xref: /petsc/src/tao/bound/impls/bnk/bnk.c (revision ca964c4924045cf1d9552da2e7805dac7209e511)
1 #include <petsctaolinesearch.h>
2 #include <../src/tao/bound/impls/bnk/bnk.h>
3 
4 #include <petscksp.h>
5 
6 static const char *BNK_PC[64] = {"none", "ahess", "bfgs", "petsc"};
7 static const char *BFGS_SCALE[64] = {"ahess", "phess", "bfgs"};
8 static const char *BNK_INIT[64] = {"constant", "direction", "interpolation"};
9 static const char *BNK_UPDATE[64] = {"step", "reduction", "interpolation"};
10 static const char *BNK_AS[64] = {"none", "bertsekas"};
11 
12 /*------------------------------------------------------------*/
13 
14 /* Routine for BFGS preconditioner */
15 
16 PetscErrorCode MatLMVMSolveShell(PC pc, Vec b, Vec x)
17 {
18   PetscErrorCode ierr;
19   Mat            M;
20 
21   PetscFunctionBegin;
22   PetscValidHeaderSpecific(pc,PC_CLASSID,1);
23   PetscValidHeaderSpecific(b,VEC_CLASSID,2);
24   PetscValidHeaderSpecific(x,VEC_CLASSID,3);
25   ierr = PCShellGetContext(pc,(void**)&M);CHKERRQ(ierr);
26   ierr = MatLMVMSolve(M, b, x);CHKERRQ(ierr);
27   PetscFunctionReturn(0);
28 }
29 
30 /*------------------------------------------------------------*/
31 
32 /* Routine for initializing the KSP solver, the BFGS preconditioner, and the initial trust radius estimation */
33 
34 PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType, PetscBool *needH)
35 {
36   PetscErrorCode               ierr;
37   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
38   PC                           pc;
39 
40   PetscReal                    f_min, ftrial, prered, actred, kappa, sigma, resnorm;
41   PetscReal                    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
42   PetscReal                    delta;
43 
44   PetscInt                     n,N,nDiff;
45 
46   PetscInt                     i_max = 5;
47   PetscInt                     j_max = 1;
48   PetscInt                     i, j;
49 
50   PetscFunctionBegin;
51   /* Project the current point onto the feasible set */
52   ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr);
53   ierr = TaoSetVariableBounds(bnk->bncg, tao->XL, tao->XU);CHKERRQ(ierr);
54   ierr = TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);CHKERRQ(ierr);
55 
56   /* Project the initial point onto the feasible region */
57   ierr = TaoBoundSolution(tao->XL, tao->XU, tao->solution, 0.0, &nDiff);CHKERRQ(ierr);
58 
59   /* Check convergence criteria */
60   ierr = TaoComputeObjectiveAndGradient(tao, tao->solution, &bnk->f, bnk->unprojected_gradient);CHKERRQ(ierr);
61   ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
62   ierr = VecCopy(bnk->unprojected_gradient, tao->gradient);CHKERRQ(ierr);
63   ierr = VecISSet(tao->gradient, bnk->active_idx, 0.0);CHKERRQ(ierr);
64   ierr = VecNorm(tao->gradient,NORM_2,&bnk->gnorm);CHKERRQ(ierr);
65 
66   /* Test the initial point for convergence */
67   ierr = VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);CHKERRQ(ierr);
68   ierr = VecNorm(bnk->W, NORM_2, &resnorm);CHKERRQ(ierr);
69   if (PetscIsInfOrNanReal(bnk->f) || PetscIsInfOrNanReal(resnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
70   ierr = TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
71   ierr = TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);CHKERRQ(ierr);
72   ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
73   if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
74 
75   /* Reset KSP stopping reason counters */
76   bnk->ksp_atol = 0;
77   bnk->ksp_rtol = 0;
78   bnk->ksp_dtol = 0;
79   bnk->ksp_ctol = 0;
80   bnk->ksp_negc = 0;
81   bnk->ksp_iter = 0;
82   bnk->ksp_othr = 0;
83 
84   /* Reset accepted step type counters */
85   bnk->tot_cg_its = 0;
86   bnk->newt = 0;
87   bnk->bfgs = 0;
88   bnk->sgrad = 0;
89   bnk->grad = 0;
90 
91   /* Initialize the Hessian perturbation */
92   bnk->pert = bnk->sval;
93 
94   /* Reset initial steplength to zero (this helps BNCG reset its direction internally) */
95   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
96 
97   /* Allocate the vectors needed for the BFGS approximation */
98   if (BNK_PC_BFGS == bnk->pc_type) {
99     if (!bnk->M) {
100       ierr = VecGetLocalSize(tao->solution,&n);CHKERRQ(ierr);
101       ierr = VecGetSize(tao->solution,&N);CHKERRQ(ierr);
102       ierr = MatCreateLMVM(((PetscObject)tao)->comm,n,N,&bnk->M);CHKERRQ(ierr);
103       ierr = MatLMVMAllocateVectors(bnk->M,tao->solution);CHKERRQ(ierr);
104     }
105     if (bnk->bfgs_scale_type != BFGS_SCALE_BFGS && !bnk->Diag) {
106       ierr = VecDuplicate(tao->solution,&bnk->Diag);CHKERRQ(ierr);
107     }
108   }
109 
110   /* Prepare the min/max vectors for safeguarding diagonal scales */
111   ierr = VecSet(bnk->Diag_min, bnk->dmin);CHKERRQ(ierr);
112   ierr = VecSet(bnk->Diag_max, bnk->dmax);CHKERRQ(ierr);
113 
114   /* Modify the preconditioner to use the bfgs approximation */
115   ierr = KSPGetPC(tao->ksp, &pc);CHKERRQ(ierr);
116   switch(bnk->pc_type) {
117   case BNK_PC_NONE:
118     ierr = PCSetType(pc, PCNONE);CHKERRQ(ierr);
119     ierr = PCSetFromOptions(pc);CHKERRQ(ierr);
120     break;
121 
122   case BNK_PC_AHESS:
123     ierr = PCSetType(pc, PCJACOBI);CHKERRQ(ierr);
124     ierr = PCSetFromOptions(pc);CHKERRQ(ierr);
125     ierr = PCJacobiSetUseAbs(pc,PETSC_TRUE);CHKERRQ(ierr);
126     break;
127 
128   case BNK_PC_BFGS:
129     ierr = PCSetType(pc, PCSHELL);CHKERRQ(ierr);
130     ierr = PCSetFromOptions(pc);CHKERRQ(ierr);
131     ierr = PCShellSetName(pc, "bfgs");CHKERRQ(ierr);
132     ierr = PCShellSetContext(pc, bnk->M);CHKERRQ(ierr);
133     ierr = PCShellSetApply(pc, MatLMVMSolveShell);CHKERRQ(ierr);
134     break;
135 
136   default:
137     /* Use the pc method set by pc_type */
138     break;
139   }
140 
141   /* Initialize trust-region radius.  The initialization is only performed
142      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
143   *needH = PETSC_TRUE;
144   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
145     switch(initType) {
146     case BNK_INIT_CONSTANT:
147       /* Use the initial radius specified */
148       tao->trust = tao->trust0;
149       break;
150 
151     case BNK_INIT_INTERPOLATION:
152       /* Use interpolation based on the initial Hessian */
153       max_radius = 0.0;
154       tao->trust = tao->trust0;
155       for (j = 0; j < j_max; ++j) {
156         f_min = bnk->f;
157         sigma = 0.0;
158 
159         if (*needH) {
160           /* Compute the Hessian at the new step, and extract the inactive subsystem */
161           ierr = TaoBNKComputeHessian(tao);CHKERRQ(ierr);
162           ierr = TaoBNKEstimateActiveSet(tao, BNK_AS_NONE);CHKERRQ(ierr);
163           ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
164           if (bnk->active_idx) {
165             ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
166           } else {
167             ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);CHKERRQ(ierr);
168           }
169           *needH = PETSC_FALSE;
170         }
171 
172         for (i = 0; i < i_max; ++i) {
173           /* Take a steepest descent step and snap it to bounds */
174           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
175           ierr = VecAXPY(tao->solution, -tao->trust/bnk->gnorm, tao->gradient);CHKERRQ(ierr);
176           ierr = TaoBoundSolution(tao->XL, tao->XU, tao->solution, 0.0, &nDiff);CHKERRQ(ierr);
177           /* Compute the step we actually accepted */
178           ierr = VecCopy(tao->solution, bnk->W);CHKERRQ(ierr);
179           ierr = VecAXPY(bnk->W, -1.0, bnk->Xold);CHKERRQ(ierr);
180           /* Compute the objective at the trial */
181           ierr = TaoComputeObjective(tao, tao->solution, &ftrial);CHKERRQ(ierr);
182           if (PetscIsInfOrNanReal(bnk->f)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
183           ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
184           if (PetscIsInfOrNanReal(ftrial)) {
185             tau = bnk->gamma1_i;
186           } else {
187             if (ftrial < f_min) {
188               f_min = ftrial;
189               sigma = -tao->trust / bnk->gnorm;
190             }
191 
192             /* Compute the predicted and actual reduction */
193             if (bnk->active_idx) {
194               ierr = VecGetSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
195               ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
196             } else {
197               bnk->X_inactive = bnk->W;
198               bnk->inactive_work = bnk->Xwork;
199             }
200             ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
201             ierr = VecDot(bnk->X_inactive, bnk->inactive_work, &prered);CHKERRQ(ierr);
202             if (bnk->active_idx) {
203               ierr = VecRestoreSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
204               ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
205             }
206             prered = tao->trust * (bnk->gnorm - 0.5 * tao->trust * prered / (bnk->gnorm * bnk->gnorm));
207             actred = bnk->f - ftrial;
208             if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
209               kappa = 1.0;
210             } else {
211               kappa = actred / prered;
212             }
213 
214             tau_1 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust + (1.0 - bnk->theta_i) * prered - actred);
215             tau_2 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust - (1.0 + bnk->theta_i) * prered + actred);
216             tau_min = PetscMin(tau_1, tau_2);
217             tau_max = PetscMax(tau_1, tau_2);
218 
219             if (PetscAbsScalar(kappa - 1.0) <= bnk->mu1_i) {
220               /*  Great agreement */
221               max_radius = PetscMax(max_radius, tao->trust);
222 
223               if (tau_max < 1.0) {
224                 tau = bnk->gamma3_i;
225               } else if (tau_max > bnk->gamma4_i) {
226                 tau = bnk->gamma4_i;
227               } else {
228                 tau = tau_max;
229               }
230             }
231             else if (PetscAbsScalar(kappa - 1.0) <= bnk->mu2_i) {
232               /*  Good agreement */
233               max_radius = PetscMax(max_radius, tao->trust);
234 
235               if (tau_max < bnk->gamma2_i) {
236                 tau = bnk->gamma2_i;
237               } else if (tau_max > bnk->gamma3_i) {
238                 tau = bnk->gamma3_i;
239               } else {
240                 tau = tau_max;
241               }
242             }
243             else {
244               /*  Not good agreement */
245               if (tau_min > 1.0) {
246                 tau = bnk->gamma2_i;
247               } else if (tau_max < bnk->gamma1_i) {
248                 tau = bnk->gamma1_i;
249               } else if ((tau_min < bnk->gamma1_i) && (tau_max >= 1.0)) {
250                 tau = bnk->gamma1_i;
251               } else if ((tau_1 >= bnk->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
252                 tau = tau_1;
253               } else if ((tau_2 >= bnk->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
254                 tau = tau_2;
255               } else {
256                 tau = tau_max;
257               }
258             }
259           }
260           tao->trust = tau * tao->trust;
261         }
262 
263         if (f_min < bnk->f) {
264           /* We accidentally found a solution better than the initial, so accept it */
265           bnk->f = f_min;
266           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
267           ierr = VecAXPY(tao->solution,sigma,tao->gradient);CHKERRQ(ierr);
268           ierr = TaoBoundSolution(tao->XL, tao->XU, tao->solution, 0.0, &nDiff);CHKERRQ(ierr);
269           ierr = VecCopy(tao->solution, tao->stepdirection);CHKERRQ(ierr);
270           ierr = VecAXPY(tao->stepdirection, -1.0, bnk->Xold);CHKERRQ(ierr);
271           ierr = TaoComputeGradient(tao,tao->solution,bnk->unprojected_gradient);CHKERRQ(ierr);
272           ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
273           ierr = VecCopy(bnk->unprojected_gradient, tao->gradient);CHKERRQ(ierr);
274           ierr = VecISSet(tao->gradient, bnk->active_idx, 0.0);CHKERRQ(ierr);
275           /* Compute gradient at the new iterate and flip switch to compute the Hessian later */
276           ierr = VecNorm(tao->gradient, NORM_2, &bnk->gnorm);CHKERRQ(ierr);
277           *needH = PETSC_TRUE;
278           /* Test the new step for convergence */
279           ierr = VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);CHKERRQ(ierr);
280           ierr = VecNorm(bnk->W, NORM_2, &resnorm);CHKERRQ(ierr);
281           if (PetscIsInfOrNanReal(resnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
282           ierr = TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
283           ierr = TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);CHKERRQ(ierr);
284           ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
285           if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
286           /* active BNCG recycling early because we have a stepdirection computed */
287           ierr = TaoBNCGSetRecycleFlag(bnk->bncg, PETSC_TRUE);CHKERRQ(ierr);
288         }
289       }
290       tao->trust = PetscMax(tao->trust, max_radius);
291 
292       /* Ensure that the trust radius is within the limits */
293       tao->trust = PetscMax(tao->trust, bnk->min_radius);
294       tao->trust = PetscMin(tao->trust, bnk->max_radius);
295       break;
296 
297     default:
298       /* Norm of the first direction will initialize radius */
299       tao->trust = 0.0;
300       break;
301     }
302   }
303 
304   /* Set initial scaling for the BFGS preconditioner
305      This step is done after computing the initial trust-region radius
306      since the function value may have decreased */
307   if (BNK_PC_BFGS == bnk->pc_type) {
308     delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
309     ierr = MatLMVMSetDelta(bnk->M,delta);CHKERRQ(ierr);
310   }
311   PetscFunctionReturn(0);
312 }
313 
314 /*------------------------------------------------------------*/
315 
316 /* Routine for computing the Hessian and preparing the preconditioner at the new iterate */
317 
318 PetscErrorCode TaoBNKComputeHessian(Tao tao)
319 {
320   PetscErrorCode               ierr;
321   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
322   PetscBool                    diagExists;
323   PetscReal                    delta;
324 
325   PetscFunctionBegin;
326   /* Compute the Hessian */
327   ierr = TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);CHKERRQ(ierr);
328   /* Add a correction to the BFGS preconditioner */
329   if (BNK_PC_BFGS == bnk->pc_type) {
330     ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
331     /* Update the BFGS diagonal scaling */
332     ierr = MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
333     if (BFGS_SCALE_AHESS == bnk->bfgs_scale_type && diagExists) {
334       ierr = MatGetDiagonal(tao->hessian, bnk->Diag);CHKERRQ(ierr);
335       ierr = VecAbs(bnk->Diag);CHKERRQ(ierr);
336       ierr = VecMedian(bnk->Diag_min, bnk->Diag, bnk->Diag_max, bnk->Diag);CHKERRQ(ierr);
337       ierr = VecReciprocal(bnk->Diag);CHKERRQ(ierr);
338       ierr = MatLMVMSetScale(bnk->M,bnk->Diag);CHKERRQ(ierr);
339     } else {
340       /* Fall back onto stand-alone BFGS scaling */
341       delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
342       ierr = MatLMVMSetDelta(bnk->M,delta);CHKERRQ(ierr);
343     }
344   }
345   PetscFunctionReturn(0);
346 }
347 
348 /*------------------------------------------------------------*/
349 
350 /* Routine for estimating the active set */
351 
352 PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
353 {
354   PetscErrorCode               ierr;
355   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
356   PetscBool                    hessComputed, diagExists;
357 
358   PetscFunctionBegin;
359   switch (asType) {
360   case BNK_AS_NONE:
361     ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
362     ierr = VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);CHKERRQ(ierr);
363     ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
364     ierr = ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);CHKERRQ(ierr);
365     break;
366 
367   case BNK_AS_BERTSEKAS:
368     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
369     if (BNK_PC_BFGS == bnk->pc_type) {
370       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
371       ierr = MatLMVMSetInactive(bnk->M, NULL);CHKERRQ(ierr);
372       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
373     } else {
374       ierr = MatAssembled(tao->hessian, &hessComputed);CHKERRQ(ierr);
375       ierr = MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
376       if (hessComputed && diagExists) {
377         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
378         ierr = MatGetDiagonal(tao->hessian, bnk->Xwork);CHKERRQ(ierr);
379         ierr = VecAbs(bnk->Xwork);CHKERRQ(ierr);
380         ierr = VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);CHKERRQ(ierr);
381         ierr = VecReciprocal(bnk->Xwork);CHKERRQ(ierr);CHKERRQ(ierr);
382         ierr = VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);CHKERRQ(ierr);
383       } else {
384         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
385         ierr = VecCopy(bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
386       }
387     }
388     ierr = VecScale(bnk->W, -1.0);CHKERRQ(ierr);
389     ierr = TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol,
390                                    &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);CHKERRQ(ierr);
391     break;
392 
393   default:
394     break;
395   }
396   PetscFunctionReturn(0);
397 }
398 
399 /*------------------------------------------------------------*/
400 
401 /* Routine for bounding the step direction */
402 
403 PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
404 {
405   PetscErrorCode               ierr;
406   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
407 
408   PetscFunctionBegin;
409   switch (asType) {
410   case BNK_AS_NONE:
411     ierr = VecISSet(step, bnk->active_idx, 0.0);CHKERRQ(ierr);
412     break;
413 
414   case BNK_AS_BERTSEKAS:
415     ierr = TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step);CHKERRQ(ierr);
416     break;
417 
418   default:
419     break;
420   }
421   PetscFunctionReturn(0);
422 }
423 
424 /*------------------------------------------------------------*/
425 
426 /* Routine for taking a finite number of BNCG iterations to
427    accelerate Newton convergence.
428 
429    In practice, this approach simply trades off Hessian evaluations
430    for more gradient evaluations.
431 */
432 
433 PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
434 {
435   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
436   PetscErrorCode               ierr;
437 
438   PetscFunctionBegin;
439   *terminate = PETSC_FALSE;
440   if (bnk->max_cg_its > 0) {
441     /* Copy the current function value (important vectors are already shared) */
442     bnk->bncg_ctx->f = bnk->f;
443     /* Take some small finite number of BNCG iterations */
444     ierr = TaoSolve(bnk->bncg);CHKERRQ(ierr);
445     /* Add the number of gradient and function evaluations to the total */
446     tao->nfuncs += bnk->bncg->nfuncs;
447     tao->nfuncgrads += bnk->bncg->nfuncgrads;
448     tao->ngrads += bnk->bncg->ngrads;
449     tao->nhess += bnk->bncg->nhess;
450     bnk->tot_cg_its += bnk->bncg->niter;
451     /* Extract the BNCG function value out and save it into BNK */
452     bnk->f = bnk->bncg_ctx->f;
453     if (bnk->bncg->reason == TAO_CONVERGED_GATOL || bnk->bncg->reason == TAO_CONVERGED_GRTOL || bnk->bncg->reason == TAO_CONVERGED_GTTOL || bnk->bncg->reason == TAO_CONVERGED_MINF) {
454       *terminate = PETSC_TRUE;
455     } else {
456       ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);
457     }
458   }
459   PetscFunctionReturn(0);
460 }
461 
462 /*------------------------------------------------------------*/
463 
464 /* Routine for computing the Newton step. */
465 
466 PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason)
467 {
468   PetscErrorCode               ierr;
469   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
470 
471   PetscReal                    delta;
472   PetscInt                     bfgsUpdates = 0;
473   PetscInt                     kspits;
474   PetscBool                    diagExists;
475 
476   PetscFunctionBegin;
477   /* If there are no inactive variables left, save some computation and return an adjusted zero step
478      that has (l-x) and (u-x) for lower and upper bounded variables. */
479   if (!bnk->inactive_idx) {
480     ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
481     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
482     PetscFunctionReturn(0);
483   }
484 
485   /* Prepare the reduced sub-matrices for the inactive set */
486   if (BNK_PC_BFGS == bnk->pc_type) {
487     ierr = MatLMVMSetInactive(bnk->M, bnk->inactive_idx);CHKERRQ(ierr);
488   }
489   if (bnk->active_idx) {
490     ierr = MatDestroy(&bnk->H_inactive);
491     ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
492     if (tao->hessian == tao->hessian_pre) {
493       bnk->Hpre_inactive = bnk->H_inactive;
494     } else {
495       ierr = MatDestroy(&bnk->Hpre_inactive);
496       ierr = MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive);CHKERRQ(ierr);
497     }
498   } else {
499     ierr = MatDestroy(&bnk->H_inactive);
500     ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);
501     if (tao->hessian == tao->hessian_pre) {
502       bnk->Hpre_inactive = bnk->H_inactive;
503     } else {
504       ierr = MatDestroy(&bnk->Hpre_inactive);
505       ierr = MatDuplicate(tao->hessian_pre, MAT_COPY_VALUES, &bnk->Hpre_inactive);
506     }
507   }
508 
509   /* Shift the reduced Hessian matrix */
510   if ((shift) && (bnk->pert > 0)) {
511     ierr = MatShift(bnk->H_inactive, bnk->pert);CHKERRQ(ierr);
512     if (bnk->H_inactive != bnk->Hpre_inactive) {
513       ierr = MatShift(bnk->Hpre_inactive, bnk->pert);CHKERRQ(ierr);
514     }
515   }
516 
517   /* Update the diagonal scaling for the BFGS preconditioner, this time with the Hessian perturbation */
518   if ((BNK_PC_BFGS == bnk->pc_type) && (BFGS_SCALE_PHESS == bnk->bfgs_scale_type)) {
519     ierr = MatHasOperation(bnk->H_inactive, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
520     if (diagExists) {
521       /* Obtain diagonal for the bfgs preconditioner  */
522       ierr = VecSet(bnk->Diag, 1.0);CHKERRQ(ierr);
523       if (bnk->active_idx) {
524         ierr = VecGetSubVector(bnk->Diag, bnk->inactive_idx, &bnk->Diag_red);CHKERRQ(ierr);
525       } else {
526         bnk->Diag_red = bnk->Diag;
527       }
528       ierr = MatGetDiagonal(bnk->H_inactive, bnk->Diag_red);CHKERRQ(ierr);
529       if (bnk->active_idx) {
530         ierr = VecRestoreSubVector(bnk->Diag, bnk->inactive_idx, &bnk->Diag_red);CHKERRQ(ierr);
531       }
532       ierr = VecAbs(bnk->Diag);CHKERRQ(ierr);
533       ierr = VecMedian(bnk->Diag_min, bnk->Diag, bnk->Diag_max, bnk->Diag);CHKERRQ(ierr);
534       ierr = VecReciprocal(bnk->Diag);CHKERRQ(ierr);
535       ierr = MatLMVMSetScale(bnk->M,bnk->Diag);CHKERRQ(ierr);
536     } else {
537       /* Fall back onto stand-alone BFGS scaling */
538       delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
539       ierr = MatLMVMSetDelta(bnk->M,delta);CHKERRQ(ierr);
540     }
541   }
542 
543   /* Solve the Newton system of equations */
544   tao->ksp_its = 0;
545   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
546   ierr = KSPReset(tao->ksp);CHKERRQ(ierr);
547   ierr = KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);CHKERRQ(ierr);
548   ierr = VecCopy(bnk->unprojected_gradient, bnk->Gwork);CHKERRQ(ierr);
549   if (bnk->active_idx) {
550     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
551     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
552   } else {
553     bnk->G_inactive = bnk->unprojected_gradient;
554     bnk->X_inactive = tao->stepdirection;
555   }
556   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
557     ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
558     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
559     ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
560     tao->ksp_its+=kspits;
561     tao->ksp_tot_its+=kspits;
562     ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
563 
564     if (0.0 == tao->trust) {
565       /* Radius was uninitialized; use the norm of the direction */
566       if (bnk->dnorm > 0.0) {
567         tao->trust = bnk->dnorm;
568 
569         /* Modify the radius if it is too large or small */
570         tao->trust = PetscMax(tao->trust, bnk->min_radius);
571         tao->trust = PetscMin(tao->trust, bnk->max_radius);
572       } else {
573         /* The direction was bad; set radius to default value and re-solve
574            the trust-region subproblem to get a direction */
575         tao->trust = tao->trust0;
576 
577         /* Modify the radius if it is too large or small */
578         tao->trust = PetscMax(tao->trust, bnk->min_radius);
579         tao->trust = PetscMin(tao->trust, bnk->max_radius);
580 
581         ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
582         ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
583         ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
584         tao->ksp_its+=kspits;
585         tao->ksp_tot_its+=kspits;
586         ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
587 
588         if (bnk->dnorm == 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial direction zero");
589       }
590     }
591   } else {
592     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
593     ierr = KSPGetIterationNumber(tao->ksp, &kspits);CHKERRQ(ierr);
594     tao->ksp_its += kspits;
595     tao->ksp_tot_its+=kspits;
596   }
597   /* Restore sub vectors back */
598   if (bnk->active_idx) {
599     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
600     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
601   }
602   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
603   ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
604   ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
605 
606   /* Record convergence reasons */
607   ierr = KSPGetConvergedReason(tao->ksp, ksp_reason);CHKERRQ(ierr);
608   if (KSP_CONVERGED_ATOL == *ksp_reason) {
609     ++bnk->ksp_atol;
610   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
611     ++bnk->ksp_rtol;
612   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
613     ++bnk->ksp_ctol;
614   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
615     ++bnk->ksp_negc;
616   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
617     ++bnk->ksp_dtol;
618   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
619     ++bnk->ksp_iter;
620   } else {
621     ++bnk->ksp_othr;
622   }
623 
624   /* Make sure the BFGS preconditioner is healthy */
625   if (bnk->pc_type == BNK_PC_BFGS) {
626     ierr = MatLMVMGetUpdates(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
627     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 1)) {
628       /* Preconditioner is numerically indefinite; reset the approximation. */
629       delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
630       ierr = MatLMVMSetDelta(bnk->M,delta);CHKERRQ(ierr);
631       ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
632       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
633     }
634   }
635   PetscFunctionReturn(0);
636 }
637 
638 /*------------------------------------------------------------*/
639 
640 /* Routine for recomputing the predicted reduction for a given step vector */
641 
642 PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
643 {
644   PetscErrorCode               ierr;
645   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
646 
647   PetscFunctionBegin;
648   /* Extract subvectors associated with the inactive set */
649   if (bnk->active_idx){
650     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
651     ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
652     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
653   } else {
654     bnk->X_inactive = tao->stepdirection;
655     bnk->inactive_work = bnk->Xwork;
656     bnk->G_inactive = bnk->Gwork;
657   }
658   /* Recompute the predicted decrease based on the quadratic model */
659   ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
660   ierr = VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive);CHKERRQ(ierr);
661   ierr = VecDot(bnk->inactive_work, bnk->X_inactive, prered);
662   /* Restore the sub vectors */
663   if (bnk->active_idx){
664     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
665     ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
666     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
667   }
668   PetscFunctionReturn(0);
669 }
670 
671 /*------------------------------------------------------------*/
672 
673 /* Routine for ensuring that the Newton step is a descent direction.
674 
675    The step direction falls back onto BFGS, scaled gradient and gradient steps
676    in the event that the Newton step fails the test.
677 */
678 
679 PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
680 {
681   PetscErrorCode               ierr;
682   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
683 
684   PetscReal                    gdx, delta, e_min;
685   PetscInt                     bfgsUpdates;
686 
687   PetscFunctionBegin;
688   ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
689   if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
690     /* Newton step is not descent or direction produced Inf or NaN
691        Update the perturbation for next time */
692     if (bnk->pert <= 0.0) {
693       /* Initialize the perturbation */
694       bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
695       if (bnk->is_gltr) {
696         ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
697         bnk->pert = PetscMax(bnk->pert, -e_min);
698       }
699     } else {
700       /* Increase the perturbation */
701       bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
702     }
703 
704     if (BNK_PC_BFGS != bnk->pc_type) {
705       /* We don't have the bfgs matrix around and updated
706          Must use gradient direction in this case */
707       ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
708       *stepType = BNK_GRADIENT;
709     } else {
710       /* Attempt to use the BFGS direction */
711       ierr = MatLMVMSetInactive(bnk->M, NULL);CHKERRQ(ierr);
712       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
713 
714       /* Check for success (descent direction)
715          NOTE: Negative gdx here means not a descent direction because
716          the fall-back step is missing a negative sign. */
717       ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
718       if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
719         /* BFGS direction is not descent or direction produced not a number
720            We can assert bfgsUpdates > 1 in this case because
721            the first solve produces the scaled gradient direction,
722            which is guaranteed to be descent */
723 
724         /* Use steepest descent direction (scaled) */
725         delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
726         ierr = MatLMVMSetDelta(bnk->M, delta);CHKERRQ(ierr);
727         ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
728         ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
729         ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
730 
731         *stepType = BNK_SCALED_GRADIENT;
732       } else {
733         ierr = MatLMVMGetUpdates(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
734         if (1 == bfgsUpdates) {
735           /* The first BFGS direction is always the scaled gradient */
736           *stepType = BNK_SCALED_GRADIENT;
737         } else {
738           *stepType = BNK_BFGS;
739         }
740       }
741     }
742     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
743     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
744     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
745   } else {
746     /* Computed Newton step is descent */
747     switch (ksp_reason) {
748     case KSP_DIVERGED_NANORINF:
749     case KSP_DIVERGED_BREAKDOWN:
750     case KSP_DIVERGED_INDEFINITE_MAT:
751     case KSP_DIVERGED_INDEFINITE_PC:
752     case KSP_CONVERGED_CG_NEG_CURVE:
753       /* Matrix or preconditioner is indefinite; increase perturbation */
754       if (bnk->pert <= 0.0) {
755         /* Initialize the perturbation */
756         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
757         if (bnk->is_gltr) {
758           ierr = KSPCGGLTRGetMinEig(tao->ksp, &e_min);CHKERRQ(ierr);
759           bnk->pert = PetscMax(bnk->pert, -e_min);
760         }
761       } else {
762         /* Increase the perturbation */
763         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
764       }
765       break;
766 
767     default:
768       /* Newton step computation is good; decrease perturbation */
769       bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
770       if (bnk->pert < bnk->pmin) {
771         bnk->pert = 0.0;
772       }
773       break;
774     }
775     *stepType = BNK_NEWTON;
776   }
777   PetscFunctionReturn(0);
778 }
779 
780 /*------------------------------------------------------------*/
781 
782 /* Routine for performing a bound-projected More-Thuente line search.
783 
784   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
785   Newton step does not produce a valid step length.
786 */
787 
788 PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
789 {
790   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
791   PetscErrorCode ierr;
792   TaoLineSearchConvergedReason ls_reason;
793 
794   PetscReal      e_min, gdx, delta;
795   PetscInt       bfgsUpdates;
796 
797   PetscFunctionBegin;
798   /* Perform the linesearch */
799   ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
800   ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
801 
802   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_GRADIENT) {
803     /* Linesearch failed, revert solution */
804     bnk->f = bnk->fold;
805     ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
806     ierr = VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);CHKERRQ(ierr);
807 
808     switch(*stepType) {
809     case BNK_NEWTON:
810       /* Failed to obtain acceptable iterate with Newton step
811          Update the perturbation for next time */
812       if (bnk->pert <= 0.0) {
813         /* Initialize the perturbation */
814         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
815         if (bnk->is_gltr) {
816           ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
817           bnk->pert = PetscMax(bnk->pert, -e_min);
818         }
819       } else {
820         /* Increase the perturbation */
821         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
822       }
823 
824       if (BNK_PC_BFGS != bnk->pc_type) {
825         /* We don't have the bfgs matrix around and being updated
826            Must use gradient direction in this case */
827         ierr = VecCopy(bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
828         *stepType = BNK_GRADIENT;
829       } else {
830         /* Attempt to use the BFGS direction */
831         ierr = MatLMVMSetInactive(bnk->M, NULL);CHKERRQ(ierr);
832         ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
833         /* Check for success (descent direction)
834            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
835         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
836         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
837           /* BFGS direction is not descent or direction produced not a number
838              We can assert bfgsUpdates > 1 in this case
839              Use steepest descent direction (scaled) */
840           delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
841           ierr = MatLMVMSetDelta(bnk->M, delta);CHKERRQ(ierr);
842           ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
843           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
844           ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
845 
846           bfgsUpdates = 1;
847           *stepType = BNK_SCALED_GRADIENT;
848         } else {
849           ierr = MatLMVMGetUpdates(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
850           if (1 == bfgsUpdates) {
851             /* The first BFGS direction is always the scaled gradient */
852             *stepType = BNK_SCALED_GRADIENT;
853           } else {
854             *stepType = BNK_BFGS;
855           }
856         }
857       }
858       break;
859 
860     case BNK_BFGS:
861       /* Can only enter if pc_type == BNK_PC_BFGS
862          Failed to obtain acceptable iterate with BFGS step
863          Attempt to use the scaled gradient direction */
864       delta = 2.0 * PetscMax(1.0, PetscAbsScalar(bnk->f)) / (bnk->gnorm*bnk->gnorm);
865       ierr = MatLMVMSetDelta(bnk->M, delta);CHKERRQ(ierr);
866       ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
867       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
868       ierr = MatLMVMSetInactive(bnk->M, NULL);CHKERRQ(ierr);
869       ierr = MatLMVMSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
870 
871       bfgsUpdates = 1;
872       *stepType = BNK_SCALED_GRADIENT;
873       break;
874 
875     case BNK_SCALED_GRADIENT:
876       /* Can only enter if pc_type == BNK_PC_BFGS
877          The scaled gradient step did not produce a new iterate;
878          reset the BFGS matrix and attemp to use the gradient direction. */
879       ierr = MatLMVMSetScale(bnk->M,0);CHKERRQ(ierr);
880       ierr = MatLMVMSetDelta(bnk->M,1.0);CHKERRQ(ierr);
881       ierr = MatLMVMReset(bnk->M);CHKERRQ(ierr);
882       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
883       ierr = VecCopy(bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
884 
885       bfgsUpdates = 1;
886       *stepType = BNK_GRADIENT;
887       break;
888     }
889     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
890     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
891     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
892 
893     /* Perform one last line search with the fall-back step */
894     ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
895     ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
896   }
897   *reason = ls_reason;
898   PetscFunctionReturn(0);
899 }
900 
901 /*------------------------------------------------------------*/
902 
903 /* Routine for updating the trust radius.
904 
905   Function features three different update methods:
906   1) Line-search step length based
907   2) Predicted decrease on the CG quadratic model
908   3) Interpolation
909 */
910 
911 PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
912 {
913   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
914   PetscErrorCode ierr;
915 
916   PetscReal      step, kappa;
917   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;
918 
919   PetscFunctionBegin;
920   /* Update trust region radius */
921   *accept = PETSC_FALSE;
922   switch(updateType) {
923   case BNK_UPDATE_STEP:
924     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
925     if (stepType == BNK_NEWTON) {
926       ierr = TaoLineSearchGetStepLength(tao->linesearch, &step);CHKERRQ(ierr);
927       if (step < bnk->nu1) {
928         /* Very bad step taken; reduce radius */
929         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
930       } else if (step < bnk->nu2) {
931         /* Reasonably bad step taken; reduce radius */
932         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
933       } else if (step < bnk->nu3) {
934         /*  Reasonable step was taken; leave radius alone */
935         if (bnk->omega3 < 1.0) {
936           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
937         } else if (bnk->omega3 > 1.0) {
938           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
939         }
940       } else if (step < bnk->nu4) {
941         /*  Full step taken; increase the radius */
942         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
943       } else {
944         /*  More than full step taken; increase the radius */
945         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
946       }
947     } else {
948       /*  Newton step was not good; reduce the radius */
949       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
950     }
951     break;
952 
953   case BNK_UPDATE_REDUCTION:
954     if (stepType == BNK_NEWTON) {
955       if (prered < 0.0) {
956         /* The predicted reduction has the wrong sign.  This cannot
957            happen in infinite precision arithmetic.  Step should
958            be rejected! */
959         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
960       } else {
961         if (PetscIsInfOrNanReal(actred)) {
962           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
963         } else {
964           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
965             kappa = 1.0;
966           } else {
967             kappa = actred / prered;
968           }
969 
970           /* Accept or reject the step and update radius */
971           if (kappa < bnk->eta1) {
972             /* Reject the step */
973             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
974           } else {
975             /* Accept the step */
976             *accept = PETSC_TRUE;
977             /* Update the trust region radius only if the computed step is at the trust radius boundary */
978             if (bnk->dnorm == tao->trust) {
979               if (kappa < bnk->eta2) {
980                 /* Marginal bad step */
981                 tao->trust = bnk->alpha2 * tao->trust;
982               } else if (kappa < bnk->eta3) {
983                 /* Reasonable step */
984                 tao->trust = bnk->alpha3 * tao->trust;
985               } else if (kappa < bnk->eta4) {
986                 /* Good step */
987                 tao->trust = bnk->alpha4 * tao->trust;
988               } else {
989                 /* Very good step */
990                 tao->trust = bnk->alpha5 * tao->trust;
991               }
992             }
993           }
994         }
995       }
996     } else {
997       /*  Newton step was not good; reduce the radius */
998       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
999     }
1000     break;
1001 
1002   default:
1003     if (stepType == BNK_NEWTON) {
1004       if (prered < 0.0) {
1005         /*  The predicted reduction has the wrong sign.  This cannot */
1006         /*  happen in infinite precision arithmetic.  Step should */
1007         /*  be rejected! */
1008         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
1009       } else {
1010         if (PetscIsInfOrNanReal(actred)) {
1011           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
1012         } else {
1013           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
1014             kappa = 1.0;
1015           } else {
1016             kappa = actred / prered;
1017           }
1018 
1019           ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
1020           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
1021           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
1022           tau_min = PetscMin(tau_1, tau_2);
1023           tau_max = PetscMax(tau_1, tau_2);
1024 
1025           if (kappa >= 1.0 - bnk->mu1) {
1026             /*  Great agreement */
1027             *accept = PETSC_TRUE;
1028             if (tau_max < 1.0) {
1029               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
1030             } else if (tau_max > bnk->gamma4) {
1031               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
1032             } else {
1033               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
1034             }
1035           } else if (kappa >= 1.0 - bnk->mu2) {
1036             /*  Good agreement */
1037             *accept = PETSC_TRUE;
1038             if (tau_max < bnk->gamma2) {
1039               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
1040             } else if (tau_max > bnk->gamma3) {
1041               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
1042             } else if (tau_max < 1.0) {
1043               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
1044             } else {
1045               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
1046             }
1047           } else {
1048             /*  Not good agreement */
1049             if (tau_min > 1.0) {
1050               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
1051             } else if (tau_max < bnk->gamma1) {
1052               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
1053             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
1054               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
1055             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
1056               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
1057             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
1058               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
1059             } else {
1060               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
1061             }
1062           }
1063         }
1064       }
1065     } else {
1066       /*  Newton step was not good; reduce the radius */
1067       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
1068     }
1069     break;
1070   }
1071   /* Make sure the radius does not violate min and max settings */
1072   tao->trust = PetscMin(tao->trust, bnk->max_radius);
1073   tao->trust = PetscMax(tao->trust, bnk->min_radius);
1074   PetscFunctionReturn(0);
1075 }
1076 
1077 /* ---------------------------------------------------------- */
1078 
1079 PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
1080 {
1081   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1082 
1083   PetscFunctionBegin;
1084   switch (stepType) {
1085   case BNK_NEWTON:
1086     ++bnk->newt;
1087     break;
1088   case BNK_BFGS:
1089     ++bnk->bfgs;
1090     break;
1091   case BNK_SCALED_GRADIENT:
1092     ++bnk->sgrad;
1093     break;
1094   case BNK_GRADIENT:
1095     ++bnk->grad;
1096     break;
1097   default:
1098     break;
1099   }
1100   PetscFunctionReturn(0);
1101 }
1102 
1103 /* ---------------------------------------------------------- */
1104 
1105 PetscErrorCode TaoSetUp_BNK(Tao tao)
1106 {
1107   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1108   PetscErrorCode ierr;
1109   KSPType        ksp_type;
1110   PetscInt       i;
1111 
1112   PetscFunctionBegin;
1113   if (!tao->gradient) {
1114     ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);
1115   }
1116   if (!tao->stepdirection) {
1117     ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);
1118   }
1119   if (!bnk->W) {
1120     ierr = VecDuplicate(tao->solution,&bnk->W);CHKERRQ(ierr);
1121   }
1122   if (!bnk->Xold) {
1123     ierr = VecDuplicate(tao->solution,&bnk->Xold);CHKERRQ(ierr);
1124   }
1125   if (!bnk->Gold) {
1126     ierr = VecDuplicate(tao->solution,&bnk->Gold);CHKERRQ(ierr);
1127   }
1128   if (!bnk->Xwork) {
1129     ierr = VecDuplicate(tao->solution,&bnk->Xwork);CHKERRQ(ierr);
1130   }
1131   if (!bnk->Gwork) {
1132     ierr = VecDuplicate(tao->solution,&bnk->Gwork);CHKERRQ(ierr);
1133   }
1134   if (!bnk->unprojected_gradient) {
1135     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient);CHKERRQ(ierr);
1136   }
1137   if (!bnk->unprojected_gradient_old) {
1138     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1139   }
1140   if (!bnk->Diag_min) {
1141     ierr = VecDuplicate(tao->solution,&bnk->Diag_min);CHKERRQ(ierr);
1142   }
1143   if (!bnk->Diag_max) {
1144     ierr = VecDuplicate(tao->solution,&bnk->Diag_max);CHKERRQ(ierr);
1145   }
1146   if (bnk->max_cg_its > 0) {
1147     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
1148     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
1149     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old));CHKERRQ(ierr);
1150     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old);CHKERRQ(ierr);
1151     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
1152     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient));CHKERRQ(ierr);
1153     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient);CHKERRQ(ierr);
1154     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
1155     ierr = PetscObjectReference((PetscObject)(bnk->Gold));CHKERRQ(ierr);
1156     ierr = VecDestroy(&bnk->bncg_ctx->G_old);CHKERRQ(ierr);
1157     bnk->bncg_ctx->G_old = bnk->Gold;
1158     ierr = PetscObjectReference((PetscObject)(tao->gradient));CHKERRQ(ierr);
1159     ierr = VecDestroy(&bnk->bncg->gradient);CHKERRQ(ierr);
1160     bnk->bncg->gradient = tao->gradient;
1161     ierr = PetscObjectReference((PetscObject)(tao->stepdirection));CHKERRQ(ierr);
1162     ierr = VecDestroy(&bnk->bncg->stepdirection);CHKERRQ(ierr);
1163     bnk->bncg->stepdirection = tao->stepdirection;
1164     ierr = TaoSetInitialVector(bnk->bncg, tao->solution);CHKERRQ(ierr);
1165     /* Copy over some settings from BNK into BNCG */
1166     ierr = TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its);CHKERRQ(ierr);
1167     ierr = TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol);CHKERRQ(ierr);
1168     ierr = TaoSetFunctionLowerBound(bnk->bncg, tao->fmin);CHKERRQ(ierr);
1169     ierr = TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP);CHKERRQ(ierr);
1170     ierr = TaoSetObjectiveRoutine(bnk->bncg, tao->ops->computeobjective, tao->user_objP);CHKERRQ(ierr);
1171     ierr = TaoSetGradientRoutine(bnk->bncg, tao->ops->computegradient, tao->user_gradP);CHKERRQ(ierr);
1172     ierr = TaoSetObjectiveAndGradientRoutine(bnk->bncg, tao->ops->computeobjectiveandgradient, tao->user_objgradP);CHKERRQ(ierr);
1173     ierr = PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg));CHKERRQ(ierr);
1174     for (i=0; i<tao->numbermonitors; ++i) {
1175       ierr = TaoSetMonitor(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);CHKERRQ(ierr);
1176       ierr = PetscObjectReference((PetscObject)(tao->monitorcontext[i]));CHKERRQ(ierr);
1177     }
1178   }
1179   bnk->Diag = 0;
1180   bnk->Diag_red = 0;
1181   bnk->X_inactive = 0;
1182   bnk->G_inactive = 0;
1183   bnk->inactive_work = 0;
1184   bnk->active_work = 0;
1185   bnk->inactive_idx = 0;
1186   bnk->active_idx = 0;
1187   bnk->active_lower = 0;
1188   bnk->active_upper = 0;
1189   bnk->active_fixed = 0;
1190   bnk->M = 0;
1191   bnk->H_inactive = 0;
1192   bnk->Hpre_inactive = 0;
1193   ierr = KSPGetType(tao->ksp,&ksp_type);CHKERRQ(ierr);
1194   ierr = PetscStrcmp(ksp_type,KSPCGNASH,&bnk->is_nash);CHKERRQ(ierr);
1195   ierr = PetscStrcmp(ksp_type,KSPCGSTCG,&bnk->is_stcg);CHKERRQ(ierr);
1196   ierr = PetscStrcmp(ksp_type,KSPCGGLTR,&bnk->is_gltr);CHKERRQ(ierr);
1197   PetscFunctionReturn(0);
1198 }
1199 
1200 /*------------------------------------------------------------*/
1201 
1202 static PetscErrorCode TaoDestroy_BNK(Tao tao)
1203 {
1204   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1205   PetscErrorCode ierr;
1206 
1207   PetscFunctionBegin;
1208   if (tao->setupcalled) {
1209     ierr = VecDestroy(&bnk->W);CHKERRQ(ierr);
1210     ierr = VecDestroy(&bnk->Xold);CHKERRQ(ierr);
1211     ierr = VecDestroy(&bnk->Gold);CHKERRQ(ierr);
1212     ierr = VecDestroy(&bnk->Xwork);CHKERRQ(ierr);
1213     ierr = VecDestroy(&bnk->Gwork);CHKERRQ(ierr);
1214     ierr = VecDestroy(&bnk->unprojected_gradient);CHKERRQ(ierr);
1215     ierr = VecDestroy(&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1216     ierr = VecDestroy(&bnk->Diag_min);CHKERRQ(ierr);
1217     ierr = VecDestroy(&bnk->Diag_max);CHKERRQ(ierr);
1218   }
1219   ierr = ISDestroy(&bnk->active_lower);CHKERRQ(ierr);
1220   ierr = ISDestroy(&bnk->active_upper);CHKERRQ(ierr);
1221   ierr = ISDestroy(&bnk->active_fixed);CHKERRQ(ierr);
1222   ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
1223   ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
1224   ierr = VecDestroy(&bnk->Diag);CHKERRQ(ierr);
1225   ierr = MatDestroy(&bnk->M);CHKERRQ(ierr);
1226   if (bnk->Hpre_inactive != tao->hessian_pre && bnk->Hpre_inactive != bnk->H_inactive) {
1227     ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
1228   }
1229   if (bnk->H_inactive != tao->hessian) {
1230     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
1231   }
1232   if (bnk->max_cg_its > 0) {
1233     ierr = TaoDestroy(&bnk->bncg);CHKERRQ(ierr);
1234   }
1235   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1236   PetscFunctionReturn(0);
1237 }
1238 
1239 /*------------------------------------------------------------*/
1240 
1241 static PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
1242 {
1243   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1244   PetscErrorCode ierr;
1245 
1246   PetscFunctionBegin;
1247   ierr = PetscOptionsHead(PetscOptionsObject,"Newton line search method for unconstrained optimization");CHKERRQ(ierr);
1248   ierr = PetscOptionsEList("-tao_bnk_pc_type", "pc type", "", BNK_PC, BNK_PC_TYPES, BNK_PC[bnk->pc_type], &bnk->pc_type, 0);CHKERRQ(ierr);
1249   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);
1250   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);
1251   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);
1252   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);
1253   ierr = PetscOptionsReal("-tao_bnk_sval", "perturbation starting value", "", bnk->sval, &bnk->sval,NULL);CHKERRQ(ierr);
1254   ierr = PetscOptionsReal("-tao_bnk_imin", "minimum initial perturbation", "", bnk->imin, &bnk->imin,NULL);CHKERRQ(ierr);
1255   ierr = PetscOptionsReal("-tao_bnk_imax", "maximum initial perturbation", "", bnk->imax, &bnk->imax,NULL);CHKERRQ(ierr);
1256   ierr = PetscOptionsReal("-tao_bnk_imfac", "initial merit factor", "", bnk->imfac, &bnk->imfac,NULL);CHKERRQ(ierr);
1257   ierr = PetscOptionsReal("-tao_bnk_pmin", "minimum perturbation", "", bnk->pmin, &bnk->pmin,NULL);CHKERRQ(ierr);
1258   ierr = PetscOptionsReal("-tao_bnk_pmax", "maximum perturbation", "", bnk->pmax, &bnk->pmax,NULL);CHKERRQ(ierr);
1259   ierr = PetscOptionsReal("-tao_bnk_pgfac", "growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);CHKERRQ(ierr);
1260   ierr = PetscOptionsReal("-tao_bnk_psfac", "shrink factor", "", bnk->psfac, &bnk->psfac,NULL);CHKERRQ(ierr);
1261   ierr = PetscOptionsReal("-tao_bnk_pmgfac", "merit growth factor", "", bnk->pmgfac, &bnk->pmgfac,NULL);CHKERRQ(ierr);
1262   ierr = PetscOptionsReal("-tao_bnk_pmsfac", "merit shrink factor", "", bnk->pmsfac, &bnk->pmsfac,NULL);CHKERRQ(ierr);
1263   ierr = PetscOptionsReal("-tao_bnk_eta1", "poor steplength; reduce radius", "", bnk->eta1, &bnk->eta1,NULL);CHKERRQ(ierr);
1264   ierr = PetscOptionsReal("-tao_bnk_eta2", "reasonable steplength; leave radius alone", "", bnk->eta2, &bnk->eta2,NULL);CHKERRQ(ierr);
1265   ierr = PetscOptionsReal("-tao_bnk_eta3", "good steplength; increase radius", "", bnk->eta3, &bnk->eta3,NULL);CHKERRQ(ierr);
1266   ierr = PetscOptionsReal("-tao_bnk_eta4", "excellent steplength; greatly increase radius", "", bnk->eta4, &bnk->eta4,NULL);CHKERRQ(ierr);
1267   ierr = PetscOptionsReal("-tao_bnk_alpha1", "", "", bnk->alpha1, &bnk->alpha1,NULL);CHKERRQ(ierr);
1268   ierr = PetscOptionsReal("-tao_bnk_alpha2", "", "", bnk->alpha2, &bnk->alpha2,NULL);CHKERRQ(ierr);
1269   ierr = PetscOptionsReal("-tao_bnk_alpha3", "", "", bnk->alpha3, &bnk->alpha3,NULL);CHKERRQ(ierr);
1270   ierr = PetscOptionsReal("-tao_bnk_alpha4", "", "", bnk->alpha4, &bnk->alpha4,NULL);CHKERRQ(ierr);
1271   ierr = PetscOptionsReal("-tao_bnk_alpha5", "", "", bnk->alpha5, &bnk->alpha5,NULL);CHKERRQ(ierr);
1272   ierr = PetscOptionsReal("-tao_bnk_nu1", "poor steplength; reduce radius", "", bnk->nu1, &bnk->nu1,NULL);CHKERRQ(ierr);
1273   ierr = PetscOptionsReal("-tao_bnk_nu2", "reasonable steplength; leave radius alone", "", bnk->nu2, &bnk->nu2,NULL);CHKERRQ(ierr);
1274   ierr = PetscOptionsReal("-tao_bnk_nu3", "good steplength; increase radius", "", bnk->nu3, &bnk->nu3,NULL);CHKERRQ(ierr);
1275   ierr = PetscOptionsReal("-tao_bnk_nu4", "excellent steplength; greatly increase radius", "", bnk->nu4, &bnk->nu4,NULL);CHKERRQ(ierr);
1276   ierr = PetscOptionsReal("-tao_bnk_omega1", "", "", bnk->omega1, &bnk->omega1,NULL);CHKERRQ(ierr);
1277   ierr = PetscOptionsReal("-tao_bnk_omega2", "", "", bnk->omega2, &bnk->omega2,NULL);CHKERRQ(ierr);
1278   ierr = PetscOptionsReal("-tao_bnk_omega3", "", "", bnk->omega3, &bnk->omega3,NULL);CHKERRQ(ierr);
1279   ierr = PetscOptionsReal("-tao_bnk_omega4", "", "", bnk->omega4, &bnk->omega4,NULL);CHKERRQ(ierr);
1280   ierr = PetscOptionsReal("-tao_bnk_omega5", "", "", bnk->omega5, &bnk->omega5,NULL);CHKERRQ(ierr);
1281   ierr = PetscOptionsReal("-tao_bnk_mu1_i", "", "", bnk->mu1_i, &bnk->mu1_i,NULL);CHKERRQ(ierr);
1282   ierr = PetscOptionsReal("-tao_bnk_mu2_i", "", "", bnk->mu2_i, &bnk->mu2_i,NULL);CHKERRQ(ierr);
1283   ierr = PetscOptionsReal("-tao_bnk_gamma1_i", "", "", bnk->gamma1_i, &bnk->gamma1_i,NULL);CHKERRQ(ierr);
1284   ierr = PetscOptionsReal("-tao_bnk_gamma2_i", "", "", bnk->gamma2_i, &bnk->gamma2_i,NULL);CHKERRQ(ierr);
1285   ierr = PetscOptionsReal("-tao_bnk_gamma3_i", "", "", bnk->gamma3_i, &bnk->gamma3_i,NULL);CHKERRQ(ierr);
1286   ierr = PetscOptionsReal("-tao_bnk_gamma4_i", "", "", bnk->gamma4_i, &bnk->gamma4_i,NULL);CHKERRQ(ierr);
1287   ierr = PetscOptionsReal("-tao_bnk_theta_i", "", "", bnk->theta_i, &bnk->theta_i,NULL);CHKERRQ(ierr);
1288   ierr = PetscOptionsReal("-tao_bnk_mu1", "", "", bnk->mu1, &bnk->mu1,NULL);CHKERRQ(ierr);
1289   ierr = PetscOptionsReal("-tao_bnk_mu2", "", "", bnk->mu2, &bnk->mu2,NULL);CHKERRQ(ierr);
1290   ierr = PetscOptionsReal("-tao_bnk_gamma1", "", "", bnk->gamma1, &bnk->gamma1,NULL);CHKERRQ(ierr);
1291   ierr = PetscOptionsReal("-tao_bnk_gamma2", "", "", bnk->gamma2, &bnk->gamma2,NULL);CHKERRQ(ierr);
1292   ierr = PetscOptionsReal("-tao_bnk_gamma3", "", "", bnk->gamma3, &bnk->gamma3,NULL);CHKERRQ(ierr);
1293   ierr = PetscOptionsReal("-tao_bnk_gamma4", "", "", bnk->gamma4, &bnk->gamma4,NULL);CHKERRQ(ierr);
1294   ierr = PetscOptionsReal("-tao_bnk_theta", "", "", bnk->theta, &bnk->theta,NULL);CHKERRQ(ierr);
1295   ierr = PetscOptionsReal("-tao_bnk_min_radius", "lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);CHKERRQ(ierr);
1296   ierr = PetscOptionsReal("-tao_bnk_max_radius", "upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);CHKERRQ(ierr);
1297   ierr = PetscOptionsReal("-tao_bnk_epsilon", "tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);CHKERRQ(ierr);
1298   ierr = PetscOptionsReal("-tao_bnk_as_tol", "initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);CHKERRQ(ierr);
1299   ierr = PetscOptionsReal("-tao_bnk_as_step", "step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);CHKERRQ(ierr);
1300   ierr = PetscOptionsInt("-tao_bnk_max_cg_its", "number of BNCG iterations to take for each Newton step", "", bnk->max_cg_its, &bnk->max_cg_its,NULL);CHKERRQ(ierr);
1301   ierr = PetscOptionsTail();CHKERRQ(ierr);
1302   ierr = TaoSetFromOptions(bnk->bncg);
1303   ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr);
1304   ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr);
1305   PetscFunctionReturn(0);
1306 }
1307 
1308 /*------------------------------------------------------------*/
1309 
1310 static PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
1311 {
1312   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1313   PetscInt       nrejects;
1314   PetscBool      isascii;
1315   PetscErrorCode ierr;
1316 
1317   PetscFunctionBegin;
1318   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1319   if (isascii) {
1320     ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
1321     if (BNK_PC_BFGS == bnk->pc_type && bnk->M) {
1322       ierr = MatLMVMGetRejects(bnk->M,&nrejects);CHKERRQ(ierr);
1323       ierr = PetscViewerASCIIPrintf(viewer, "Rejected matrix updates: %D\n",nrejects);CHKERRQ(ierr);
1324     }
1325     ierr = PetscViewerASCIIPrintf(viewer, "CG steps: %D\n", bnk->tot_cg_its);CHKERRQ(ierr);
1326     ierr = PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);CHKERRQ(ierr);
1327     ierr = PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);CHKERRQ(ierr);
1328     ierr = PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);CHKERRQ(ierr);
1329     ierr = PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);CHKERRQ(ierr);
1330     ierr = PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");CHKERRQ(ierr);
1331     ierr = PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);CHKERRQ(ierr);
1332     ierr = PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);CHKERRQ(ierr);
1333     ierr = PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);CHKERRQ(ierr);
1334     ierr = PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);CHKERRQ(ierr);
1335     ierr = PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);CHKERRQ(ierr);
1336     ierr = PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);CHKERRQ(ierr);
1337     ierr = PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);CHKERRQ(ierr);
1338     ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
1339   }
1340   PetscFunctionReturn(0);
1341 }
1342 
1343 /* ---------------------------------------------------------- */
1344 
1345 /*MC
1346   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
1347   At each iteration, the BNK methods solve the symmetric
1348   system of equations to obtain the step diretion dk:
1349               Hk dk = -gk
1350   for free variables only. The step can be globalized either through
1351   trust-region methods, or a line search, or a heuristic mixture of both.
1352 
1353     Options Database Keys:
1354 + -tao_bnk_pc_type - "none","ahess","bfgs","petsc"
1355 . -tao_bnk_bfgs_scale_type - "ahess","phess","bfgs"
1356 . -tao_bnk_init_type - "constant","direction","interpolation"
1357 . -tao_bnk_update_type - "step","direction","interpolation"
1358 . -tao_bnk_as_type - "none","bertsekas"
1359 . -tao_bnk_sval - perturbation starting value
1360 . -tao_bnk_imin - minimum initial perturbation
1361 . -tao_bnk_imax - maximum initial perturbation
1362 . -tao_bnk_pmin - minimum perturbation
1363 . -tao_bnk_pmax - maximum perturbation
1364 . -tao_bnk_pgfac - growth factor
1365 . -tao_bnk_psfac - shrink factor
1366 . -tao_bnk_imfac - initial merit factor
1367 . -tao_bnk_pmgfac - merit growth factor
1368 . -tao_bnk_pmsfac - merit shrink factor
1369 . -tao_bnk_eta1 - poor steplength; reduce radius
1370 . -tao_bnk_eta2 - reasonable steplength; leave radius
1371 . -tao_bnk_eta3 - good steplength; increase readius
1372 . -tao_bnk_eta4 - excellent steplength; greatly increase radius
1373 . -tao_bnk_alpha1 - alpha1 reduction
1374 . -tao_bnk_alpha2 - alpha2 reduction
1375 . -tao_bnk_alpha3 - alpha3 reduction
1376 . -tao_bnk_alpha4 - alpha4 reduction
1377 . -tao_bnk_alpha - alpha5 reduction
1378 . -tao_bnk_mu1 - mu1 interpolation update
1379 . -tao_bnk_mu2 - mu2 interpolation update
1380 . -tao_bnk_gamma1 - gamma1 interpolation update
1381 . -tao_bnk_gamma2 - gamma2 interpolation update
1382 . -tao_bnk_gamma3 - gamma3 interpolation update
1383 . -tao_bnk_gamma4 - gamma4 interpolation update
1384 . -tao_bnk_theta - theta interpolation update
1385 . -tao_bnk_omega1 - omega1 step update
1386 . -tao_bnk_omega2 - omega2 step update
1387 . -tao_bnk_omega3 - omega3 step update
1388 . -tao_bnk_omega4 - omega4 step update
1389 . -tao_bnk_omega5 - omega5 step update
1390 . -tao_bnk_mu1_i -  mu1 interpolation init factor
1391 . -tao_bnk_mu2_i -  mu2 interpolation init factor
1392 . -tao_bnk_gamma1_i -  gamma1 interpolation init factor
1393 . -tao_bnk_gamma2_i -  gamma2 interpolation init factor
1394 . -tao_bnk_gamma3_i -  gamma3 interpolation init factor
1395 . -tao_bnk_gamma4_i -  gamma4 interpolation init factor
1396 . -tao_bnk_theta_i -  theta interpolation init factor
1397 - -tao_bnk_bound_tol -  initial tolerance used in estimating bounded active variables
1398 
1399   Level: beginner
1400 M*/
1401 
1402 PetscErrorCode TaoCreate_BNK(Tao tao)
1403 {
1404   TAO_BNK        *bnk;
1405   const char     *morethuente_type = TAOLINESEARCHMT;
1406   PetscErrorCode ierr;
1407 
1408   PetscFunctionBegin;
1409   ierr = PetscNewLog(tao,&bnk);CHKERRQ(ierr);
1410 
1411   tao->ops->setup = TaoSetUp_BNK;
1412   tao->ops->view = TaoView_BNK;
1413   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
1414   tao->ops->destroy = TaoDestroy_BNK;
1415 
1416   /*  Override default settings (unless already changed) */
1417   if (!tao->max_it_changed) tao->max_it = 50;
1418   if (!tao->trust0_changed) tao->trust0 = 100.0;
1419 
1420   tao->data = (void*)bnk;
1421 
1422   /*  Hessian shifting parameters */
1423   bnk->sval   = 0.0;
1424   bnk->imin   = 1.0e-4;
1425   bnk->imax   = 1.0e+2;
1426   bnk->imfac  = 1.0e-1;
1427 
1428   bnk->pmin   = 1.0e-12;
1429   bnk->pmax   = 1.0e+2;
1430   bnk->pgfac  = 1.0e+1;
1431   bnk->psfac  = 4.0e-1;
1432   bnk->pmgfac = 1.0e-1;
1433   bnk->pmsfac = 1.0e-1;
1434 
1435   /*  Default values for trust-region radius update based on steplength */
1436   bnk->nu1 = 0.25;
1437   bnk->nu2 = 0.50;
1438   bnk->nu3 = 1.00;
1439   bnk->nu4 = 1.25;
1440 
1441   bnk->omega1 = 0.25;
1442   bnk->omega2 = 0.50;
1443   bnk->omega3 = 1.00;
1444   bnk->omega4 = 2.00;
1445   bnk->omega5 = 4.00;
1446 
1447   /*  Default values for trust-region radius update based on reduction */
1448   bnk->eta1 = 1.0e-4;
1449   bnk->eta2 = 0.25;
1450   bnk->eta3 = 0.50;
1451   bnk->eta4 = 0.90;
1452 
1453   bnk->alpha1 = 0.25;
1454   bnk->alpha2 = 0.50;
1455   bnk->alpha3 = 1.00;
1456   bnk->alpha4 = 2.00;
1457   bnk->alpha5 = 4.00;
1458 
1459   /*  Default values for trust-region radius update based on interpolation */
1460   bnk->mu1 = 0.10;
1461   bnk->mu2 = 0.50;
1462 
1463   bnk->gamma1 = 0.25;
1464   bnk->gamma2 = 0.50;
1465   bnk->gamma3 = 2.00;
1466   bnk->gamma4 = 4.00;
1467 
1468   bnk->theta = 0.05;
1469 
1470   /*  Default values for trust region initialization based on interpolation */
1471   bnk->mu1_i = 0.35;
1472   bnk->mu2_i = 0.50;
1473 
1474   bnk->gamma1_i = 0.0625;
1475   bnk->gamma2_i = 0.5;
1476   bnk->gamma3_i = 2.0;
1477   bnk->gamma4_i = 5.0;
1478 
1479   bnk->theta_i = 0.25;
1480 
1481   /*  Remaining parameters */
1482   bnk->max_cg_its = 0;
1483   bnk->min_radius = 1.0e-10;
1484   bnk->max_radius = 1.0e10;
1485   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
1486   bnk->as_tol = 1.0e-3;
1487   bnk->as_step = 1.0e-3;
1488   bnk->dmin = 1.0e-6;
1489   bnk->dmax = 1.0e6;
1490 
1491   bnk->pc_type         = BNK_PC_AHESS;
1492   bnk->bfgs_scale_type = BFGS_SCALE_PHESS;
1493   bnk->init_type       = BNK_INIT_INTERPOLATION;
1494   bnk->update_type     = BNK_UPDATE_INTERPOLATION;
1495   bnk->as_type         = BNK_AS_BERTSEKAS;
1496 
1497   /* Create the embedded BNCG solver */
1498   ierr = TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg);CHKERRQ(ierr);
1499   ierr = PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1);CHKERRQ(ierr);
1500   ierr = TaoSetOptionsPrefix(bnk->bncg, "tao_bnk_");CHKERRQ(ierr);
1501   ierr = TaoSetType(bnk->bncg, TAOBNCG);CHKERRQ(ierr);
1502 
1503   /* Create the line search */
1504   ierr = TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);CHKERRQ(ierr);
1505   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr);
1506   ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr);
1507   ierr = TaoLineSearchSetType(tao->linesearch,morethuente_type);CHKERRQ(ierr);
1508   ierr = TaoLineSearchUseTaoRoutines(tao->linesearch,tao);CHKERRQ(ierr);
1509 
1510   /*  Set linear solver to default for symmetric matrices */
1511   ierr = KSPCreate(((PetscObject)tao)->comm,&tao->ksp);CHKERRQ(ierr);
1512   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr);
1513   ierr = KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);CHKERRQ(ierr);
1514   ierr = KSPSetType(tao->ksp,KSPCGSTCG);CHKERRQ(ierr);
1515   PetscFunctionReturn(0);
1516 }
1517