---|
426 | { int i, j; |
---|
427 | for (i = 1; i <= lp->m; i++) |
---|
428 | glp_set_rii(lp, i, round2n(glp_get_rii(lp, i))); |
---|
429 | for (j = 1; j <= lp->n; j++) |
---|
430 | glp_set_sjj(lp, j, round2n(glp_get_sjj(lp, j))); |
---|
431 | min_aij = min_mat_aij(lp, 1); |
---|
432 | max_aij = max_mat_aij(lp, 1); |
---|
433 | ratio = max_aij / min_aij; |
---|
434 | xprintf(fmt, "2N", min_aij, max_aij, ratio); |
---|
435 | } |
---|
436 | done: return; |
---|
437 | } |
---|
438 | |
---|
439 | /*********************************************************************** |
---|
440 | * NAME |
---|
441 | * |
---|
442 | * glp_scale_prob - scale problem data |
---|
443 | * |
---|
444 | * SYNOPSIS |
---|
445 | * |
---|
446 | * void glp_scale_prob(glp_prob *lp, int flags); |
---|
447 | * |
---|
448 | * DESCRIPTION |
---|
449 | * |
---|
450 | * The routine glp_scale_prob performs automatic scaling of problem |
---|
451 | * data for the specified problem object. |
---|
452 | * |
---|
453 | * The parameter flags specifies scaling options used by the routine. |
---|
454 | * Options can be combined with the bitwise OR operator and may be the |
---|
455 | * following: |
---|
456 | * |
---|
457 | * GLP_SF_GM perform geometric mean scaling; |
---|
458 | * GLP_SF_EQ perform equilibration scaling; |
---|
459 | * GLP_SF_2N round scale factors to nearest power of two; |
---|
460 | * GLP_SF_SKIP skip scaling, if the problem is well scaled. |
---|
461 | * |
---|
462 | * The parameter flags may be specified as GLP_SF_AUTO, in which case |
---|
463 | * the routine chooses scaling options automatically. */ |
---|
464 | |
---|
465 | void glp_scale_prob(glp_prob *lp, int flags) |
---|
466 | { if (flags & ~(GLP_SF_GM | GLP_SF_EQ | GLP_SF_2N | GLP_SF_SKIP | |
---|
467 | GLP_SF_AUTO)) |
---|
468 | xerror("glp_scale_prob: flags = 0x%02X; invalid scaling option" |
---|
469 | "s\n", flags); |
---|
470 | if (flags & GLP_SF_AUTO) |
---|
471 | flags = (GLP_SF_GM | GLP_SF_EQ | GLP_SF_SKIP); |
---|
472 | scale_prob(lp, flags); |
---|
473 | return; |
---|
474 | } |
---|
475 | |
---|
476 | /* eof */ |
---|