xref: /petsc/src/tao/bound/impls/bnk/bnk.c (revision 9b6ef8482b0295dfac0fe7cdd25b1a30d1be2d60) 
1 #include <petsctaolinesearch.h>
2 #include <../src/tao/bound/impls/bnk/bnk.h>
3 
4 #include <petscksp.h>
5 
6 /* Routine for BFGS preconditioner */
7 
8 PetscErrorCode MatLMVMSolveShell(PC pc, Vec b, Vec x)
9 {
10   PetscErrorCode ierr;
11   Mat            M;
12 
13   PetscFunctionBegin;
14   PetscValidHeaderSpecific(pc,PC_CLASSID,1);
15   PetscValidHeaderSpecific(b,VEC_CLASSID,2);
16   PetscValidHeaderSpecific(x,VEC_CLASSID,3);
17   ierr = PCShellGetContext(pc,(void**)&M);CHKERRQ(ierr);
18   ierr = MatLMVMSolveInactive(M, b, x);CHKERRQ(ierr);
19   PetscFunctionReturn(0);
20 }
21 
22 /*------------------------------------------------------------*/
23 
24 /* Routine for initializing the KSP solver, the BFGS preconditioner, and the initial trust radius estimation */
25 
26 PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType)
27 {
28   PetscErrorCode               ierr;
29   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
30   PC                           pc;
31 
32   PetscReal                    f_min, ftrial, prered, actred, kappa, sigma;
33   PetscReal                    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
34   PetscReal                    delta;
35 
36   PetscInt                     n,N,needH = 1;
37 
38   PetscInt                     i_max = 5;
39   PetscInt                     j_max = 1;
40   PetscInt                     i, j;
41 
42   PetscFunctionBegin;
43   /*   Project the current point onto the feasible set */
44   ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr);
45   ierr = TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);CHKERRQ(ierr);
46 
47   /* Project the initial point onto the feasible region */
48   ierr = VecMedian(tao->XL,tao->solution,tao->XU,tao->solution);CHKERRQ(ierr);
49 
50   /* Check convergence criteria */
51   ierr = TaoComputeObjectiveAndGradient(tao, tao->solution, &bnk->f, bnk->unprojected_gradient);CHKERRQ(ierr);
52   ierr = VecBoundGradientProjection(bnk->unprojected_gradient,tao->solution,tao->XL,tao->XU,tao->gradient);CHKERRQ(ierr);
53   ierr = TaoGradientNorm(tao, tao->gradient,NORM_2,&bnk->gnorm);CHKERRQ(ierr);
54   if (PetscIsInfOrNanReal(bnk->f) || PetscIsInfOrNanReal(bnk->gnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
55 
56   /* Number of times ksp stopped because of these reasons */
57   bnk->ksp_atol = 0;
58   bnk->ksp_rtol = 0;
59   bnk->ksp_dtol = 0;
60   bnk->ksp_ctol = 0;
61   bnk->ksp_negc = 0;
62   bnk->ksp_iter = 0;
63   bnk->ksp_othr = 0;
64 
65   /* Initialize the Hessian perturbation */
66   bnk->pert = bnk->sval;
67 
68   /* Initialize trust-region radius when using nash, stcg, or gltr
69      Command automatically ignored for other methods
70      Will be reset during the first iteration
71   */
72   ierr = KSPCGSetRadius(tao->ksp,bnk->max_radius);CHKERRQ(ierr);
73   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
74     if (tao->trust0 < 0.0) SETERRQ(PETSC_COMM_SELF,1,"Initial radius negative");
75     tao->trust = tao->trust0;
76     tao->trust = PetscMax(tao->trust, bnk->min_radius);
77     tao->trust = PetscMin(tao->trust, bnk->max_radius);
78   }
79 
80   ierr = TaoLogConvergenceHistory(tao,bnk->f,bnk->gnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
81   ierr = TaoMonitor(tao,tao->niter,bnk->f,bnk->gnorm,0.0,1.0);CHKERRQ(ierr);
82   ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
83   if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
84 
85   /* Get vectors we will need */
86   if (BNK_PC_BFGS == bnk->pc_type && !bnk->M) {
87     ierr = VecGetLocalSize(tao->solution,&n);CHKERRQ(ierr);
88     ierr = VecGetSize(tao->solution,&N);CHKERRQ(ierr);
89     ierr = MatCreateLMVM(((PetscObject)tao)->comm,n,N,&bnk->M);CHKERRQ(ierr);
90     ierr = MatLMVMAllocateVectors(bnk->M,tao->solution);CHKERRQ(ierr);
91   }
92 
93   /* create vectors for the limited memory preconditioner */
94   if ((BNK_PC_BFGS == bnk->pc_type) && (BFGS_SCALE_BFGS != bnk->bfgs_scale_type)) {
95     if (!bnk->Diag) {ierr = VecDuplicate(tao->solution,&bnk->Diag);CHKERRQ(ierr);}
96     if (!bnk->Diag_min) {ierr = VecDuplicate(tao->solution,&bnk->Diag_min);CHKERRQ(ierr);}
97     if (!bnk->Diag_max) {ierr = VecDuplicate(tao->solution,&bnk->Diag_max);CHKERRQ(ierr);}
98     ierr = VecSet(bnk->Diag_min, bnk->dmin);CHKERRQ(ierr);
99     ierr = VecSet(bnk->Diag_max, bnk->dmax);CHKERRQ(ierr);
100   }
101 
102   /* Modify the preconditioner to use the bfgs approximation */
103   ierr = KSPGetPC(tao->ksp, &pc);CHKERRQ(ierr);
104   switch(bnk->pc_type) {
105   case BNK_PC_NONE:
106     ierr = PCSetType(pc, PCNONE);CHKERRQ(ierr);
107     ierr = PCSetFromOptions(pc);CHKERRQ(ierr);
108     break;
109 
110   case BNK_PC_AHESS:
111     ierr = PCSetType(pc, PCJACOBI);CHKERRQ(ierr);
112     ierr = PCSetFromOptions(pc);CHKERRQ(ierr);
113     ierr = PCJacobiSetUseAbs(pc,PETSC_TRUE);CHKERRQ(ierr);
114     break;
115 
116   case BNK_PC_BFGS:
117     ierr = PCSetType(pc, PCSHELL);CHKERRQ(ierr);
118     ierr = PCSetFromOptions(pc);CHKERRQ(ierr);
119     ierr = PCShellSetName(pc, "bfgs");CHKERRQ(ierr);
120     ierr = PCShellSetContext(pc, bnk->M);CHKERRQ(ierr);
121     ierr = PCShellSetApply(pc, MatLMVMSolveShell);CHKERRQ(ierr);
122     break;
123 
124   default:
125     /* Use the pc method set by pc_type */
126     break;
127   }
128 
129   /* Initialize trust-region radius.  The initialization is only performed
130      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
131   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
132     switch(initType) {
133     case BNK_INIT_CONSTANT:
134       /* Use the initial radius specified */
135       break;
136 
137     case BNK_INIT_INTERPOLATION:
138       /* Use the initial radius specified */
139       max_radius = 0.0;
140 
141       for (j = 0; j < j_max; ++j) {
142         f_min = bnk->f;
143         sigma = 0.0;
144 
145         if (needH) {
146           /* Compute the Hessian at the new step, and extract the inactive subsystem */
147           ierr = TaoComputeHessian(tao, tao->solution,tao->hessian,tao->hessian_pre);CHKERRQ(ierr);
148           if (bnk->inactive_idx) {
149             ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
150             ierr = VecWhichInactive(tao->XL,tao->solution,bnk->unprojected_gradient,tao->XU,PETSC_TRUE,&bnk->inactive_idx);CHKERRQ(ierr);
151             ierr = TaoMatGetSubMat(tao->hessian, bnk->inactive_idx, bnk->Xwork, TAO_SUBSET_MASK, &bnk->H_inactive);CHKERRQ(ierr);
152           } else {
153             ierr = MatDestroy(&bnk->H_inactive);
154             ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);
155           }
156           needH = 0;
157         }
158 
159         for (i = 0; i < i_max; ++i) {
160           /* Take a steepest descent step and snap it to bounds */
161           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
162           ierr = VecAXPY(tao->solution, -tao->trust/bnk->gnorm, tao->gradient);CHKERRQ(ierr);
163           ierr = VecMedian(tao->XL, tao->solution, tao->XU, tao->solution);CHKERRQ(ierr);
164           /* Recompute the step after bound snapping so that it can be used in predicted decrease calculation later */
165           ierr = VecCopy(tao->solution, bnk->W);CHKERRQ(ierr);
166           ierr = VecAXPY(bnk->W, -1.0, bnk->Xold);CHKERRQ(ierr);
167           /* Compute the objective at the trial */
168           ierr = TaoComputeObjective(tao, tao->solution, &ftrial);CHKERRQ(ierr);
169           ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
170           if (PetscIsInfOrNanReal(ftrial)) {
171             tau = bnk->gamma1_i;
172           } else {
173             if (ftrial < f_min) {
174               f_min = ftrial;
175               sigma = -tao->trust / bnk->gnorm;
176             }
177             /* Compute the predicted and actual reduction */
178             ierr = MatMult(bnk->H_inactive, tao->gradient, bnk->W);CHKERRQ(ierr);
179             ierr = VecDot(tao->gradient, bnk->W, &prered);CHKERRQ(ierr);
180             prered = tao->trust * (bnk->gnorm - 0.5 * tao->trust * prered / (bnk->gnorm * bnk->gnorm));
181             actred = bnk->f - ftrial;
182             if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
183               kappa = 1.0;
184             } else {
185               kappa = actred / prered;
186             }
187 
188             tau_1 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust + (1.0 - bnk->theta_i) * prered - actred);
189             tau_2 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust - (1.0 + bnk->theta_i) * prered + actred);
190             tau_min = PetscMin(tau_1, tau_2);
191             tau_max = PetscMax(tau_1, tau_2);
192 
193             if (PetscAbsScalar(kappa - 1.0) <= bnk->mu1_i) {
194               /* Great agreement */
195               max_radius = PetscMax(max_radius, tao->trust);
196 
197               if (tau_max < 1.0) {
198                 tau = bnk->gamma3_i;
199               } else if (tau_max > bnk->gamma4_i) {
200                 tau = bnk->gamma4_i;
201               } else if (tau_1 >= 1.0 && tau_1 <= bnk->gamma4_i && tau_2 < 1.0) {
202                 tau = tau_1;
203               } else if (tau_2 >= 1.0 && tau_2 <= bnk->gamma4_i && tau_1 < 1.0) {
204                 tau = tau_2;
205               } else {
206                 tau = tau_max;
207               }
208             } else if (PetscAbsScalar(kappa - 1.0) <= bnk->mu2_i) {
209               /* Good agreement */
210               max_radius = PetscMax(max_radius, tao->trust);
211 
212               if (tau_max < bnk->gamma2_i) {
213                 tau = bnk->gamma2_i;
214               } else if (tau_max > bnk->gamma3_i) {
215                 tau = bnk->gamma3_i;
216               } else {
217                 tau = tau_max;
218               }
219             } else {
220               /* Not good agreement */
221               if (tau_min > 1.0) {
222                 tau = bnk->gamma2_i;
223               } else if (tau_max < bnk->gamma1_i) {
224                 tau = bnk->gamma1_i;
225               } else if ((tau_min < bnk->gamma1_i) && (tau_max >= 1.0)) {
226                 tau = bnk->gamma1_i;
227               } else if ((tau_1 >= bnk->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
228                 tau = tau_1;
229               } else if ((tau_2 >= bnk->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
230                 tau = tau_2;
231               } else {
232                 tau = tau_max;
233               }
234             }
235           }
236           tao->trust = tau * tao->trust;
237         }
238 
239         if (f_min < bnk->f) {
240           bnk->f = f_min;
241           ierr = VecAXPY(tao->solution,sigma,tao->gradient);CHKERRQ(ierr);
242           ierr = VecMedian(tao->XL, tao->solution, tao->XU, tao->solution);CHKERRQ(ierr);
243           ierr = TaoComputeGradient(tao,tao->solution,bnk->unprojected_gradient);CHKERRQ(ierr);
244           ierr = VecBoundGradientProjection(bnk->unprojected_gradient,tao->solution,tao->XL,tao->XU,tao->gradient);CHKERRQ(ierr);
245 
246           ierr = TaoGradientNorm(tao, tao->gradient,NORM_2,&bnk->gnorm);CHKERRQ(ierr);
247           if (PetscIsInfOrNanReal(bnk->gnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute gradient generated Inf or NaN");
248           needH = 1;
249 
250           ierr = TaoLogConvergenceHistory(tao,bnk->f,bnk->gnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
251           ierr = TaoMonitor(tao,tao->niter,bnk->f,bnk->gnorm,0.0,1.0);CHKERRQ(ierr);
252           ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
253           if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
254         }
255       }
256       tao->trust = PetscMax(tao->trust, max_radius);
257 
258       /* Modify the radius if it is too large or small */
259       tao->trust = PetscMax(tao->trust, bnk->min_radius);
260       tao->trust = PetscMin(tao->trust, bnk->max_radius);
261       break;
262 
263     default:
264       /* Norm of the first direction will initialize radius */
265       tao->trust = 0.0;
266       break;
267     }
268   }
269 
270   /* Set initial scaling for the BFGS preconditioner
271      This step is done after computing the initial trust-region radius
272      since the function value may have decreased */
273   if (BNK_PC_BFGS == bnk->pc_type) {
274     delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
275     ierr = MatLMVMSetDelta(bnk->M,delta);CHKERRQ(ierr);
276   }
277 
278   /* Set counter for gradient/reset steps*/
279   bnk->newt = 0;
280   bnk->bfgs = 0;
281   bnk->sgrad = 0;
282   bnk->grad = 0;
283   PetscFunctionReturn(0);
284 }
285 
286 /*------------------------------------------------------------*/
287 
288 /* Routine for computing the Hessian and preparing the preconditioner at the new iterate */
289 
290 PetscErrorCode TaoBNKComputeHessian(Tao tao)
291 {
292   PetscErrorCode               ierr;
293   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
294 
295   PetscFunctionBegin;
296   /* Compute the Hessian */
297   ierr = TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);CHKERRQ(ierr);
298   /* Add a correction to the BFGS preconditioner */
299   if (BNK_PC_BFGS == bnk->pc_type) {
300     ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
301     /* Update the BFGS diagonal scaling */
302     if (BFGS_SCALE_AHESS == bnk->bfgs_scale_type) {
303       ierr = MatGetDiagonal(tao->hessian, bnk->Diag);CHKERRQ(ierr);
304       ierr = VecAbs(bnk->Diag);CHKERRQ(ierr);
305       ierr = VecMedian(bnk->Diag_min, bnk->Diag, bnk->Diag_max, bnk->Diag);CHKERRQ(ierr);
306       ierr = VecReciprocal(bnk->Diag);CHKERRQ(ierr);
307       ierr = MatLMVMSetScale(bnk->M,bnk->Diag);CHKERRQ(ierr);
308     }
309   }
310   PetscFunctionReturn(0);
311 }
312 
313 /*------------------------------------------------------------*/
314 
315 /* Routine for estimating the active set */
316 
317 PetscErrorCode TaoBNKEstimateActiveSet(Tao tao)
318 {
319   PetscErrorCode               ierr;
320   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
321 
322   PetscFunctionBegin;
323   switch (bnk->as_type) {
324   case BNK_AS_NONE:
325     ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
326     ierr = VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);CHKERRQ(ierr);
327     ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
328     ierr = ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);CHKERRQ(ierr);
329     break;
330 
331   case BNK_AS_BERTSEKAS:
332     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
333     if (BNK_PC_BFGS == bnk->pc_type) {
334       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
335       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
336     } else {
337       /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
338       ierr = MatGetDiagonal(tao->hessian, bnk->Xwork);CHKERRQ(ierr);
339       ierr = VecAbs(bnk->Xwork);CHKERRQ(ierr);
340       ierr = VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);CHKERRQ(ierr);
341       ierr = VecReciprocal(bnk->Xwork);CHKERRQ(ierr);CHKERRQ(ierr);
342       ierr = VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);CHKERRQ(ierr);
343     }
344     ierr = VecScale(bnk->W, -1.0);CHKERRQ(ierr);
345     ierr = TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->as_step, &bnk->as_tol, &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);CHKERRQ(ierr);
346 
347   default:
348     break;
349   }
350   PetscFunctionReturn(0);
351 }
352 
353 /*------------------------------------------------------------*/
354 
355 /* Routine for bounding the step direction */
356 
357 PetscErrorCode TaoBNKBoundStep(Tao tao, Vec step)
358 {
359   PetscErrorCode               ierr;
360   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
361 
362   PetscFunctionBegin;
363   if (bnk->active_idx) {
364     switch (bnk->as_type) {
365     case BNK_AS_NONE:
366       if (bnk->active_idx) {
367         ierr = VecGetSubVector(step, bnk->active_idx, &bnk->active_work);CHKERRQ(ierr);
368         ierr = VecSet(bnk->active_work, 0.0);CHKERRQ(ierr);
369         ierr = VecRestoreSubVector(step, bnk->active_idx, &bnk->active_work);CHKERRQ(ierr);
370       }
371       break;
372 
373     case BNK_AS_BERTSEKAS:
374       ierr = TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, step);CHKERRQ(ierr);
375       break;
376 
377     default:
378       break;
379     }
380   }
381   PetscFunctionReturn(0);
382 }
383 
384 /*------------------------------------------------------------*/
385 
386 /* Routine for computing the Newton step.
387 
388   If the safeguard is enabled, the Newton step is verified to be a
389   descent direction, with fallbacks onto BFGS, scaled gradient, and unscaled
390   gradient steps if/when necessary.
391 
392   The function reports back on which type of step has ultimately been stored
393   under tao->stepdirection.
394 */
395 
396 PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason)
397 {
398   PetscErrorCode               ierr;
399   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
400 
401   PetscReal                    delta;
402   PetscInt                     bfgsUpdates = 0;
403   PetscInt                     kspits;
404 
405   PetscFunctionBegin;
406   /* Determine the active and inactive sets */
407   ierr = TaoBNKEstimateActiveSet(tao);CHKERRQ(ierr);
408 
409   /* Prepare masked matrices for the inactive set */
410   if (BNK_PC_BFGS == bnk->pc_type) { ierr = MatLMVMSetInactive(bnk->M, bnk->inactive_idx);CHKERRQ(ierr); }
411   if (bnk->inactive_idx) {
412     ierr = TaoMatGetSubMat(tao->hessian, bnk->inactive_idx, bnk->Xwork, TAO_SUBSET_MASK, &bnk->H_inactive);CHKERRQ(ierr);
413     if (tao->hessian == tao->hessian_pre) {
414       bnk->Hpre_inactive = bnk->H_inactive;
415     } else {
416       ierr = TaoMatGetSubMat(tao->hessian_pre, bnk->inactive_idx, bnk->Xwork, TAO_SUBSET_MASK, &bnk->Hpre_inactive);CHKERRQ(ierr);
417     }
418   } else {
419     ierr = MatDestroy(&bnk->H_inactive);
420     ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);
421     if (tao->hessian == tao->hessian_pre) {
422       bnk->Hpre_inactive = bnk->H_inactive;
423     } else {
424       ierr = MatDestroy(&bnk->Hpre_inactive);
425       ierr = MatDuplicate(tao->hessian_pre, MAT_COPY_VALUES, &bnk->Hpre_inactive);
426     }
427   }
428 
429   /* Shift the reduced Hessian matrix */
430   if ((shift) && (bnk->pert > 0)) {
431     ierr = MatShift(bnk->H_inactive, bnk->pert);CHKERRQ(ierr);
432     if (bnk->H_inactive != bnk->Hpre_inactive) {
433       ierr = MatShift(bnk->Hpre_inactive, bnk->pert);CHKERRQ(ierr);
434     }
435   }
436 
437   /* Update the diagonal scaling for the BFGS preconditioner, this time with the Hessian perturbation */
438   if ((BNK_PC_BFGS == bnk->pc_type) && (BFGS_SCALE_PHESS == bnk->bfgs_scale_type)) {
439     /* Obtain diagonal for the bfgs preconditioner  */
440     ierr = MatGetDiagonal(bnk->H_inactive, bnk->Diag);CHKERRQ(ierr);
441     ierr = VecAbs(bnk->Diag);CHKERRQ(ierr);
442     ierr = VecMedian(bnk->Diag_min, bnk->Diag, bnk->Diag_max, bnk->Diag);CHKERRQ(ierr);
443     ierr = VecReciprocal(bnk->Diag);CHKERRQ(ierr);
444     ierr = MatLMVMSetScale(bnk->M,bnk->Diag);CHKERRQ(ierr);
445   }
446 
447   /* Solve the Newton system of equations */
448   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
449   ierr = KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);CHKERRQ(ierr);
450   ierr = VecCopy(bnk->unprojected_gradient, bnk->G_inactive);CHKERRQ(ierr);
451   if (bnk->active_idx) {
452     ierr = VecGetSubVector(bnk->G_inactive, bnk->active_idx, &bnk->active_work);CHKERRQ(ierr);
453     ierr = VecSet(bnk->active_work, 0.0);CHKERRQ(ierr);
454     ierr = VecRestoreSubVector(bnk->G_inactive, bnk->active_idx, &bnk->active_work);CHKERRQ(ierr);
455   }
456   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
457     ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
458     ierr = KSPSolve(tao->ksp, bnk->G_inactive, tao->stepdirection);CHKERRQ(ierr);
459     ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
460     tao->ksp_its+=kspits;
461     tao->ksp_tot_its+=kspits;
462     ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
463 
464     if (0.0 == tao->trust) {
465       /* Radius was uninitialized; use the norm of the direction */
466       if (bnk->dnorm > 0.0) {
467         tao->trust = bnk->dnorm;
468 
469         /* Modify the radius if it is too large or small */
470         tao->trust = PetscMax(tao->trust, bnk->min_radius);
471         tao->trust = PetscMin(tao->trust, bnk->max_radius);
472       } else {
473         /* The direction was bad; set radius to default value and re-solve
474            the trust-region subproblem to get a direction */
475         tao->trust = tao->trust0;
476 
477         /* Modify the radius if it is too large or small */
478         tao->trust = PetscMax(tao->trust, bnk->min_radius);
479         tao->trust = PetscMin(tao->trust, bnk->max_radius);
480 
481         ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
482         ierr = KSPSolve(tao->ksp, bnk->G_inactive, tao->stepdirection);CHKERRQ(ierr);
483         ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
484         tao->ksp_its+=kspits;
485         tao->ksp_tot_its+=kspits;
486         ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
487 
488         if (bnk->dnorm == 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial direction zero");
489       }
490     }
491   } else {
492     ierr = KSPSolve(tao->ksp, bnk->G_inactive, tao->stepdirection);CHKERRQ(ierr);
493     ierr = KSPGetIterationNumber(tao->ksp, &kspits);CHKERRQ(ierr);
494     tao->ksp_its += kspits;
495     tao->ksp_tot_its+=kspits;
496   }
497   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
498   ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
499   ierr = TaoBNKBoundStep(tao, tao->stepdirection);CHKERRQ(ierr);
500 
501   /* Record convergence reasons */
502   ierr = KSPGetConvergedReason(tao->ksp, ksp_reason);CHKERRQ(ierr);
503   if (KSP_CONVERGED_ATOL == *ksp_reason) {
504     ++bnk->ksp_atol;
505   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
506     ++bnk->ksp_rtol;
507   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
508     ++bnk->ksp_ctol;
509   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
510     ++bnk->ksp_negc;
511   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
512     ++bnk->ksp_dtol;
513   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
514     ++bnk->ksp_iter;
515   } else {
516     ++bnk->ksp_othr;
517   }
518 
519   /* Make sure the BFGS preconditioner is healthy */
520   if (bnk->pc_type == BNK_PC_BFGS) {
521     ierr = MatLMVMGetUpdates(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
522     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 1)) {
523       /* Preconditioner is numerically indefinite; reset the approximation. */
524       delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
525       ierr = MatLMVMSetDelta(bnk->M,delta);CHKERRQ(ierr);
526       ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
527       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
528     }
529   }
530   PetscFunctionReturn(0);
531 }
532 
533 /*------------------------------------------------------------*/
534 
535 /* Routine for ensuring that the Newton step is a descent direction.
536 
537    The step direction falls back onto BFGS, scaled gradient and gradient steps
538    in the event that the Newton step fails the test.
539 */
540 
541 PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
542 {
543   PetscErrorCode               ierr;
544   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
545 
546   PetscReal                    gdx, delta, e_min;
547   PetscInt                     bfgsUpdates;
548 
549   PetscFunctionBegin;
550   ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
551   if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
552     /* Newton step is not descent or direction produced Inf or NaN
553        Update the perturbation for next time */
554     if (bnk->pert <= 0.0) {
555       /* Initialize the perturbation */
556       bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
557       if (bnk->is_gltr) {
558         ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
559         bnk->pert = PetscMax(bnk->pert, -e_min);
560       }
561     } else {
562       /* Increase the perturbation */
563       bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
564     }
565 
566     if (BNK_PC_BFGS != bnk->pc_type) {
567       /* We don't have the bfgs matrix around and updated
568          Must use gradient direction in this case */
569       ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
570       *stepType = BNK_GRADIENT;
571     } else {
572       /* Attempt to use the BFGS direction */
573       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
574 
575       /* Check for success (descent direction)
576          NOTE: Negative gdx here means not a descent direction because
577          the fall-back step is missing a negative sign. */
578       ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
579       if ((gdx <= 0) || PetscIsInfOrNanReal(gdx)) {
580         /* BFGS direction is not descent or direction produced not a number
581            We can assert bfgsUpdates > 1 in this case because
582            the first solve produces the scaled gradient direction,
583            which is guaranteed to be descent */
584 
585         /* Use steepest descent direction (scaled) */
586         delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
587         ierr = MatLMVMSetDelta(bnk->M, delta);CHKERRQ(ierr);
588         ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
589         ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
590         ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
591 
592         *stepType = BNK_SCALED_GRADIENT;
593       } else {
594         ierr = MatLMVMGetUpdates(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
595         if (1 == bfgsUpdates) {
596           /* The first BFGS direction is always the scaled gradient */
597           *stepType = BNK_SCALED_GRADIENT;
598         } else {
599           *stepType = BNK_BFGS;
600         }
601       }
602     }
603     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
604     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
605     ierr = TaoBNKBoundStep(tao, tao->stepdirection);CHKERRQ(ierr);
606   } else {
607     /* Computed Newton step is descent */
608     switch (ksp_reason) {
609     case KSP_DIVERGED_NANORINF:
610     case KSP_DIVERGED_BREAKDOWN:
611     case KSP_DIVERGED_INDEFINITE_MAT:
612     case KSP_DIVERGED_INDEFINITE_PC:
613     case KSP_CONVERGED_CG_NEG_CURVE:
614       /* Matrix or preconditioner is indefinite; increase perturbation */
615       if (bnk->pert <= 0.0) {
616         /* Initialize the perturbation */
617         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
618         if (bnk->is_gltr) {
619           ierr = KSPCGGLTRGetMinEig(tao->ksp, &e_min);CHKERRQ(ierr);
620           bnk->pert = PetscMax(bnk->pert, -e_min);
621         }
622       } else {
623         /* Increase the perturbation */
624         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
625       }
626       break;
627 
628     default:
629       /* Newton step computation is good; decrease perturbation */
630       bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
631       if (bnk->pert < bnk->pmin) {
632         bnk->pert = 0.0;
633       }
634       break;
635     }
636     *stepType = BNK_NEWTON;
637   }
638   PetscFunctionReturn(0);
639 }
640 
641 /*------------------------------------------------------------*/
642 
643 /* Routine for performing a bound-projected More-Thuente line search.
644 
645   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
646   Newton step does not produce a valid step length.
647 */
648 
649 PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
650 {
651   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
652   PetscErrorCode ierr;
653   TaoLineSearchConvergedReason ls_reason;
654 
655   PetscReal      e_min, gdx, delta;
656   PetscInt       bfgsUpdates;
657 
658   PetscFunctionBegin;
659   /* Perform the linesearch */
660   ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
661   ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
662 
663   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && (stepType != BNK_GRADIENT || stepType !=BNK_SCALED_GRADIENT)) {
664     /* Linesearch failed, revert solution */
665     bnk->f = bnk->fold;
666     ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
667     ierr = VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);CHKERRQ(ierr);
668 
669     switch(stepType) {
670     case BNK_NEWTON:
671       /* Failed to obtain acceptable iterate with Newton step
672          Update the perturbation for next time */
673       if (bnk->pert <= 0.0) {
674         /* Initialize the perturbation */
675         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
676         if (bnk->is_gltr) {
677           ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
678           bnk->pert = PetscMax(bnk->pert, -e_min);
679         }
680       } else {
681         /* Increase the perturbation */
682         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
683       }
684 
685       if (BNK_PC_BFGS != bnk->pc_type) {
686         /* We don't have the bfgs matrix around and being updated
687            Must use gradient direction in this case */
688         ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
689         stepType = BNK_GRADIENT;
690       } else {
691         /* Attempt to use the BFGS direction */
692         ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
693         /* Check for success (descent direction)
694            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
695         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
696         if ((gdx <= 0) || PetscIsInfOrNanReal(gdx)) {
697           /* BFGS direction is not descent or direction produced not a number
698              We can assert bfgsUpdates > 1 in this case
699              Use steepest descent direction (scaled) */
700           delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
701           ierr = MatLMVMSetDelta(bnk->M, delta);CHKERRQ(ierr);
702           ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
703           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
704           ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
705 
706           bfgsUpdates = 1;
707           stepType = BNK_SCALED_GRADIENT;
708         } else {
709           ierr = MatLMVMGetUpdates(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
710           if (1 == bfgsUpdates) {
711             /* The first BFGS direction is always the scaled gradient */
712             stepType = BNK_SCALED_GRADIENT;
713           } else {
714             stepType = BNK_BFGS;
715           }
716         }
717       }
718       break;
719 
720     case BNK_BFGS:
721       /* Can only enter if pc_type == BNK_PC_BFGS
722          Failed to obtain acceptable iterate with BFGS step
723          Attempt to use the scaled gradient direction */
724       delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
725       ierr = MatLMVMSetDelta(bnk->M, delta);CHKERRQ(ierr);
726       ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
727       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
728       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
729 
730       bfgsUpdates = 1;
731       stepType = BNK_SCALED_GRADIENT;
732       break;
733 
734     case BNK_SCALED_GRADIENT:
735       /* Can only enter if pc_type == BNK_PC_BFGS
736          The scaled gradient step did not produce a new iterate;
737          attemp to use the gradient direction.
738          Need to make sure we are not using a different diagonal scaling */
739       ierr = MatLMVMSetScale(bnk->M,0);CHKERRQ(ierr);
740       ierr = MatLMVMSetDelta(bnk->M,1.0);CHKERRQ(ierr);
741       ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
742       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
743       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
744 
745       bfgsUpdates = 1;
746       stepType = BNK_GRADIENT;
747       break;
748     }
749     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
750     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
751     ierr = TaoBNKBoundStep(tao, tao->stepdirection);CHKERRQ(ierr);
752 
753     /* Perform one last line search with the fall-back step */
754     ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
755     ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
756   }
757   *reason = ls_reason;
758   PetscFunctionReturn(0);
759 }
760 
761 /*------------------------------------------------------------*/
762 
763 /* Routine for updating the trust radius.
764 
765   Function features three different update methods:
766   1) Line-search step length based
767   2) Predicted decrease on the CG quadratic model
768   3) Interpolation
769 */
770 
771 PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
772 {
773   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
774   PetscErrorCode ierr;
775 
776   PetscReal      step, kappa;
777   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;
778 
779   PetscFunctionBegin;
780   /* Update trust region radius */
781   *accept = PETSC_FALSE;
782   switch(updateType) {
783   case BNK_UPDATE_STEP:
784     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
785     if (stepType == BNK_NEWTON) {
786       ierr = TaoLineSearchGetStepLength(tao->linesearch, &step);CHKERRQ(ierr);
787       if (step < bnk->nu1) {
788         /* Very bad step taken; reduce radius */
789         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
790       } else if (step < bnk->nu2) {
791         /* Reasonably bad step taken; reduce radius */
792         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
793       } else if (step < bnk->nu3) {
794         /*  Reasonable step was taken; leave radius alone */
795         if (bnk->omega3 < 1.0) {
796           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
797         } else if (bnk->omega3 > 1.0) {
798           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
799         }
800       } else if (step < bnk->nu4) {
801         /*  Full step taken; increase the radius */
802         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
803       } else {
804         /*  More than full step taken; increase the radius */
805         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
806       }
807     } else {
808       /*  Newton step was not good; reduce the radius */
809       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
810     }
811     break;
812 
813   case BNK_UPDATE_REDUCTION:
814     if (stepType == BNK_NEWTON) {
815       if (prered < 0.0) {
816         /* The predicted reduction has the wrong sign.  This cannot
817            happen in infinite precision arithmetic.  Step should
818            be rejected! */
819         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
820       }
821       else {
822         if (PetscIsInfOrNanReal(actred)) {
823           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
824         } else {
825           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) &&
826               (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
827             kappa = 1.0;
828           }
829           else {
830             kappa = actred / prered;
831           }
832 
833           /* Accept or reject the step and update radius */
834           if (kappa < bnk->eta1) {
835             /* Reject the step */
836             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
837           }
838           else {
839             /* Accept the step */
840             *accept = PETSC_TRUE;
841             /* Update the trust region radius only if the computed step is at the trust radius boundary */
842             if (bnk->dnorm == tao->trust) {
843               if (kappa < bnk->eta2) {
844                 /* Marginal bad step */
845                 tao->trust = bnk->alpha2 * tao->trust;
846               }
847               else if (kappa < bnk->eta3) {
848                 /* Reasonable step */
849                 tao->trust = bnk->alpha3 * tao->trust;
850               }
851               else if (kappa < bnk->eta4) {
852                 /* Good step */
853                 tao->trust = bnk->alpha4 * tao->trust;
854               }
855               else {
856                 /* Very good step */
857                 tao->trust = bnk->alpha5 * tao->trust;
858               }
859             }
860           }
861         }
862       }
863     } else {
864       /*  Newton step was not good; reduce the radius */
865       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
866     }
867     break;
868 
869   default:
870     if (stepType == BNK_NEWTON) {
871       if (prered < 0.0) {
872         /*  The predicted reduction has the wrong sign.  This cannot */
873         /*  happen in infinite precision arithmetic.  Step should */
874         /*  be rejected! */
875         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
876       } else {
877         if (PetscIsInfOrNanReal(actred)) {
878           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
879         } else {
880           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
881             kappa = 1.0;
882           } else {
883             kappa = actred / prered;
884           }
885 
886           ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
887           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
888           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
889           tau_min = PetscMin(tau_1, tau_2);
890           tau_max = PetscMax(tau_1, tau_2);
891 
892           if (kappa >= 1.0 - bnk->mu1) {
893             /*  Great agreement */
894             *accept = PETSC_TRUE;
895             if (tau_max < 1.0) {
896               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
897             } else if (tau_max > bnk->gamma4) {
898               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
899             } else {
900               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
901             }
902           } else if (kappa >= 1.0 - bnk->mu2) {
903             /*  Good agreement */
904             *accept = PETSC_TRUE;
905             if (tau_max < bnk->gamma2) {
906               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
907             } else if (tau_max > bnk->gamma3) {
908               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
909             } else if (tau_max < 1.0) {
910               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
911             } else {
912               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
913             }
914           } else {
915             /*  Not good agreement */
916             if (tau_min > 1.0) {
917               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
918             } else if (tau_max < bnk->gamma1) {
919               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
920             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
921               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
922             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
923               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
924             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
925               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
926             } else {
927               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
928             }
929           }
930         }
931       }
932     } else {
933       /*  Newton step was not good; reduce the radius */
934       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
935     }
936     break;
937   }
938   /* Make sure the radius does not violate min and max settings */
939   tao->trust = PetscMin(tao->trust, bnk->max_radius);
940   tao->trust = PetscMax(tao->trust, bnk->min_radius);
941   PetscFunctionReturn(0);
942 }
943 
944 /* ---------------------------------------------------------- */
945 
946 PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
947 {
948   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
949 
950   PetscFunctionBegin;
951   switch (stepType) {
952   case BNK_NEWTON:
953     ++bnk->newt;
954     break;
955   case BNK_BFGS:
956     ++bnk->bfgs;
957     break;
958   case BNK_SCALED_GRADIENT:
959     ++bnk->sgrad;
960     break;
961   case BNK_GRADIENT:
962     ++bnk->grad;
963     break;
964   default:
965     break;
966   }
967   PetscFunctionReturn(0);
968 }
969 
970 /* ---------------------------------------------------------- */
971 
972 PetscErrorCode TaoSetUp_BNK(Tao tao)
973 {
974   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
975   PetscErrorCode ierr;
976   KSPType        ksp_type;
977 
978   PetscFunctionBegin;
979   if (!tao->gradient) {ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);}
980   if (!tao->stepdirection) {ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);}
981   if (!bnk->W) {ierr = VecDuplicate(tao->solution,&bnk->W);CHKERRQ(ierr);}
982   if (!bnk->Xold) {ierr = VecDuplicate(tao->solution,&bnk->Xold);CHKERRQ(ierr);}
983   if (!bnk->Gold) {ierr = VecDuplicate(tao->solution,&bnk->Gold);CHKERRQ(ierr);}
984   if (!bnk->Xwork) {ierr = VecDuplicate(tao->solution,&bnk->Xwork);CHKERRQ(ierr);}
985   if (!bnk->Gwork) {ierr = VecDuplicate(tao->solution,&bnk->Gwork);CHKERRQ(ierr);}
986   if (!bnk->unprojected_gradient) {ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient);CHKERRQ(ierr);}
987   if (!bnk->unprojected_gradient_old) {ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);CHKERRQ(ierr);}
988   if (!bnk->G_inactive) {ierr = VecDuplicate(tao->solution,&bnk->G_inactive);CHKERRQ(ierr);}
989   bnk->Diag = 0;
990   bnk->Diag_min = 0;
991   bnk->Diag_max = 0;
992   bnk->inactive_work = 0;
993   bnk->active_work = 0;
994   bnk->inactive_idx = 0;
995   bnk->active_idx = 0;
996   bnk->active_lower = 0;
997   bnk->active_upper = 0;
998   bnk->active_fixed = 0;
999   bnk->M = 0;
1000   bnk->H_inactive = 0;
1001   bnk->Hpre_inactive = 0;
1002   ierr = KSPGetType(tao->ksp,&ksp_type);CHKERRQ(ierr);
1003   ierr = PetscStrcmp(ksp_type,KSPCGNASH,&bnk->is_nash);CHKERRQ(ierr);
1004   ierr = PetscStrcmp(ksp_type,KSPCGSTCG,&bnk->is_stcg);CHKERRQ(ierr);
1005   ierr = PetscStrcmp(ksp_type,KSPCGGLTR,&bnk->is_gltr);CHKERRQ(ierr);
1006   PetscFunctionReturn(0);
1007 }
1008 
1009 /*------------------------------------------------------------*/
1010 
1011 static PetscErrorCode TaoDestroy_BNK(Tao tao)
1012 {
1013   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1014   PetscErrorCode ierr;
1015 
1016   PetscFunctionBegin;
1017   if (tao->setupcalled) {
1018     ierr = VecDestroy(&bnk->W);CHKERRQ(ierr);
1019     ierr = VecDestroy(&bnk->Xold);CHKERRQ(ierr);
1020     ierr = VecDestroy(&bnk->Gold);CHKERRQ(ierr);
1021     ierr = VecDestroy(&bnk->Xwork);CHKERRQ(ierr);
1022     ierr = VecDestroy(&bnk->Gwork);CHKERRQ(ierr);
1023     ierr = VecDestroy(&bnk->unprojected_gradient);CHKERRQ(ierr);
1024     ierr = VecDestroy(&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1025     ierr = VecDestroy(&bnk->G_inactive);CHKERRQ(ierr);
1026   }
1027   ierr = VecDestroy(&bnk->Diag);CHKERRQ(ierr);
1028   ierr = VecDestroy(&bnk->Diag_min);CHKERRQ(ierr);
1029   ierr = VecDestroy(&bnk->Diag_max);CHKERRQ(ierr);
1030   ierr = MatDestroy(&bnk->M);CHKERRQ(ierr);
1031   if (bnk->Hpre_inactive != bnk->H_inactive) {ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);}
1032   ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
1033   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1034   PetscFunctionReturn(0);
1035 }
1036 
1037 /*------------------------------------------------------------*/
1038 
1039 static PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
1040 {
1041   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1042   PetscErrorCode ierr;
1043 
1044   PetscFunctionBegin;
1045   ierr = PetscOptionsHead(PetscOptionsObject,"Newton line search method for unconstrained optimization");CHKERRQ(ierr);
1046   ierr = PetscOptionsEList("-tao_bnk_pc_type", "pc type", "", BNK_PC, BNK_PC_TYPES, BNK_PC[bnk->pc_type], &bnk->pc_type, 0);CHKERRQ(ierr);
1047   ierr = PetscOptionsEList("-tao_bnk_bfgs_scale_type", "bfgs scale type", "", BFGS_SCALE, BFGS_SCALE_TYPES, BFGS_SCALE[bnk->bfgs_scale_type], &bnk->bfgs_scale_type, 0);CHKERRQ(ierr);
1048   ierr = PetscOptionsEList("-tao_bnk_init_type", "radius initialization type", "", BNK_INIT, BNK_INIT_TYPES, BNK_INIT[bnk->init_type], &bnk->init_type, 0);CHKERRQ(ierr);
1049   ierr = PetscOptionsEList("-tao_bnk_update_type", "radius update type", "", BNK_UPDATE, BNK_UPDATE_TYPES, BNK_UPDATE[bnk->update_type], &bnk->update_type, 0);CHKERRQ(ierr);
1050   ierr = PetscOptionsEList("-tao_bnk_as_type", "active set estimation method", "", BNK_AS, BNK_AS_TYPES, BNK_AS[bnk->as_type], &bnk->as_type, 0);CHKERRQ(ierr);
1051   ierr = PetscOptionsReal("-tao_bnk_sval", "perturbation starting value", "", bnk->sval, &bnk->sval,NULL);CHKERRQ(ierr);
1052   ierr = PetscOptionsReal("-tao_bnk_imin", "minimum initial perturbation", "", bnk->imin, &bnk->imin,NULL);CHKERRQ(ierr);
1053   ierr = PetscOptionsReal("-tao_bnk_imax", "maximum initial perturbation", "", bnk->imax, &bnk->imax,NULL);CHKERRQ(ierr);
1054   ierr = PetscOptionsReal("-tao_bnk_imfac", "initial merit factor", "", bnk->imfac, &bnk->imfac,NULL);CHKERRQ(ierr);
1055   ierr = PetscOptionsReal("-tao_bnk_pmin", "minimum perturbation", "", bnk->pmin, &bnk->pmin,NULL);CHKERRQ(ierr);
1056   ierr = PetscOptionsReal("-tao_bnk_pmax", "maximum perturbation", "", bnk->pmax, &bnk->pmax,NULL);CHKERRQ(ierr);
1057   ierr = PetscOptionsReal("-tao_bnk_pgfac", "growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);CHKERRQ(ierr);
1058   ierr = PetscOptionsReal("-tao_bnk_psfac", "shrink factor", "", bnk->psfac, &bnk->psfac,NULL);CHKERRQ(ierr);
1059   ierr = PetscOptionsReal("-tao_bnk_pmgfac", "merit growth factor", "", bnk->pmgfac, &bnk->pmgfac,NULL);CHKERRQ(ierr);
1060   ierr = PetscOptionsReal("-tao_bnk_pmsfac", "merit shrink factor", "", bnk->pmsfac, &bnk->pmsfac,NULL);CHKERRQ(ierr);
1061   ierr = PetscOptionsReal("-tao_bnk_eta1", "poor steplength; reduce radius", "", bnk->eta1, &bnk->eta1,NULL);CHKERRQ(ierr);
1062   ierr = PetscOptionsReal("-tao_bnk_eta2", "reasonable steplength; leave radius alone", "", bnk->eta2, &bnk->eta2,NULL);CHKERRQ(ierr);
1063   ierr = PetscOptionsReal("-tao_bnk_eta3", "good steplength; increase radius", "", bnk->eta3, &bnk->eta3,NULL);CHKERRQ(ierr);
1064   ierr = PetscOptionsReal("-tao_bnk_eta4", "excellent steplength; greatly increase radius", "", bnk->eta4, &bnk->eta4,NULL);CHKERRQ(ierr);
1065   ierr = PetscOptionsReal("-tao_bnk_alpha1", "", "", bnk->alpha1, &bnk->alpha1,NULL);CHKERRQ(ierr);
1066   ierr = PetscOptionsReal("-tao_bnk_alpha2", "", "", bnk->alpha2, &bnk->alpha2,NULL);CHKERRQ(ierr);
1067   ierr = PetscOptionsReal("-tao_bnk_alpha3", "", "", bnk->alpha3, &bnk->alpha3,NULL);CHKERRQ(ierr);
1068   ierr = PetscOptionsReal("-tao_bnk_alpha4", "", "", bnk->alpha4, &bnk->alpha4,NULL);CHKERRQ(ierr);
1069   ierr = PetscOptionsReal("-tao_bnk_alpha5", "", "", bnk->alpha5, &bnk->alpha5,NULL);CHKERRQ(ierr);
1070   ierr = PetscOptionsReal("-tao_bnk_nu1", "poor steplength; reduce radius", "", bnk->nu1, &bnk->nu1,NULL);CHKERRQ(ierr);
1071   ierr = PetscOptionsReal("-tao_bnk_nu2", "reasonable steplength; leave radius alone", "", bnk->nu2, &bnk->nu2,NULL);CHKERRQ(ierr);
1072   ierr = PetscOptionsReal("-tao_bnk_nu3", "good steplength; increase radius", "", bnk->nu3, &bnk->nu3,NULL);CHKERRQ(ierr);
1073   ierr = PetscOptionsReal("-tao_bnk_nu4", "excellent steplength; greatly increase radius", "", bnk->nu4, &bnk->nu4,NULL);CHKERRQ(ierr);
1074   ierr = PetscOptionsReal("-tao_bnk_omega1", "", "", bnk->omega1, &bnk->omega1,NULL);CHKERRQ(ierr);
1075   ierr = PetscOptionsReal("-tao_bnk_omega2", "", "", bnk->omega2, &bnk->omega2,NULL);CHKERRQ(ierr);
1076   ierr = PetscOptionsReal("-tao_bnk_omega3", "", "", bnk->omega3, &bnk->omega3,NULL);CHKERRQ(ierr);
1077   ierr = PetscOptionsReal("-tao_bnk_omega4", "", "", bnk->omega4, &bnk->omega4,NULL);CHKERRQ(ierr);
1078   ierr = PetscOptionsReal("-tao_bnk_omega5", "", "", bnk->omega5, &bnk->omega5,NULL);CHKERRQ(ierr);
1079   ierr = PetscOptionsReal("-tao_bnk_mu1_i", "", "", bnk->mu1_i, &bnk->mu1_i,NULL);CHKERRQ(ierr);
1080   ierr = PetscOptionsReal("-tao_bnk_mu2_i", "", "", bnk->mu2_i, &bnk->mu2_i,NULL);CHKERRQ(ierr);
1081   ierr = PetscOptionsReal("-tao_bnk_gamma1_i", "", "", bnk->gamma1_i, &bnk->gamma1_i,NULL);CHKERRQ(ierr);
1082   ierr = PetscOptionsReal("-tao_bnk_gamma2_i", "", "", bnk->gamma2_i, &bnk->gamma2_i,NULL);CHKERRQ(ierr);
1083   ierr = PetscOptionsReal("-tao_bnk_gamma3_i", "", "", bnk->gamma3_i, &bnk->gamma3_i,NULL);CHKERRQ(ierr);
1084   ierr = PetscOptionsReal("-tao_bnk_gamma4_i", "", "", bnk->gamma4_i, &bnk->gamma4_i,NULL);CHKERRQ(ierr);
1085   ierr = PetscOptionsReal("-tao_bnk_theta_i", "", "", bnk->theta_i, &bnk->theta_i,NULL);CHKERRQ(ierr);
1086   ierr = PetscOptionsReal("-tao_bnk_mu1", "", "", bnk->mu1, &bnk->mu1,NULL);CHKERRQ(ierr);
1087   ierr = PetscOptionsReal("-tao_bnk_mu2", "", "", bnk->mu2, &bnk->mu2,NULL);CHKERRQ(ierr);
1088   ierr = PetscOptionsReal("-tao_bnk_gamma1", "", "", bnk->gamma1, &bnk->gamma1,NULL);CHKERRQ(ierr);
1089   ierr = PetscOptionsReal("-tao_bnk_gamma2", "", "", bnk->gamma2, &bnk->gamma2,NULL);CHKERRQ(ierr);
1090   ierr = PetscOptionsReal("-tao_bnk_gamma3", "", "", bnk->gamma3, &bnk->gamma3,NULL);CHKERRQ(ierr);
1091   ierr = PetscOptionsReal("-tao_bnk_gamma4", "", "", bnk->gamma4, &bnk->gamma4,NULL);CHKERRQ(ierr);
1092   ierr = PetscOptionsReal("-tao_bnk_theta", "", "", bnk->theta, &bnk->theta,NULL);CHKERRQ(ierr);
1093   ierr = PetscOptionsReal("-tao_bnk_min_radius", "lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);CHKERRQ(ierr);
1094   ierr = PetscOptionsReal("-tao_bnk_max_radius", "upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);CHKERRQ(ierr);
1095   ierr = PetscOptionsReal("-tao_bnk_epsilon", "tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);CHKERRQ(ierr);
1096   ierr = PetscOptionsReal("-tao_bnk_as_tol", "initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);CHKERRQ(ierr);
1097   ierr = PetscOptionsReal("-tao_bnk_as_step", "step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);CHKERRQ(ierr);
1098   ierr = PetscOptionsTail();CHKERRQ(ierr);
1099   ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr);
1100   ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr);
1101   PetscFunctionReturn(0);
1102 }
1103 
1104 /*------------------------------------------------------------*/
1105 
1106 static PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
1107 {
1108   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1109   PetscInt       nrejects;
1110   PetscBool      isascii;
1111   PetscErrorCode ierr;
1112 
1113   PetscFunctionBegin;
1114   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1115   if (isascii) {
1116     ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
1117     if (BNK_PC_BFGS == bnk->pc_type && bnk->M) {
1118       ierr = MatLMVMGetRejects(bnk->M,&nrejects);CHKERRQ(ierr);
1119       ierr = PetscViewerASCIIPrintf(viewer, "Rejected matrix updates: %D\n",nrejects);CHKERRQ(ierr);
1120     }
1121     ierr = PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);CHKERRQ(ierr);
1122     ierr = PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);CHKERRQ(ierr);
1123     ierr = PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);CHKERRQ(ierr);
1124     ierr = PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);CHKERRQ(ierr);
1125     ierr = PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");CHKERRQ(ierr);
1126     ierr = PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);CHKERRQ(ierr);
1127     ierr = PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);CHKERRQ(ierr);
1128     ierr = PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);CHKERRQ(ierr);
1129     ierr = PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);CHKERRQ(ierr);
1130     ierr = PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);CHKERRQ(ierr);
1131     ierr = PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);CHKERRQ(ierr);
1132     ierr = PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);CHKERRQ(ierr);
1133     ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
1134   }
1135   PetscFunctionReturn(0);
1136 }
1137 
1138 /* ---------------------------------------------------------- */
1139 
1140 /*MC
1141   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
1142   At each iteration, the BNK methods solve the symmetric
1143   system of equations to obtain the step diretion dk:
1144               Hk dk = -gk
1145   for free variables only. The step can be globalized either through
1146   trust-region methods, or a line search, or a heuristic mixture of both.
1147 
1148     Options Database Keys:
1149 + -tao_bnk_pc_type - "none","ahess","bfgs","petsc"
1150 . -tao_bnk_bfgs_scale_type - "ahess","phess","bfgs"
1151 . -tao_bnk_init_type - "constant","direction","interpolation"
1152 . -tao_bnk_update_type - "step","direction","interpolation"
1153 . -tao_bnk_as_type - "none","bertsekas"
1154 . -tao_bnk_sval - perturbation starting value
1155 . -tao_bnk_imin - minimum initial perturbation
1156 . -tao_bnk_imax - maximum initial perturbation
1157 . -tao_bnk_pmin - minimum perturbation
1158 . -tao_bnk_pmax - maximum perturbation
1159 . -tao_bnk_pgfac - growth factor
1160 . -tao_bnk_psfac - shrink factor
1161 . -tao_bnk_imfac - initial merit factor
1162 . -tao_bnk_pmgfac - merit growth factor
1163 . -tao_bnk_pmsfac - merit shrink factor
1164 . -tao_bnk_eta1 - poor steplength; reduce radius
1165 . -tao_bnk_eta2 - reasonable steplength; leave radius
1166 . -tao_bnk_eta3 - good steplength; increase readius
1167 . -tao_bnk_eta4 - excellent steplength; greatly increase radius
1168 . -tao_bnk_alpha1 - alpha1 reduction
1169 . -tao_bnk_alpha2 - alpha2 reduction
1170 . -tao_bnk_alpha3 - alpha3 reduction
1171 . -tao_bnk_alpha4 - alpha4 reduction
1172 . -tao_bnk_alpha - alpha5 reduction
1173 . -tao_bnk_mu1 - mu1 interpolation update
1174 . -tao_bnk_mu2 - mu2 interpolation update
1175 . -tao_bnk_gamma1 - gamma1 interpolation update
1176 . -tao_bnk_gamma2 - gamma2 interpolation update
1177 . -tao_bnk_gamma3 - gamma3 interpolation update
1178 . -tao_bnk_gamma4 - gamma4 interpolation update
1179 . -tao_bnk_theta - theta interpolation update
1180 . -tao_bnk_omega1 - omega1 step update
1181 . -tao_bnk_omega2 - omega2 step update
1182 . -tao_bnk_omega3 - omega3 step update
1183 . -tao_bnk_omega4 - omega4 step update
1184 . -tao_bnk_omega5 - omega5 step update
1185 . -tao_bnk_mu1_i -  mu1 interpolation init factor
1186 . -tao_bnk_mu2_i -  mu2 interpolation init factor
1187 . -tao_bnk_gamma1_i -  gamma1 interpolation init factor
1188 . -tao_bnk_gamma2_i -  gamma2 interpolation init factor
1189 . -tao_bnk_gamma3_i -  gamma3 interpolation init factor
1190 . -tao_bnk_gamma4_i -  gamma4 interpolation init factor
1191 . -tao_bnk_theta_i -  theta interpolation init factor
1192 - -tao_bnk_bound_tol -  initial tolerance used in estimating bounded active variables
1193 
1194   Level: beginner
1195 M*/
1196 
1197 PetscErrorCode TaoCreate_BNK(Tao tao)
1198 {
1199   TAO_BNK        *bnk;
1200   const char     *morethuente_type = TAOLINESEARCHMT;
1201   PetscErrorCode ierr;
1202 
1203   PetscFunctionBegin;
1204   ierr = PetscNewLog(tao,&bnk);CHKERRQ(ierr);
1205 
1206   tao->ops->setup = TaoSetUp_BNK;
1207   tao->ops->view = TaoView_BNK;
1208   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
1209   tao->ops->destroy = TaoDestroy_BNK;
1210 
1211   /*  Override default settings (unless already changed) */
1212   if (!tao->max_it_changed) tao->max_it = 50;
1213   if (!tao->trust0_changed) tao->trust0 = 100.0;
1214 
1215   tao->data = (void*)bnk;
1216 
1217   /*  Hessian shifting parameters */
1218   bnk->sval   = 0.0;
1219   bnk->imin   = 1.0e-4;
1220   bnk->imax   = 1.0e+2;
1221   bnk->imfac  = 1.0e-1;
1222 
1223   bnk->pmin   = 1.0e-12;
1224   bnk->pmax   = 1.0e+2;
1225   bnk->pgfac  = 1.0e+1;
1226   bnk->psfac  = 4.0e-1;
1227   bnk->pmgfac = 1.0e-1;
1228   bnk->pmsfac = 1.0e-1;
1229 
1230   /*  Default values for trust-region radius update based on steplength */
1231   bnk->nu1 = 0.25;
1232   bnk->nu2 = 0.50;
1233   bnk->nu3 = 1.00;
1234   bnk->nu4 = 1.25;
1235 
1236   bnk->omega1 = 0.25;
1237   bnk->omega2 = 0.50;
1238   bnk->omega3 = 1.00;
1239   bnk->omega4 = 2.00;
1240   bnk->omega5 = 4.00;
1241 
1242   /*  Default values for trust-region radius update based on reduction */
1243   bnk->eta1 = 1.0e-4;
1244   bnk->eta2 = 0.25;
1245   bnk->eta3 = 0.50;
1246   bnk->eta4 = 0.90;
1247 
1248   bnk->alpha1 = 0.25;
1249   bnk->alpha2 = 0.50;
1250   bnk->alpha3 = 1.00;
1251   bnk->alpha4 = 2.00;
1252   bnk->alpha5 = 4.00;
1253 
1254   /*  Default values for trust-region radius update based on interpolation */
1255   bnk->mu1 = 0.10;
1256   bnk->mu2 = 0.50;
1257 
1258   bnk->gamma1 = 0.25;
1259   bnk->gamma2 = 0.50;
1260   bnk->gamma3 = 2.00;
1261   bnk->gamma4 = 4.00;
1262 
1263   bnk->theta = 0.05;
1264 
1265   /*  Default values for trust region initialization based on interpolation */
1266   bnk->mu1_i = 0.35;
1267   bnk->mu2_i = 0.50;
1268 
1269   bnk->gamma1_i = 0.0625;
1270   bnk->gamma2_i = 0.5;
1271   bnk->gamma3_i = 2.0;
1272   bnk->gamma4_i = 5.0;
1273 
1274   bnk->theta_i = 0.25;
1275 
1276   /*  Remaining parameters */
1277   bnk->min_radius = 1.0e-10;
1278   bnk->max_radius = 1.0e10;
1279   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
1280   bnk->as_tol = 1.0e-3;
1281   bnk->as_step = 1.0e-3;
1282   bnk->dmin = 1.0e-6;
1283   bnk->dmax = 1.0e6;
1284 
1285   bnk->pc_type         = BNK_PC_BFGS;
1286   bnk->bfgs_scale_type = BFGS_SCALE_PHESS;
1287   bnk->init_type       = BNK_INIT_INTERPOLATION;
1288   bnk->update_type     = BNK_UPDATE_INTERPOLATION;
1289   bnk->as_type         = BNK_AS_BERTSEKAS;
1290 
1291   ierr = TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);CHKERRQ(ierr);
1292   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr);
1293   ierr = TaoLineSearchSetType(tao->linesearch,morethuente_type);CHKERRQ(ierr);
1294   ierr = TaoLineSearchUseTaoRoutines(tao->linesearch,tao);CHKERRQ(ierr);
1295   ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr);
1296 
1297   /*  Set linear solver to default for symmetric matrices */
1298   ierr = KSPCreate(((PetscObject)tao)->comm,&tao->ksp);CHKERRQ(ierr);
1299   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr);
1300   ierr = KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);CHKERRQ(ierr);
1301   ierr = KSPSetType(tao->ksp,KSPCGSTCG);CHKERRQ(ierr);
1302   PetscFunctionReturn(0);
1303 }
1304