Actual source code: ntl.c

  1: #include <../src/tao/unconstrained/impls/ntl/ntlimpl.h>

  3: #include <petscksp.h>

  5: #define NTL_INIT_CONSTANT      0
  6: #define NTL_INIT_DIRECTION     1
  7: #define NTL_INIT_INTERPOLATION 2
  8: #define NTL_INIT_TYPES         3

 10: #define NTL_UPDATE_REDUCTION     0
 11: #define NTL_UPDATE_INTERPOLATION 1
 12: #define NTL_UPDATE_TYPES         2

 14: static const char *NTL_INIT[64] = {"constant", "direction", "interpolation"};

 16: static const char *NTL_UPDATE[64] = {"reduction", "interpolation"};

 18: /* Implements Newton's Method with a trust-region, line-search approach for
 19:    solving unconstrained minimization problems.  A More'-Thuente line search
 20:    is used to guarantee that the bfgs preconditioner remains positive
 21:    definite. */

 23: #define NTL_NEWTON          0
 24: #define NTL_BFGS            1
 25: #define NTL_SCALED_GRADIENT 2
 26: #define NTL_GRADIENT        3

 28: static PetscErrorCode TaoSolve_NTL(Tao tao)
 29: {
 30:   TAO_NTL                     *tl = (TAO_NTL *)tao->data;
 31:   KSPType                      ksp_type;
 32:   PetscBool                    is_nash, is_stcg, is_gltr, is_bfgs, is_jacobi, is_symmetric, sym_set;
 33:   KSPConvergedReason           ksp_reason;
 34:   PC                           pc;
 35:   TaoLineSearchConvergedReason ls_reason;

 37:   PetscReal fmin, ftrial, prered, actred, kappa, sigma;
 38:   PetscReal tau, tau_1, tau_2, tau_max, tau_min, max_radius;
 39:   PetscReal f, fold, gdx, gnorm;
 40:   PetscReal step = 1.0;

 42:   PetscReal norm_d = 0.0;
 43:   PetscInt  stepType;
 44:   PetscInt  its;

 46:   PetscInt bfgsUpdates = 0;
 47:   PetscInt needH;

 49:   PetscInt i_max = 5;
 50:   PetscInt j_max = 1;
 51:   PetscInt i, j, n, N;

 53:   PetscInt tr_reject;

 55:   PetscFunctionBegin;
 56:   if (tao->XL || tao->XU || tao->ops->computebounds) PetscCall(PetscInfo(tao, "WARNING: Variable bounds have been set but will be ignored by ntl algorithm\n"));

 58:   PetscCall(KSPGetType(tao->ksp, &ksp_type));
 59:   PetscCall(PetscStrcmp(ksp_type, KSPNASH, &is_nash));
 60:   PetscCall(PetscStrcmp(ksp_type, KSPSTCG, &is_stcg));
 61:   PetscCall(PetscStrcmp(ksp_type, KSPGLTR, &is_gltr));
 62:   PetscCheck(is_nash || is_stcg || is_gltr, PetscObjectComm((PetscObject)tao), PETSC_ERR_SUP, "TAO_NTR requires nash, stcg, or gltr for the KSP");

 64:   /* Initialize the radius and modify if it is too large or small */
 65:   tao->trust = tao->trust0;
 66:   tao->trust = PetscMax(tao->trust, tl->min_radius);
 67:   tao->trust = PetscMin(tao->trust, tl->max_radius);

 69:   /* Allocate the vectors needed for the BFGS approximation */
 70:   PetscCall(KSPGetPC(tao->ksp, &pc));
 71:   PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCLMVM, &is_bfgs));
 72:   PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCJACOBI, &is_jacobi));
 73:   if (is_bfgs) {
 74:     tl->bfgs_pre = pc;
 75:     PetscCall(PCLMVMGetMatLMVM(tl->bfgs_pre, &tl->M));
 76:     PetscCall(VecGetLocalSize(tao->solution, &n));
 77:     PetscCall(VecGetSize(tao->solution, &N));
 78:     PetscCall(MatSetSizes(tl->M, n, n, N, N));
 79:     PetscCall(MatLMVMAllocate(tl->M, tao->solution, tao->gradient));
 80:     PetscCall(MatIsSymmetricKnown(tl->M, &sym_set, &is_symmetric));
 81:     PetscCheck(sym_set && is_symmetric, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_INCOMP, "LMVM matrix in the LMVM preconditioner must be symmetric.");
 82:   } else if (is_jacobi) PetscCall(PCJacobiSetUseAbs(pc, PETSC_TRUE));

 84:   /* Check convergence criteria */
 85:   PetscCall(TaoComputeObjectiveAndGradient(tao, tao->solution, &f, tao->gradient));
 86:   PetscCall(VecNorm(tao->gradient, NORM_2, &gnorm));
 87:   PetscCheck(!PetscIsInfOrNanReal(f) && !PetscIsInfOrNanReal(gnorm), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
 88:   needH = 1;

 90:   tao->reason = TAO_CONTINUE_ITERATING;
 91:   PetscCall(TaoLogConvergenceHistory(tao, f, gnorm, 0.0, tao->ksp_its));
 92:   PetscCall(TaoMonitor(tao, tao->niter, f, gnorm, 0.0, step));
 93:   PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
 94:   if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(PETSC_SUCCESS);

 96:   /* Initialize trust-region radius */
 97:   switch (tl->init_type) {
 98:   case NTL_INIT_CONSTANT:
 99:     /* Use the initial radius specified */
100:     break;

102:   case NTL_INIT_INTERPOLATION:
103:     /* Use the initial radius specified */
104:     max_radius = 0.0;

106:     for (j = 0; j < j_max; ++j) {
107:       fmin  = f;
108:       sigma = 0.0;

110:       if (needH) {
111:         PetscCall(TaoComputeHessian(tao, tao->solution, tao->hessian, tao->hessian_pre));
112:         needH = 0;
113:       }

115:       for (i = 0; i < i_max; ++i) {
116:         PetscCall(VecCopy(tao->solution, tl->W));
117:         PetscCall(VecAXPY(tl->W, -tao->trust / gnorm, tao->gradient));

119:         PetscCall(TaoComputeObjective(tao, tl->W, &ftrial));
120:         if (PetscIsInfOrNanReal(ftrial)) {
121:           tau = tl->gamma1_i;
122:         } else {
123:           if (ftrial < fmin) {
124:             fmin  = ftrial;
125:             sigma = -tao->trust / gnorm;
126:           }

128:           PetscCall(MatMult(tao->hessian, tao->gradient, tao->stepdirection));
129:           PetscCall(VecDot(tao->gradient, tao->stepdirection, &prered));

131:           prered = tao->trust * (gnorm - 0.5 * tao->trust * prered / (gnorm * gnorm));
132:           actred = f - ftrial;
133:           if ((PetscAbsScalar(actred) <= tl->epsilon) && (PetscAbsScalar(prered) <= tl->epsilon)) {
134:             kappa = 1.0;
135:           } else {
136:             kappa = actred / prered;
137:           }

139:           tau_1   = tl->theta_i * gnorm * tao->trust / (tl->theta_i * gnorm * tao->trust + (1.0 - tl->theta_i) * prered - actred);
140:           tau_2   = tl->theta_i * gnorm * tao->trust / (tl->theta_i * gnorm * tao->trust - (1.0 + tl->theta_i) * prered + actred);
141:           tau_min = PetscMin(tau_1, tau_2);
142:           tau_max = PetscMax(tau_1, tau_2);

144:           if (PetscAbsScalar(kappa - (PetscReal)1.0) <= tl->mu1_i) {
145:             /* Great agreement */
146:             max_radius = PetscMax(max_radius, tao->trust);

148:             if (tau_max < 1.0) {
149:               tau = tl->gamma3_i;
150:             } else if (tau_max > tl->gamma4_i) {
151:               tau = tl->gamma4_i;
152:             } else if (tau_1 >= 1.0 && tau_1 <= tl->gamma4_i && tau_2 < 1.0) {
153:               tau = tau_1;
154:             } else if (tau_2 >= 1.0 && tau_2 <= tl->gamma4_i && tau_1 < 1.0) {
155:               tau = tau_2;
156:             } else {
157:               tau = tau_max;
158:             }
159:           } else if (PetscAbsScalar(kappa - (PetscReal)1.0) <= tl->mu2_i) {
160:             /* Good agreement */
161:             max_radius = PetscMax(max_radius, tao->trust);

163:             if (tau_max < tl->gamma2_i) {
164:               tau = tl->gamma2_i;
165:             } else if (tau_max > tl->gamma3_i) {
166:               tau = tl->gamma3_i;
167:             } else {
168:               tau = tau_max;
169:             }
170:           } else {
171:             /* Not good agreement */
172:             if (tau_min > 1.0) {
173:               tau = tl->gamma2_i;
174:             } else if (tau_max < tl->gamma1_i) {
175:               tau = tl->gamma1_i;
176:             } else if ((tau_min < tl->gamma1_i) && (tau_max >= 1.0)) {
177:               tau = tl->gamma1_i;
178:             } else if ((tau_1 >= tl->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < tl->gamma1_i) || (tau_2 >= 1.0))) {
179:               tau = tau_1;
180:             } else if ((tau_2 >= tl->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < tl->gamma1_i) || (tau_2 >= 1.0))) {
181:               tau = tau_2;
182:             } else {
183:               tau = tau_max;
184:             }
185:           }
186:         }
187:         tao->trust = tau * tao->trust;
188:       }

190:       if (fmin < f) {
191:         f = fmin;
192:         PetscCall(VecAXPY(tao->solution, sigma, tao->gradient));
193:         PetscCall(TaoComputeGradient(tao, tao->solution, tao->gradient));

195:         PetscCall(VecNorm(tao->gradient, NORM_2, &gnorm));
196:         PetscCheck(!PetscIsInfOrNanReal(f) && !PetscIsInfOrNanReal(gnorm), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
197:         needH = 1;

199:         PetscCall(TaoLogConvergenceHistory(tao, f, gnorm, 0.0, tao->ksp_its));
200:         PetscCall(TaoMonitor(tao, tao->niter, f, gnorm, 0.0, step));
201:         PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
202:         if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(PETSC_SUCCESS);
203:       }
204:     }
205:     tao->trust = PetscMax(tao->trust, max_radius);

207:     /* Modify the radius if it is too large or small */
208:     tao->trust = PetscMax(tao->trust, tl->min_radius);
209:     tao->trust = PetscMin(tao->trust, tl->max_radius);
210:     break;

212:   default:
213:     /* Norm of the first direction will initialize radius */
214:     tao->trust = 0.0;
215:     break;
216:   }

218:   /* Set counter for gradient/reset steps */
219:   tl->ntrust = 0;
220:   tl->newt   = 0;
221:   tl->bfgs   = 0;
222:   tl->grad   = 0;

224:   /* Have not converged; continue with Newton method */
225:   while (tao->reason == TAO_CONTINUE_ITERATING) {
226:     /* Call general purpose update function */
227:     if (tao->ops->update) {
228:       PetscUseTypeMethod(tao, update, tao->niter, tao->user_update);
229:       PetscCall(TaoComputeObjective(tao, tao->solution, &f));
230:     }
231:     ++tao->niter;
232:     tao->ksp_its = 0;
233:     /* Compute the Hessian */
234:     if (needH) PetscCall(TaoComputeHessian(tao, tao->solution, tao->hessian, tao->hessian_pre));

236:     if (tl->bfgs_pre) {
237:       /* Update the limited memory preconditioner */
238:       PetscCall(MatLMVMUpdate(tl->M, tao->solution, tao->gradient));
239:       ++bfgsUpdates;
240:     }
241:     PetscCall(KSPSetOperators(tao->ksp, tao->hessian, tao->hessian_pre));
242:     /* Solve the Newton system of equations */
243:     PetscCall(KSPCGSetRadius(tao->ksp, tl->max_radius));
244:     PetscCall(KSPSolve(tao->ksp, tao->gradient, tao->stepdirection));
245:     PetscCall(KSPGetIterationNumber(tao->ksp, &its));
246:     tao->ksp_its += its;
247:     tao->ksp_tot_its += its;
248:     PetscCall(KSPCGGetNormD(tao->ksp, &norm_d));

250:     if (0.0 == tao->trust) {
251:       /* Radius was uninitialized; use the norm of the direction */
252:       if (norm_d > 0.0) {
253:         tao->trust = norm_d;

255:         /* Modify the radius if it is too large or small */
256:         tao->trust = PetscMax(tao->trust, tl->min_radius);
257:         tao->trust = PetscMin(tao->trust, tl->max_radius);
258:       } else {
259:         /* The direction was bad; set radius to default value and re-solve
260:            the trust-region subproblem to get a direction */
261:         tao->trust = tao->trust0;

263:         /* Modify the radius if it is too large or small */
264:         tao->trust = PetscMax(tao->trust, tl->min_radius);
265:         tao->trust = PetscMin(tao->trust, tl->max_radius);

267:         PetscCall(KSPCGSetRadius(tao->ksp, tl->max_radius));
268:         PetscCall(KSPSolve(tao->ksp, tao->gradient, tao->stepdirection));
269:         PetscCall(KSPGetIterationNumber(tao->ksp, &its));
270:         tao->ksp_its += its;
271:         tao->ksp_tot_its += its;
272:         PetscCall(KSPCGGetNormD(tao->ksp, &norm_d));

274:         PetscCheck(norm_d != 0.0, PetscObjectComm((PetscObject)tao), PETSC_ERR_PLIB, "Initial direction zero");
275:       }
276:     }

278:     PetscCall(VecScale(tao->stepdirection, -1.0));
279:     PetscCall(KSPGetConvergedReason(tao->ksp, &ksp_reason));
280:     if ((KSP_DIVERGED_INDEFINITE_PC == ksp_reason) && (tl->bfgs_pre)) {
281:       /* Preconditioner is numerically indefinite; reset the
282:          approximate if using BFGS preconditioning. */
283:       PetscCall(MatLMVMReset(tl->M, PETSC_FALSE));
284:       PetscCall(MatLMVMUpdate(tl->M, tao->solution, tao->gradient));
285:       bfgsUpdates = 1;
286:     }

288:     /* Check trust-region reduction conditions */
289:     tr_reject = 0;
290:     if (NTL_UPDATE_REDUCTION == tl->update_type) {
291:       /* Get predicted reduction */
292:       PetscCall(KSPCGGetObjFcn(tao->ksp, &prered));
293:       if (prered >= 0.0) {
294:         /* The predicted reduction has the wrong sign.  This cannot
295:            happen in infinite precision arithmetic.  Step should
296:            be rejected! */
297:         tao->trust = tl->alpha1 * PetscMin(tao->trust, norm_d);
298:         tr_reject  = 1;
299:       } else {
300:         /* Compute trial step and function value */
301:         PetscCall(VecCopy(tao->solution, tl->W));
302:         PetscCall(VecAXPY(tl->W, 1.0, tao->stepdirection));
303:         PetscCall(TaoComputeObjective(tao, tl->W, &ftrial));

305:         if (PetscIsInfOrNanReal(ftrial)) {
306:           tao->trust = tl->alpha1 * PetscMin(tao->trust, norm_d);
307:           tr_reject  = 1;
308:         } else {
309:           /* Compute and actual reduction */
310:           actred = f - ftrial;
311:           prered = -prered;
312:           if ((PetscAbsScalar(actred) <= tl->epsilon) && (PetscAbsScalar(prered) <= tl->epsilon)) {
313:             kappa = 1.0;
314:           } else {
315:             kappa = actred / prered;
316:           }

318:           /* Accept of reject the step and update radius */
319:           if (kappa < tl->eta1) {
320:             /* Reject the step */
321:             tao->trust = tl->alpha1 * PetscMin(tao->trust, norm_d);
322:             tr_reject  = 1;
323:           } else {
324:             /* Accept the step */
325:             if (kappa < tl->eta2) {
326:               /* Marginal bad step */
327:               tao->trust = tl->alpha2 * PetscMin(tao->trust, norm_d);
328:             } else if (kappa < tl->eta3) {
329:               /* Reasonable step */
330:               tao->trust = tl->alpha3 * tao->trust;
331:             } else if (kappa < tl->eta4) {
332:               /* Good step */
333:               tao->trust = PetscMax(tl->alpha4 * norm_d, tao->trust);
334:             } else {
335:               /* Very good step */
336:               tao->trust = PetscMax(tl->alpha5 * norm_d, tao->trust);
337:             }
338:           }
339:         }
340:       }
341:     } else {
342:       /* Get predicted reduction */
343:       PetscCall(KSPCGGetObjFcn(tao->ksp, &prered));
344:       if (prered >= 0.0) {
345:         /* The predicted reduction has the wrong sign.  This cannot
346:            happen in infinite precision arithmetic.  Step should
347:            be rejected! */
348:         tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
349:         tr_reject  = 1;
350:       } else {
351:         PetscCall(VecCopy(tao->solution, tl->W));
352:         PetscCall(VecAXPY(tl->W, 1.0, tao->stepdirection));
353:         PetscCall(TaoComputeObjective(tao, tl->W, &ftrial));
354:         if (PetscIsInfOrNanReal(ftrial)) {
355:           tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
356:           tr_reject  = 1;
357:         } else {
358:           PetscCall(VecDot(tao->gradient, tao->stepdirection, &gdx));

360:           actred = f - ftrial;
361:           prered = -prered;
362:           if ((PetscAbsScalar(actred) <= tl->epsilon) && (PetscAbsScalar(prered) <= tl->epsilon)) {
363:             kappa = 1.0;
364:           } else {
365:             kappa = actred / prered;
366:           }

368:           tau_1   = tl->theta * gdx / (tl->theta * gdx - (1.0 - tl->theta) * prered + actred);
369:           tau_2   = tl->theta * gdx / (tl->theta * gdx + (1.0 + tl->theta) * prered - actred);
370:           tau_min = PetscMin(tau_1, tau_2);
371:           tau_max = PetscMax(tau_1, tau_2);

373:           if (kappa >= 1.0 - tl->mu1) {
374:             /* Great agreement; accept step and update radius */
375:             if (tau_max < 1.0) {
376:               tao->trust = PetscMax(tao->trust, tl->gamma3 * norm_d);
377:             } else if (tau_max > tl->gamma4) {
378:               tao->trust = PetscMax(tao->trust, tl->gamma4 * norm_d);
379:             } else {
380:               tao->trust = PetscMax(tao->trust, tau_max * norm_d);
381:             }
382:           } else if (kappa >= 1.0 - tl->mu2) {
383:             /* Good agreement */

385:             if (tau_max < tl->gamma2) {
386:               tao->trust = tl->gamma2 * PetscMin(tao->trust, norm_d);
387:             } else if (tau_max > tl->gamma3) {
388:               tao->trust = PetscMax(tao->trust, tl->gamma3 * norm_d);
389:             } else if (tau_max < 1.0) {
390:               tao->trust = tau_max * PetscMin(tao->trust, norm_d);
391:             } else {
392:               tao->trust = PetscMax(tao->trust, tau_max * norm_d);
393:             }
394:           } else {
395:             /* Not good agreement */
396:             if (tau_min > 1.0) {
397:               tao->trust = tl->gamma2 * PetscMin(tao->trust, norm_d);
398:             } else if (tau_max < tl->gamma1) {
399:               tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
400:             } else if ((tau_min < tl->gamma1) && (tau_max >= 1.0)) {
401:               tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
402:             } else if ((tau_1 >= tl->gamma1) && (tau_1 < 1.0) && ((tau_2 < tl->gamma1) || (tau_2 >= 1.0))) {
403:               tao->trust = tau_1 * PetscMin(tao->trust, norm_d);
404:             } else if ((tau_2 >= tl->gamma1) && (tau_2 < 1.0) && ((tau_1 < tl->gamma1) || (tau_2 >= 1.0))) {
405:               tao->trust = tau_2 * PetscMin(tao->trust, norm_d);
406:             } else {
407:               tao->trust = tau_max * PetscMin(tao->trust, norm_d);
408:             }
409:             tr_reject = 1;
410:           }
411:         }
412:       }
413:     }

415:     if (tr_reject) {
416:       /* The trust-region constraints rejected the step.  Apply a linesearch.
417:          Check for descent direction. */
418:       PetscCall(VecDot(tao->stepdirection, tao->gradient, &gdx));
419:       if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
420:         /* Newton step is not descent or direction produced Inf or NaN */

422:         if (!tl->bfgs_pre) {
423:           /* We don't have the bfgs matrix around and updated
424:              Must use gradient direction in this case */
425:           PetscCall(VecCopy(tao->gradient, tao->stepdirection));
426:           PetscCall(VecScale(tao->stepdirection, -1.0));
427:           ++tl->grad;
428:           stepType = NTL_GRADIENT;
429:         } else {
430:           /* Attempt to use the BFGS direction */
431:           PetscCall(MatSolve(tl->M, tao->gradient, tao->stepdirection));
432:           PetscCall(VecScale(tao->stepdirection, -1.0));

434:           /* Check for success (descent direction) */
435:           PetscCall(VecDot(tao->stepdirection, tao->gradient, &gdx));
436:           if ((gdx >= 0) || PetscIsInfOrNanReal(gdx)) {
437:             /* BFGS direction is not descent or direction produced not a number
438:                We can assert bfgsUpdates > 1 in this case because
439:                the first solve produces the scaled gradient direction,
440:                which is guaranteed to be descent */

442:             /* Use steepest descent direction (scaled) */
443:             PetscCall(MatLMVMReset(tl->M, PETSC_FALSE));
444:             PetscCall(MatLMVMUpdate(tl->M, tao->solution, tao->gradient));
445:             PetscCall(MatSolve(tl->M, tao->gradient, tao->stepdirection));
446:             PetscCall(VecScale(tao->stepdirection, -1.0));

448:             bfgsUpdates = 1;
449:             ++tl->grad;
450:             stepType = NTL_GRADIENT;
451:           } else {
452:             PetscCall(MatLMVMGetUpdateCount(tl->M, &bfgsUpdates));
453:             if (1 == bfgsUpdates) {
454:               /* The first BFGS direction is always the scaled gradient */
455:               ++tl->grad;
456:               stepType = NTL_GRADIENT;
457:             } else {
458:               ++tl->bfgs;
459:               stepType = NTL_BFGS;
460:             }
461:           }
462:         }
463:       } else {
464:         /* Computed Newton step is descent */
465:         ++tl->newt;
466:         stepType = NTL_NEWTON;
467:       }

469:       /* Perform the linesearch */
470:       fold = f;
471:       PetscCall(VecCopy(tao->solution, tl->Xold));
472:       PetscCall(VecCopy(tao->gradient, tl->Gold));

474:       step = 1.0;
475:       PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &f, tao->gradient, tao->stepdirection, &step, &ls_reason));
476:       PetscCall(TaoAddLineSearchCounts(tao));

478:       while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && stepType != NTL_GRADIENT) { /* Linesearch failed */
479:         /* Linesearch failed */
480:         f = fold;
481:         PetscCall(VecCopy(tl->Xold, tao->solution));
482:         PetscCall(VecCopy(tl->Gold, tao->gradient));

484:         switch (stepType) {
485:         case NTL_NEWTON:
486:           /* Failed to obtain acceptable iterate with Newton step */

488:           if (tl->bfgs_pre) {
489:             /* We don't have the bfgs matrix around and being updated
490:                Must use gradient direction in this case */
491:             PetscCall(VecCopy(tao->gradient, tao->stepdirection));
492:             ++tl->grad;
493:             stepType = NTL_GRADIENT;
494:           } else {
495:             /* Attempt to use the BFGS direction */
496:             PetscCall(MatSolve(tl->M, tao->gradient, tao->stepdirection));

498:             /* Check for success (descent direction) */
499:             PetscCall(VecDot(tao->stepdirection, tao->gradient, &gdx));
500:             if ((gdx <= 0) || PetscIsInfOrNanReal(gdx)) {
501:               /* BFGS direction is not descent or direction produced
502:                  not a number.  We can assert bfgsUpdates > 1 in this case
503:                  Use steepest descent direction (scaled) */
504:               PetscCall(MatLMVMReset(tl->M, PETSC_FALSE));
505:               PetscCall(MatLMVMUpdate(tl->M, tao->solution, tao->gradient));
506:               PetscCall(MatSolve(tl->M, tao->gradient, tao->stepdirection));

508:               bfgsUpdates = 1;
509:               ++tl->grad;
510:               stepType = NTL_GRADIENT;
511:             } else {
512:               PetscCall(MatLMVMGetUpdateCount(tl->M, &bfgsUpdates));
513:               if (1 == bfgsUpdates) {
514:                 /* The first BFGS direction is always the scaled gradient */
515:                 ++tl->grad;
516:                 stepType = NTL_GRADIENT;
517:               } else {
518:                 ++tl->bfgs;
519:                 stepType = NTL_BFGS;
520:               }
521:             }
522:           }
523:           break;

525:         case NTL_BFGS:
526:           /* Can only enter if pc_type == NTL_PC_BFGS
527:              Failed to obtain acceptable iterate with BFGS step
528:              Attempt to use the scaled gradient direction */
529:           PetscCall(MatLMVMReset(tl->M, PETSC_FALSE));
530:           PetscCall(MatLMVMUpdate(tl->M, tao->solution, tao->gradient));
531:           PetscCall(MatSolve(tl->M, tao->gradient, tao->stepdirection));

533:           bfgsUpdates = 1;
534:           ++tl->grad;
535:           stepType = NTL_GRADIENT;
536:           break;
537:         }
538:         PetscCall(VecScale(tao->stepdirection, -1.0));

540:         /* This may be incorrect; linesearch has values for stepmax and stepmin
541:            that should be reset. */
542:         step = 1.0;
543:         PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &f, tao->gradient, tao->stepdirection, &step, &ls_reason));
544:         PetscCall(TaoAddLineSearchCounts(tao));
545:       }

547:       if (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER) {
548:         /* Failed to find an improving point */
549:         f = fold;
550:         PetscCall(VecCopy(tl->Xold, tao->solution));
551:         PetscCall(VecCopy(tl->Gold, tao->gradient));
552:         tao->trust  = 0.0;
553:         step        = 0.0;
554:         tao->reason = TAO_DIVERGED_LS_FAILURE;
555:         break;
556:       } else if (stepType == NTL_NEWTON) {
557:         if (step < tl->nu1) {
558:           /* Very bad step taken; reduce radius */
559:           tao->trust = tl->omega1 * PetscMin(norm_d, tao->trust);
560:         } else if (step < tl->nu2) {
561:           /* Reasonably bad step taken; reduce radius */
562:           tao->trust = tl->omega2 * PetscMin(norm_d, tao->trust);
563:         } else if (step < tl->nu3) {
564:           /* Reasonable step was taken; leave radius alone */
565:           if (tl->omega3 < 1.0) {
566:             tao->trust = tl->omega3 * PetscMin(norm_d, tao->trust);
567:           } else if (tl->omega3 > 1.0) {
568:             tao->trust = PetscMax(tl->omega3 * norm_d, tao->trust);
569:           }
570:         } else if (step < tl->nu4) {
571:           /* Full step taken; increase the radius */
572:           tao->trust = PetscMax(tl->omega4 * norm_d, tao->trust);
573:         } else {
574:           /* More than full step taken; increase the radius */
575:           tao->trust = PetscMax(tl->omega5 * norm_d, tao->trust);
576:         }
577:       } else {
578:         /* Newton step was not good; reduce the radius */
579:         tao->trust = tl->omega1 * PetscMin(norm_d, tao->trust);
580:       }
581:     } else {
582:       /* Trust-region step is accepted */
583:       PetscCall(VecCopy(tl->W, tao->solution));
584:       f = ftrial;
585:       PetscCall(TaoComputeGradient(tao, tao->solution, tao->gradient));
586:       ++tl->ntrust;
587:     }

589:     /* The radius may have been increased; modify if it is too large */
590:     tao->trust = PetscMin(tao->trust, tl->max_radius);

592:     /* Check for converged */
593:     PetscCall(VecNorm(tao->gradient, NORM_2, &gnorm));
594:     PetscCheck(!PetscIsInfOrNanReal(f) && !PetscIsInfOrNanReal(gnorm), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated Not-a-Number");
595:     needH = 1;

597:     PetscCall(TaoLogConvergenceHistory(tao, f, gnorm, 0.0, tao->ksp_its));
598:     PetscCall(TaoMonitor(tao, tao->niter, f, gnorm, 0.0, step));
599:     PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
600:   }
601:   PetscFunctionReturn(PETSC_SUCCESS);
602: }

604: /* ---------------------------------------------------------- */
605: static PetscErrorCode TaoSetUp_NTL(Tao tao)
606: {
607:   TAO_NTL *tl = (TAO_NTL *)tao->data;

609:   PetscFunctionBegin;
610:   if (!tao->gradient) PetscCall(VecDuplicate(tao->solution, &tao->gradient));
611:   if (!tao->stepdirection) PetscCall(VecDuplicate(tao->solution, &tao->stepdirection));
612:   if (!tl->W) PetscCall(VecDuplicate(tao->solution, &tl->W));
613:   if (!tl->Xold) PetscCall(VecDuplicate(tao->solution, &tl->Xold));
614:   if (!tl->Gold) PetscCall(VecDuplicate(tao->solution, &tl->Gold));
615:   tl->bfgs_pre = NULL;
616:   tl->M        = NULL;
617:   PetscFunctionReturn(PETSC_SUCCESS);
618: }

620: /*------------------------------------------------------------*/
621: static PetscErrorCode TaoDestroy_NTL(Tao tao)
622: {
623:   TAO_NTL *tl = (TAO_NTL *)tao->data;

625:   PetscFunctionBegin;
626:   if (tao->setupcalled) {
627:     PetscCall(VecDestroy(&tl->W));
628:     PetscCall(VecDestroy(&tl->Xold));
629:     PetscCall(VecDestroy(&tl->Gold));
630:   }
631:   PetscCall(KSPDestroy(&tao->ksp));
632:   PetscCall(PetscFree(tao->data));
633:   PetscFunctionReturn(PETSC_SUCCESS);
634: }

636: /*------------------------------------------------------------*/
637: static PetscErrorCode TaoSetFromOptions_NTL(Tao tao, PetscOptionItems *PetscOptionsObject)
638: {
639:   TAO_NTL *tl = (TAO_NTL *)tao->data;

641:   PetscFunctionBegin;
642:   PetscOptionsHeadBegin(PetscOptionsObject, "Newton trust region with line search method for unconstrained optimization");
643:   PetscCall(PetscOptionsEList("-tao_ntl_init_type", "radius initialization type", "", NTL_INIT, NTL_INIT_TYPES, NTL_INIT[tl->init_type], &tl->init_type, NULL));
644:   PetscCall(PetscOptionsEList("-tao_ntl_update_type", "radius update type", "", NTL_UPDATE, NTL_UPDATE_TYPES, NTL_UPDATE[tl->update_type], &tl->update_type, NULL));
645:   PetscCall(PetscOptionsReal("-tao_ntl_eta1", "poor steplength; reduce radius", "", tl->eta1, &tl->eta1, NULL));
646:   PetscCall(PetscOptionsReal("-tao_ntl_eta2", "reasonable steplength; leave radius alone", "", tl->eta2, &tl->eta2, NULL));
647:   PetscCall(PetscOptionsReal("-tao_ntl_eta3", "good steplength; increase radius", "", tl->eta3, &tl->eta3, NULL));
648:   PetscCall(PetscOptionsReal("-tao_ntl_eta4", "excellent steplength; greatly increase radius", "", tl->eta4, &tl->eta4, NULL));
649:   PetscCall(PetscOptionsReal("-tao_ntl_alpha1", "", "", tl->alpha1, &tl->alpha1, NULL));
650:   PetscCall(PetscOptionsReal("-tao_ntl_alpha2", "", "", tl->alpha2, &tl->alpha2, NULL));
651:   PetscCall(PetscOptionsReal("-tao_ntl_alpha3", "", "", tl->alpha3, &tl->alpha3, NULL));
652:   PetscCall(PetscOptionsReal("-tao_ntl_alpha4", "", "", tl->alpha4, &tl->alpha4, NULL));
653:   PetscCall(PetscOptionsReal("-tao_ntl_alpha5", "", "", tl->alpha5, &tl->alpha5, NULL));
654:   PetscCall(PetscOptionsReal("-tao_ntl_nu1", "poor steplength; reduce radius", "", tl->nu1, &tl->nu1, NULL));
655:   PetscCall(PetscOptionsReal("-tao_ntl_nu2", "reasonable steplength; leave radius alone", "", tl->nu2, &tl->nu2, NULL));
656:   PetscCall(PetscOptionsReal("-tao_ntl_nu3", "good steplength; increase radius", "", tl->nu3, &tl->nu3, NULL));
657:   PetscCall(PetscOptionsReal("-tao_ntl_nu4", "excellent steplength; greatly increase radius", "", tl->nu4, &tl->nu4, NULL));
658:   PetscCall(PetscOptionsReal("-tao_ntl_omega1", "", "", tl->omega1, &tl->omega1, NULL));
659:   PetscCall(PetscOptionsReal("-tao_ntl_omega2", "", "", tl->omega2, &tl->omega2, NULL));
660:   PetscCall(PetscOptionsReal("-tao_ntl_omega3", "", "", tl->omega3, &tl->omega3, NULL));
661:   PetscCall(PetscOptionsReal("-tao_ntl_omega4", "", "", tl->omega4, &tl->omega4, NULL));
662:   PetscCall(PetscOptionsReal("-tao_ntl_omega5", "", "", tl->omega5, &tl->omega5, NULL));
663:   PetscCall(PetscOptionsReal("-tao_ntl_mu1_i", "", "", tl->mu1_i, &tl->mu1_i, NULL));
664:   PetscCall(PetscOptionsReal("-tao_ntl_mu2_i", "", "", tl->mu2_i, &tl->mu2_i, NULL));
665:   PetscCall(PetscOptionsReal("-tao_ntl_gamma1_i", "", "", tl->gamma1_i, &tl->gamma1_i, NULL));
666:   PetscCall(PetscOptionsReal("-tao_ntl_gamma2_i", "", "", tl->gamma2_i, &tl->gamma2_i, NULL));
667:   PetscCall(PetscOptionsReal("-tao_ntl_gamma3_i", "", "", tl->gamma3_i, &tl->gamma3_i, NULL));
668:   PetscCall(PetscOptionsReal("-tao_ntl_gamma4_i", "", "", tl->gamma4_i, &tl->gamma4_i, NULL));
669:   PetscCall(PetscOptionsReal("-tao_ntl_theta_i", "", "", tl->theta_i, &tl->theta_i, NULL));
670:   PetscCall(PetscOptionsReal("-tao_ntl_mu1", "", "", tl->mu1, &tl->mu1, NULL));
671:   PetscCall(PetscOptionsReal("-tao_ntl_mu2", "", "", tl->mu2, &tl->mu2, NULL));
672:   PetscCall(PetscOptionsReal("-tao_ntl_gamma1", "", "", tl->gamma1, &tl->gamma1, NULL));
673:   PetscCall(PetscOptionsReal("-tao_ntl_gamma2", "", "", tl->gamma2, &tl->gamma2, NULL));
674:   PetscCall(PetscOptionsReal("-tao_ntl_gamma3", "", "", tl->gamma3, &tl->gamma3, NULL));
675:   PetscCall(PetscOptionsReal("-tao_ntl_gamma4", "", "", tl->gamma4, &tl->gamma4, NULL));
676:   PetscCall(PetscOptionsReal("-tao_ntl_theta", "", "", tl->theta, &tl->theta, NULL));
677:   PetscCall(PetscOptionsReal("-tao_ntl_min_radius", "lower bound on initial radius", "", tl->min_radius, &tl->min_radius, NULL));
678:   PetscCall(PetscOptionsReal("-tao_ntl_max_radius", "upper bound on radius", "", tl->max_radius, &tl->max_radius, NULL));
679:   PetscCall(PetscOptionsReal("-tao_ntl_epsilon", "tolerance used when computing actual and predicted reduction", "", tl->epsilon, &tl->epsilon, NULL));
680:   PetscOptionsHeadEnd();
681:   PetscCall(TaoLineSearchSetFromOptions(tao->linesearch));
682:   PetscCall(KSPSetFromOptions(tao->ksp));
683:   PetscFunctionReturn(PETSC_SUCCESS);
684: }

686: /*------------------------------------------------------------*/
687: static PetscErrorCode TaoView_NTL(Tao tao, PetscViewer viewer)
688: {
689:   TAO_NTL  *tl = (TAO_NTL *)tao->data;
690:   PetscBool isascii;

692:   PetscFunctionBegin;
693:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
694:   if (isascii) {
695:     PetscCall(PetscViewerASCIIPushTab(viewer));
696:     PetscCall(PetscViewerASCIIPrintf(viewer, "Trust-region steps: %" PetscInt_FMT "\n", tl->ntrust));
697:     PetscCall(PetscViewerASCIIPrintf(viewer, "Newton search steps: %" PetscInt_FMT "\n", tl->newt));
698:     PetscCall(PetscViewerASCIIPrintf(viewer, "BFGS search steps: %" PetscInt_FMT "\n", tl->bfgs));
699:     PetscCall(PetscViewerASCIIPrintf(viewer, "Gradient search steps: %" PetscInt_FMT "\n", tl->grad));
700:     PetscCall(PetscViewerASCIIPopTab(viewer));
701:   }
702:   PetscFunctionReturn(PETSC_SUCCESS);
703: }

705: /* ---------------------------------------------------------- */
706: /*MC
707:   TAONTL - Newton's method with trust region globalization and line search fallback.
708:   At each iteration, the Newton trust region method solves the system for d
709:   and performs a line search in the d direction:

711:             min_d  .5 dT Hk d + gkT d,  s.t.   ||d|| < Delta_k

713:   Options Database Keys:
714: + -tao_ntl_init_type - "constant","direction","interpolation"
715: . -tao_ntl_update_type - "reduction","interpolation"
716: . -tao_ntl_min_radius - lower bound on trust region radius
717: . -tao_ntl_max_radius - upper bound on trust region radius
718: . -tao_ntl_epsilon - tolerance for accepting actual / predicted reduction
719: . -tao_ntl_mu1_i - mu1 interpolation init factor
720: . -tao_ntl_mu2_i - mu2 interpolation init factor
721: . -tao_ntl_gamma1_i - gamma1 interpolation init factor
722: . -tao_ntl_gamma2_i - gamma2 interpolation init factor
723: . -tao_ntl_gamma3_i - gamma3 interpolation init factor
724: . -tao_ntl_gamma4_i - gamma4 interpolation init factor
725: . -tao_ntl_theta_i - theta1 interpolation init factor
726: . -tao_ntl_eta1 - eta1 reduction update factor
727: . -tao_ntl_eta2 - eta2 reduction update factor
728: . -tao_ntl_eta3 - eta3 reduction update factor
729: . -tao_ntl_eta4 - eta4 reduction update factor
730: . -tao_ntl_alpha1 - alpha1 reduction update factor
731: . -tao_ntl_alpha2 - alpha2 reduction update factor
732: . -tao_ntl_alpha3 - alpha3 reduction update factor
733: . -tao_ntl_alpha4 - alpha4 reduction update factor
734: . -tao_ntl_alpha4 - alpha4 reduction update factor
735: . -tao_ntl_mu1 - mu1 interpolation update
736: . -tao_ntl_mu2 - mu2 interpolation update
737: . -tao_ntl_gamma1 - gamma1 interpolcation update
738: . -tao_ntl_gamma2 - gamma2 interpolcation update
739: . -tao_ntl_gamma3 - gamma3 interpolcation update
740: . -tao_ntl_gamma4 - gamma4 interpolation update
741: - -tao_ntl_theta - theta1 interpolation update

743:   Level: beginner
744: M*/
745: PETSC_EXTERN PetscErrorCode TaoCreate_NTL(Tao tao)
746: {
747:   TAO_NTL    *tl;
748:   const char *morethuente_type = TAOLINESEARCHMT;

750:   PetscFunctionBegin;
751:   PetscCall(PetscNew(&tl));
752:   tao->ops->setup          = TaoSetUp_NTL;
753:   tao->ops->solve          = TaoSolve_NTL;
754:   tao->ops->view           = TaoView_NTL;
755:   tao->ops->setfromoptions = TaoSetFromOptions_NTL;
756:   tao->ops->destroy        = TaoDestroy_NTL;

758:   /* Override default settings (unless already changed) */
759:   if (!tao->max_it_changed) tao->max_it = 50;
760:   if (!tao->trust0_changed) tao->trust0 = 100.0;

762:   tao->data = (void *)tl;

764:   /* Default values for trust-region radius update based on steplength */
765:   tl->nu1 = 0.25;
766:   tl->nu2 = 0.50;
767:   tl->nu3 = 1.00;
768:   tl->nu4 = 1.25;

770:   tl->omega1 = 0.25;
771:   tl->omega2 = 0.50;
772:   tl->omega3 = 1.00;
773:   tl->omega4 = 2.00;
774:   tl->omega5 = 4.00;

776:   /* Default values for trust-region radius update based on reduction */
777:   tl->eta1 = 1.0e-4;
778:   tl->eta2 = 0.25;
779:   tl->eta3 = 0.50;
780:   tl->eta4 = 0.90;

782:   tl->alpha1 = 0.25;
783:   tl->alpha2 = 0.50;
784:   tl->alpha3 = 1.00;
785:   tl->alpha4 = 2.00;
786:   tl->alpha5 = 4.00;

788:   /* Default values for trust-region radius update based on interpolation */
789:   tl->mu1 = 0.10;
790:   tl->mu2 = 0.50;

792:   tl->gamma1 = 0.25;
793:   tl->gamma2 = 0.50;
794:   tl->gamma3 = 2.00;
795:   tl->gamma4 = 4.00;

797:   tl->theta = 0.05;

799:   /* Default values for trust region initialization based on interpolation */
800:   tl->mu1_i = 0.35;
801:   tl->mu2_i = 0.50;

803:   tl->gamma1_i = 0.0625;
804:   tl->gamma2_i = 0.5;
805:   tl->gamma3_i = 2.0;
806:   tl->gamma4_i = 5.0;

808:   tl->theta_i = 0.25;

810:   /* Remaining parameters */
811:   tl->min_radius = 1.0e-10;
812:   tl->max_radius = 1.0e10;
813:   tl->epsilon    = 1.0e-6;

815:   tl->init_type   = NTL_INIT_INTERPOLATION;
816:   tl->update_type = NTL_UPDATE_REDUCTION;

818:   PetscCall(TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch));
819:   PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1));
820:   PetscCall(TaoLineSearchSetType(tao->linesearch, morethuente_type));
821:   PetscCall(TaoLineSearchUseTaoRoutines(tao->linesearch, tao));
822:   PetscCall(TaoLineSearchSetOptionsPrefix(tao->linesearch, tao->hdr.prefix));
823:   PetscCall(KSPCreate(((PetscObject)tao)->comm, &tao->ksp));
824:   PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1));
825:   PetscCall(KSPSetOptionsPrefix(tao->ksp, tao->hdr.prefix));
826:   PetscCall(KSPAppendOptionsPrefix(tao->ksp, "tao_ntl_"));
827:   PetscCall(KSPSetType(tao->ksp, KSPSTCG));
828:   PetscFunctionReturn(PETSC_SUCCESS);
829: }