1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
|
C------------------------------------------------------------------------------
SUBROUTINE TRSTLP (N,M,A,B,RHO,DX,IFULL,IACT,Z,ZDOTA,VMULTC,
1 SDIRN,DXNEW,VMULTD,IPRINT)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DOUBLE PRECISION TEMP
DIMENSION A(N,*),B(*),DX(*),IACT(*),Z(N,*),ZDOTA(*),
1 VMULTC(*),SDIRN(*),DXNEW(*),VMULTD(*)
C
C This subroutine calculates an N-component vector DX by applying the
C following two stages. In the first stage, DX is set to the shortest
C vector that minimizes the greatest violation of the constraints
C A(1,K)*DX(1)+A(2,K)*DX(2)+...+A(N,K)*DX(N) .GE. B(K), K=2,3,...,M,
C subject to the Euclidean length of DX being at most RHO. If its length is
C strictly less than RHO, then we use the resultant freedom in DX to
C minimize the objective function
C -A(1,M+1)*DX(1)-A(2,M+1)*DX(2)-...-A(N,M+1)*DX(N)
C subject to no increase in any greatest constraint violation. This
C notation allows the gradient of the objective function to be regarded as
C the gradient of a constraint. Therefore the two stages are distinguished
C by MCON .EQ. M and MCON .GT. M respectively. It is possible that a
C degeneracy may prevent DX from attaining the target length RHO. Then the
C value IFULL=0 would be set, but usually IFULL=1 on return.
C
C In general NACT is the number of constraints in the active set and
C IACT(1),...,IACT(NACT) are their indices, while the remainder of IACT
C contains a permutation of the remaining constraint indices. Further, Z is
C an orthogonal matrix whose first NACT columns can be regarded as the
C result of Gram-Schmidt applied to the active constraint gradients. For
C J=1,2,...,NACT, the number ZDOTA(J) is the scalar product of the J-th
C column of Z with the gradient of the J-th active constraint. DX is the
C current vector of variables and here the residuals of the active
C constraints should be zero. Further, the active constraints have
C nonnegative Lagrange multipliers that are held at the beginning of
C VMULTC. The remainder of this vector holds the residuals of the inactive
C constraints at DX, the ordering of the components of VMULTC being in
C agreement with the permutation of the indices of the constraints that is
C in IACT. All these residuals are nonnegative, which is achieved by the
C shift RESMAX that makes the least residual zero.
C
C Initialize Z and some other variables. The value of RESMAX will be
C appropriate to DX=0, while ICON will be the index of a most violated
C constraint if RESMAX is positive. Usually during the first stage the
C vector SDIRN gives a search direction that reduces all the active
C constraint violations by one simultaneously.
C
IF (IPRINT .EQ. 3) THEN
print *, ' '
print *, 'BEFORE trstlp:'
PRINT *, ' **DX = ', (DX(I),I=1,N)
PRINT *, ' **IACT = ', (IACT(I),I=1,M+1)
PRINT *, 'M,N,RHO,IFULL =', M, N, RHO, IFULL
PRINT *, ' **A = ', ((A(I,K),I=1,N),K=1,M+1)
PRINT *, ' **B = ', (B(I),I=1,M)
PRINT *, ' **Z = ', ((Z(I,K),I=1,N),K=1,N)
PRINT *, ' **ZDOTA = ', (ZDOTA(I),I=1,N)
PRINT *, ' **VMULTC = ', (VMULTC(I),I=1,M+1)
PRINT *, ' **SDIRN = ', (SDIRN(I),I=1,N)
PRINT *, ' **DXNEW = ', (DXNEW(I),I=1,N)
PRINT *, ' **VMULTD = ', (VMULTD(I),I=1,M+1)
PRINT *, ' '
END IF
ICON=0
NACTX=0
RESOLD=0
IFULL=1
MCON=M
NACT=0
RESMAX=0.0d0
DO 20 I=1,N
DO 10 J=1,N
10 Z(I,J)=0.0d0
Z(I,I)=1.0d0
20 DX(I)=0.0d0
IF (M .GE. 1) THEN
DO 30 K=1,M
IF (B(K) .GT. RESMAX) THEN
RESMAX=B(K)
ICON=K
END IF
30 CONTINUE
DO 40 K=1,M
IACT(K)=K
40 VMULTC(K)=RESMAX-B(K)
END IF
IF (IPRINT .EQ. 3) THEN
PRINT *, ' 1. VMULTC = ', (VMULTC(I),I=1,M+1)
END IF
IF (RESMAX .EQ. 0.0d0) GOTO 480
DO 50 I=1,N
50 SDIRN(I)=0.0d0
C
C End the current stage of the calculation if 3 consecutive iterations
C have either failed to reduce the best calculated value of the objective
C function or to increase the number of active constraints since the best
C value was calculated. This strategy prevents cycling, but there is a
C remote possibility that it will cause premature termination.
C
60 OPTOLD=0.0d0
ICOUNT=0
70 IF (MCON .EQ. M) THEN
OPTNEW=RESMAX
ELSE
OPTNEW=0.0d0
DO 80 I=1,N
80 OPTNEW=OPTNEW-DX(I)*A(I,MCON)
END IF
IF (IPRINT .EQ. 3) THEN
PRINT *, ' ICOUNT, OPTNEW, OPTOLD = ', ICOUNT, OPTNEW, OPTOLD
END IF
IF (ICOUNT .EQ. 0 .OR. OPTNEW .LT. OPTOLD) THEN
OPTOLD=OPTNEW
NACTX=NACT
ICOUNT=3
ELSE IF (NACT .GT. NACTX) THEN
NACTX=NACT
ICOUNT=3
ELSE
ICOUNT=ICOUNT-1
IF (ICOUNT .EQ. 0) GOTO 490
END IF
C
C If ICON exceeds NACT, then we add the constraint with index IACT(ICON) to
C the active set. Apply Givens rotations so that the last N-NACT-1 columns
C of Z are orthogonal to the gradient of the new constraint, a scalar
C product being set to zero if its nonzero value could be due to computer
C rounding errors. The array DXNEW is used for working space.
C
IF (ICON .LE. NACT) GOTO 260
KK=IACT(ICON)
DO 90 I=1,N
90 DXNEW(I)=A(I,KK)
TOT=0.0D0
K=N
100 IF (K .GT. NACT) THEN
SP=0.0d0
SPABS=0.0d0
DO 110 I=1,N
TEMP=Z(I,K)*DXNEW(I)
SP=SP+TEMP
110 SPABS=SPABS+DABS(TEMP)
ACCA=SPABS+0.1d0*DABS(SP)
ACCB=SPABS+0.2d0*DABS(SP)
IF ((SPABS .GE. ACCA) .OR. (ACCA .GE. ACCB)) SP=0.0D0
IF (TOT .EQ. 0.0D0) THEN
TOT=SP
ELSE
KP=K+1
TEMP=DSQRT(SP*SP+TOT*TOT)
ALPHA=SP/TEMP
BETA=TOT/TEMP
TOT=TEMP
DO 120 I=1,N
TEMP=ALPHA*Z(I,K)+BETA*Z(I,KP)
Z(I,KP)=ALPHA*Z(I,KP)-BETA*Z(I,K)
120 Z(I,K)=TEMP
END IF
K=K-1
GOTO 100
END IF
C
C Add the new constraint if this can be done without a deletion from the
C active set.
C
IF (IPRINT .EQ. 3) THEN
PRINT *, '*TOT, NACT, ICON = ', TOT, NACT, ICON
END IF
IF (TOT .NE. 0.0d0) THEN
NACT=NACT+1
ZDOTA(NACT)=TOT
VMULTC(ICON)=VMULTC(NACT)
VMULTC(NACT)=0.0d0
GOTO 210
END IF
C
C The next instruction is reached if a deletion has to be made from the
C active set in order to make room for the new active constraint, because
C the new constraint gradient is a linear combination of the gradients of
C the old active constraints. Set the elements of VMULTD to the multipliers
C of the linear combination. Further, set IOUT to the index of the
C constraint to be deleted, but branch if no suitable index can be found.
C
RATIO=-1.0d0
K=NACT
130 ZDOTV=0.0d0
ZDVABS=0.0d0
DO 140 I=1,N
TEMP=Z(I,K)*DXNEW(I)
ZDOTV=ZDOTV+TEMP
140 ZDVABS=ZDVABS+DABS(TEMP)
ACCA=ZDVABS+0.1d0*DABS(ZDOTV)
ACCB=ZDVABS+0.2d0*DABS(ZDOTV)
IF (ZDVABS .LT. ACCA .AND. ACCA .LT. ACCB) THEN
TEMP=ZDOTV/ZDOTA(K)
IF (TEMP .GT. 0.0d0 .AND. IACT(K) .LE. M) THEN
TEMPA=VMULTC(K)/TEMP
IF (RATIO .LT. 0.0d0 .OR. TEMPA .LT. RATIO) THEN
RATIO=TEMPA
IOUT=K
END IF
END IF
IF (K .GE. 2) THEN
KW=IACT(K)
DO 150 I=1,N
150 DXNEW(I)=DXNEW(I)-TEMP*A(I,KW)
END IF
VMULTD(K)=TEMP
ELSE
VMULTD(K)=0.0d0
END IF
K=K-1
IF (K .GT. 0) GOTO 130
IF (IPRINT .EQ. 3) THEN
PRINT *, ' 1. VMULTD = ', (VMULTD(I),I=1,M+1)
END IF
IF (RATIO .LT. 0.0d0) GOTO 490
C
C Revise the Lagrange multipliers and reorder the active constraints so
C that the one to be replaced is at the end of the list. Also calculate the
C new value of ZDOTA(NACT) and branch if it is not acceptable.
C
DO 160 K=1,NACT
160 VMULTC(K)=DMAX1(0.0d0,VMULTC(K)-RATIO*VMULTD(K))
IF (IPRINT .EQ. 3) THEN
PRINT *, ' 2. VMULTC = ', (VMULTC(I),I=1,M+1)
END IF
IF (ICON .LT. NACT) THEN
ISAVE=IACT(ICON)
VSAVE=VMULTC(ICON)
K=ICON
170 KP=K+1
KW=IACT(KP)
SP=0.0d0
DO 180 I=1,N
180 SP=SP+Z(I,K)*A(I,KW)
TEMP=SQRT(SP*SP+ZDOTA(KP)**2)
ALPHA=ZDOTA(KP)/TEMP
BETA=SP/TEMP
ZDOTA(KP)=ALPHA*ZDOTA(K)
ZDOTA(K)=TEMP
DO 190 I=1,N
TEMP=ALPHA*Z(I,KP)+BETA*Z(I,K)
Z(I,KP)=ALPHA*Z(I,K)-BETA*Z(I,KP)
190 Z(I,K)=TEMP
IACT(K)=KW
VMULTC(K)=VMULTC(KP)
K=KP
IF (K .LT. NACT) GOTO 170
IACT(K)=ISAVE
VMULTC(K)=VSAVE
END IF
TEMP=0.0d0
DO 200 I=1,N
200 TEMP=TEMP+Z(I,NACT)*A(I,KK)
IF (TEMP .EQ. 0.0d0) GOTO 490
ZDOTA(NACT)=TEMP
VMULTC(ICON)=0.0d0
VMULTC(NACT)=RATIO
C
C Update IACT and ensure that the objective function continues to be
C treated as the last active constraint when MCON>M.
C
210 IACT(ICON)=IACT(NACT)
IACT(NACT)=KK
IF (MCON .GT. M .AND. KK .NE. MCON) THEN
K=NACT-1
SP=0.0d0
DO 220 I=1,N
220 SP=SP+Z(I,K)*A(I,KK)
TEMP=SQRT(SP*SP+ZDOTA(NACT)**2)
ALPHA=ZDOTA(NACT)/TEMP
BETA=SP/TEMP
ZDOTA(NACT)=ALPHA*ZDOTA(K)
ZDOTA(K)=TEMP
DO 230 I=1,N
TEMP=ALPHA*Z(I,NACT)+BETA*Z(I,K)
Z(I,NACT)=ALPHA*Z(I,K)-BETA*Z(I,NACT)
230 Z(I,K)=TEMP
IACT(NACT)=IACT(K)
IACT(K)=KK
TEMP=VMULTC(K)
VMULTC(K)=VMULTC(NACT)
VMULTC(NACT)=TEMP
END IF
C
C If stage one is in progress, then set SDIRN to the direction of the next
C change to the current vector of variables.
C
IF (MCON .GT. M) GOTO 320
KK=IACT(NACT)
TEMP=0.0d0
DO 240 I=1,N
240 TEMP=TEMP+SDIRN(I)*A(I,KK)
TEMP=TEMP-1.0d0
TEMP=TEMP/ZDOTA(NACT)
DO 250 I=1,N
250 SDIRN(I)=SDIRN(I)-TEMP*Z(I,NACT)
GOTO 340
C
C Delete the constraint that has the index IACT(ICON) from the active set.
C
260 IF (ICON .LT. NACT) THEN
ISAVE=IACT(ICON)
VSAVE=VMULTC(ICON)
K=ICON
270 KP=K+1
KK=IACT(KP)
SP=0.0d0
DO 280 I=1,N
280 SP=SP+Z(I,K)*A(I,KK)
TEMP=SQRT(SP*SP+ZDOTA(KP)**2)
ALPHA=ZDOTA(KP)/TEMP
BETA=SP/TEMP
ZDOTA(KP)=ALPHA*ZDOTA(K)
ZDOTA(K)=TEMP
DO 290 I=1,N
TEMP=ALPHA*Z(I,KP)+BETA*Z(I,K)
Z(I,KP)=ALPHA*Z(I,K)-BETA*Z(I,KP)
290 Z(I,K)=TEMP
IACT(K)=KK
VMULTC(K)=VMULTC(KP)
K=KP
IF (K .LT. NACT) GOTO 270
IACT(K)=ISAVE
VMULTC(K)=VSAVE
END IF
NACT=NACT-1
C
C If stage one is in progress, then set SDIRN to the direction of the next
C change to the current vector of variables.
C
IF (MCON .GT. M) GOTO 320
TEMP=0.0d0
DO 300 I=1,N
300 TEMP=TEMP+SDIRN(I)*Z(I,NACT+1)
DO 310 I=1,N
310 SDIRN(I)=SDIRN(I)-TEMP*Z(I,NACT+1)
GO TO 340
C
C Pick the next search direction of stage two.
C
320 TEMP=1.0d0/ZDOTA(NACT)
DO 330 I=1,N
330 SDIRN(I)=TEMP*Z(I,NACT)
C
C Calculate the step to the boundary of the trust region or take the step
C that reduces RESMAX to zero. The two statements below that include the
C factor 1.0E-6 prevent some harmless underflows that occurred in a test
C calculation. Further, we skip the step if it could be zero within a
C reasonable tolerance for computer rounding errors.
C
340 DD=RHO*RHO
SD=0.0d0
SS=0.0d0
DO 350 I=1,N
IF (ABS(DX(I)) .GE. 1.0E-6*RHO) DD=DD-DX(I)**2
SD=SD+DX(I)*SDIRN(I)
350 SS=SS+SDIRN(I)**2
IF (DD .LE. 0.0d0) GOTO 490
TEMP=SQRT(SS*DD)
IF (ABS(SD) .GE. 1.0E-6*TEMP) TEMP=SQRT(SS*DD+SD*SD)
STPFUL=DD/(TEMP+SD)
STEP=STPFUL
IF (MCON .EQ. M) THEN
ACCA=STEP+0.1d0*RESMAX
ACCB=STEP+0.2d0*RESMAX
IF (STEP .GE. ACCA .OR. ACCA .GE. ACCB) GOTO 480
STEP=DMIN1(STEP,RESMAX)
END IF
C
C Set DXNEW to the new variables if STEP is the steplength, and reduce
C RESMAX to the corresponding maximum residual if stage one is being done.
C Because DXNEW will be changed during the calculation of some Lagrange
C multipliers, it will be restored to the following value later.
call s360_380(DXNEW,DX,STEP,SDIRN,N,M,MCON,RESMAX,
1 NACT,IACT,B,A,RESOLD)
C
C Set VMULTD to the VMULTC vector that would occur if DX became DXNEW. A
C device is included to force VMULTD(K)=0.0 if deviations from this value
C can be attributed to computer rounding errors. First calculate the new
C Lagrange multipliers.
C
K=NACT
390 ZDOTW=0.0d0
ZDWABS=0.0d0
DO 400 I=1,N
TEMP=Z(I,K)*DXNEW(I)
ZDOTW=ZDOTW+TEMP
400 ZDWABS=ZDWABS+ABS(TEMP)
ACCA=ZDWABS+0.1d0*ABS(ZDOTW)
ACCB=ZDWABS+0.2d0*ABS(ZDOTW)
IF (ZDWABS .GE. ACCA .OR. ACCA .GE. ACCB) ZDOTW=0.0d0
VMULTD(K)=ZDOTW/ZDOTA(K)
IF (K .GE. 2) THEN
KK=IACT(K)
DO 410 I=1,N
410 DXNEW(I)=DXNEW(I)-VMULTD(K)*A(I,KK)
K=K-1
GOTO 390
END IF
IF (MCON .GT. M) VMULTD(NACT)=DMAX1(0.0d0,VMULTD(NACT))
IF (IPRINT .EQ. 3) THEN
PRINT *, ' 2. VMULTD = ', (VMULTD(I),I=1,M+1)
END IF
C
C Complete VMULTC by finding the new constraint residuals.
C
DO 420 I=1,N
420 DXNEW(I)=DX(I)+STEP*SDIRN(I)
IF (MCON .GT. NACT) THEN
KL=NACT+1
DO 440 K=KL,MCON
KK=IACT(K)
SUM=RESMAX-B(KK)
SUMABS=RESMAX+DABS(B(KK))
DO 430 I=1,N
TEMP=A(I,KK)*DXNEW(I)
SUM=SUM+TEMP
430 SUMABS=SUMABS+DABS(TEMP)
ACCA=SUMABS+0.1*DABS(SUM)
ACCB=SUMABS+0.2*DABS(SUM)
IF (SUMABS .GE. ACCA .OR. ACCA .GE. ACCB) SUM=0.0
440 VMULTD(K)=SUM
END IF
IF (IPRINT .EQ. 3) THEN
PRINT *, ' 3. VMULTD = ', (VMULTD(I),I=1,M+1)
END IF
C
C Calculate the fraction of the step from DX to DXNEW that will be taken.
C
RATIO=1.0d0
ICON=0
C
EPS = 2.2E-16
DO 450 K=1,MCON
IF (VMULTD(K) .GT. -EPS .AND. VMULTD(K) .LT. EPS) VMULTD(K)=0.0D0
IF (VMULTD(K) .LT. 0.0D0) THEN
TEMP=VMULTC(K)/(VMULTC(K)-VMULTD(K))
IF (TEMP .LT. RATIO) THEN
RATIO=TEMP
ICON=K
END IF
END IF
450 CONTINUE
C
C Update DX, VMULTC and RESMAX.
C
TEMP=1.0d0-RATIO
DO 460 I=1,N
460 DX(I)=TEMP*DX(I)+RATIO*DXNEW(I)
DO 470 K=1,MCON
470 VMULTC(K)=DMAX1(0.0d0,TEMP*VMULTC(K)+RATIO*VMULTD(K))
IF (IPRINT .EQ. 3) THEN
PRINT *, ' 3. VMULTC = ', (VMULTC(I),I=1,M+1)
END IF
IF (MCON .EQ. M) RESMAX=RESOLD+RATIO*(RESMAX-RESOLD)
IF (IPRINT .EQ. 3) THEN
PRINT *, ' RESMAX, MCON, M, ICON = ',
1 RESMAX, MCON, M, ICON
END IF
C
C If the full step is not acceptable then begin another iteration.
C Otherwise switch to stage two or end the calculation.
C
IF (ICON .GT. 0) GOTO 70
IF (STEP .EQ. STPFUL) GOTO 500
480 MCON=M+1
ICON=MCON
IACT(MCON)=MCON
VMULTC(MCON)=0.0d0
GOTO 60
C
C We employ any freedom that may be available to reduce the objective
C function before returning a DX whose length is less than RHO.
C
490 IF (MCON .EQ. M) GOTO 480
IFULL=0
500 CONTINUE
IF (IPRINT .EQ. 3) THEN
print *, ' '
print *, 'AFTER trstlp:'
PRINT *, ' **DX = ', (DX(I),I=1,N)
PRINT *, ' **IACT = ', (IACT(I),I=1,M+1)
PRINT *, 'M,N,RHO,IFULL =', M, N, RHO, IFULL
PRINT *, ' **A = ', ((A(I,K),I=1,N),K=1,M+1)
PRINT *, ' **B = ', (B(I),I=1,M)
PRINT *, ' **Z = ', ((Z(I,K),I=1,N),K=1,N)
PRINT *, ' **ZDOTA = ', (ZDOTA(I),I=1,N)
PRINT *, ' **VMULTC = ', (VMULTC(I),I=1,M+1)
PRINT *, ' **SDIRN = ', (SDIRN(I),I=1,N)
PRINT *, ' **DXNEW = ', (DXNEW(I),I=1,N)
PRINT *, ' **VMULTD = ', (VMULTD(I),I=1,M+1)
PRINT *, ' '
END IF
C 500 RETURN
END
subroutine s360_380(DXNEW,DX,STEP,SDIRN,N,M,MCON,RESMAX,
1 NACT,IACT,B,A,RESOLD)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION A(N,*),B(*),DX(*),IACT(*), SDIRN(*),DXNEW(*)
DO 360 I=1,N
360 DXNEW(I)=DX(I)+STEP*SDIRN(I)
IF (MCON .EQ. M) THEN
RESOLD=RESMAX
RESMAX=0.0d0
DO 380 K=1,NACT
KK=IACT(K)
TEMP=B(KK)
DO 370 I=1,N
370 TEMP=TEMP-A(I,KK)*DXNEW(I)
RESMAX=DMAX1(RESMAX,TEMP)
380 CONTINUE
END IF
end
|
