xref: /petsc/src/tao/bound/impls/bnk/bnk.c (revision 552a653ff41e89009a3f6713f3041aa5026d35ee)
1 #include <petsctaolinesearch.h>
2 #include <../src/tao/bound/impls/bnk/bnk.h>
3 
4 #include <petscksp.h>
5 
6 static const char *BNK_INIT[64] = {"constant", "direction", "interpolation"};
7 static const char *BNK_UPDATE[64] = {"step", "reduction", "interpolation"};
8 static const char *BNK_AS[64] = {"none", "bertsekas"};
9 
10 /*------------------------------------------------------------*/
11 
12 /* Routine for initializing the KSP solver, the BFGS preconditioner, and the initial trust radius estimation */
13 
14 PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType, PetscBool *needH)
15 {
16   PetscErrorCode               ierr;
17   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
18   PC                           pc;
19 
20   PetscReal                    f_min, ftrial, prered, actred, kappa, sigma, resnorm;
21   PetscReal                    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
22   PetscBool                    is_bfgs, is_jacobi, is_symmetric, sym_set;
23   PetscInt                     n, N, nDiff;
24   PetscInt                     i_max = 5;
25   PetscInt                     j_max = 1;
26   PetscInt                     i, j;
27 
28   PetscFunctionBegin;
29   /* Project the current point onto the feasible set */
30   ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr);
31   ierr = TaoSetVariableBounds(bnk->bncg, tao->XL, tao->XU);CHKERRQ(ierr);
32   if (tao->bounded) {
33     ierr = TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);CHKERRQ(ierr);
34   }
35 
36   /* Project the initial point onto the feasible region */
37   ierr = TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);CHKERRQ(ierr);
38 
39   /* Check convergence criteria */
40   ierr = TaoComputeObjectiveAndGradient(tao, tao->solution, &bnk->f, bnk->unprojected_gradient);CHKERRQ(ierr);
41   ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
42   ierr = VecCopy(bnk->unprojected_gradient, tao->gradient);CHKERRQ(ierr);
43   ierr = VecISSet(tao->gradient, bnk->active_idx, 0.0);CHKERRQ(ierr);
44   ierr = VecNorm(tao->gradient,NORM_2,&bnk->gnorm);CHKERRQ(ierr);
45 
46   /* Test the initial point for convergence */
47   ierr = VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);CHKERRQ(ierr);
48   ierr = VecNorm(bnk->W, NORM_2, &resnorm);CHKERRQ(ierr);
49   if (PetscIsInfOrNanReal(bnk->f) || PetscIsInfOrNanReal(resnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
50   ierr = TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
51   ierr = TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);CHKERRQ(ierr);
52   ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
53   if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
54 
55   /* Reset KSP stopping reason counters */
56   bnk->ksp_atol = 0;
57   bnk->ksp_rtol = 0;
58   bnk->ksp_dtol = 0;
59   bnk->ksp_ctol = 0;
60   bnk->ksp_negc = 0;
61   bnk->ksp_iter = 0;
62   bnk->ksp_othr = 0;
63 
64   /* Reset accepted step type counters */
65   bnk->tot_cg_its = 0;
66   bnk->newt = 0;
67   bnk->bfgs = 0;
68   bnk->sgrad = 0;
69   bnk->grad = 0;
70 
71   /* Initialize the Hessian perturbation */
72   bnk->pert = bnk->sval;
73 
74   /* Reset initial steplength to zero (this helps BNCG reset its direction internally) */
75   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
76 
77   /* Allocate the vectors needed for the BFGS approximation */
78   ierr = KSPGetPC(tao->ksp, &pc);CHKERRQ(ierr);
79   ierr = PetscObjectTypeCompare((PetscObject)pc, PCLMVM, &is_bfgs);CHKERRQ(ierr);
80   ierr = PetscObjectTypeCompare((PetscObject)pc, PCJACOBI, &is_jacobi);CHKERRQ(ierr);
81   if (is_bfgs) {
82     bnk->bfgs_pre = pc;
83     ierr = PCLMVMGetMatLMVM(bnk->bfgs_pre, &bnk->M);CHKERRQ(ierr);
84     ierr = VecGetLocalSize(tao->solution, &n);CHKERRQ(ierr);
85     ierr = VecGetSize(tao->solution, &N);CHKERRQ(ierr);
86     ierr = MatSetSizes(bnk->M, n, n, N, N);CHKERRQ(ierr);
87     ierr = MatLMVMAllocate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
88     ierr = MatIsSymmetricKnown(bnk->M, &sym_set, &is_symmetric);CHKERRQ(ierr);
89     if (!sym_set || !is_symmetric) SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_INCOMP, "LMVM matrix in the LMVM preconditioner must be symmetric.");
90   } else if (is_jacobi) {
91     ierr = PCJacobiSetUseAbs(pc,PETSC_TRUE);CHKERRQ(ierr);
92   }
93 
94   /* Prepare the min/max vectors for safeguarding diagonal scales */
95   ierr = VecSet(bnk->Diag_min, bnk->dmin);CHKERRQ(ierr);
96   ierr = VecSet(bnk->Diag_max, bnk->dmax);CHKERRQ(ierr);
97 
98   /* Initialize trust-region radius.  The initialization is only performed
99      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
100   *needH = PETSC_TRUE;
101   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
102     switch(initType) {
103     case BNK_INIT_CONSTANT:
104       /* Use the initial radius specified */
105       tao->trust = tao->trust0;
106       break;
107 
108     case BNK_INIT_INTERPOLATION:
109       /* Use interpolation based on the initial Hessian */
110       max_radius = 0.0;
111       tao->trust = tao->trust0;
112       for (j = 0; j < j_max; ++j) {
113         f_min = bnk->f;
114         sigma = 0.0;
115 
116         if (*needH) {
117           /* Compute the Hessian at the new step, and extract the inactive subsystem */
118           ierr = TaoBNKComputeHessian(tao);CHKERRQ(ierr);
119           ierr = TaoBNKEstimateActiveSet(tao, BNK_AS_NONE);CHKERRQ(ierr);
120           ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
121           if (bnk->active_idx) {
122             ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
123           } else {
124             ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);CHKERRQ(ierr);
125           }
126           *needH = PETSC_FALSE;
127         }
128 
129         for (i = 0; i < i_max; ++i) {
130           /* Take a steepest descent step and snap it to bounds */
131           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
132           ierr = VecAXPY(tao->solution, -tao->trust/bnk->gnorm, tao->gradient);CHKERRQ(ierr);
133           ierr = TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);CHKERRQ(ierr);
134           /* Compute the step we actually accepted */
135           ierr = VecCopy(tao->solution, bnk->W);CHKERRQ(ierr);
136           ierr = VecAXPY(bnk->W, -1.0, bnk->Xold);CHKERRQ(ierr);
137           /* Compute the objective at the trial */
138           ierr = TaoComputeObjective(tao, tao->solution, &ftrial);CHKERRQ(ierr);
139           if (PetscIsInfOrNanReal(bnk->f)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
140           ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
141           if (PetscIsInfOrNanReal(ftrial)) {
142             tau = bnk->gamma1_i;
143           } else {
144             if (ftrial < f_min) {
145               f_min = ftrial;
146               sigma = -tao->trust / bnk->gnorm;
147             }
148 
149             /* Compute the predicted and actual reduction */
150             if (bnk->active_idx) {
151               ierr = VecGetSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
152               ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
153             } else {
154               bnk->X_inactive = bnk->W;
155               bnk->inactive_work = bnk->Xwork;
156             }
157             ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
158             ierr = VecDot(bnk->X_inactive, bnk->inactive_work, &prered);CHKERRQ(ierr);
159             if (bnk->active_idx) {
160               ierr = VecRestoreSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
161               ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
162             }
163             prered = tao->trust * (bnk->gnorm - 0.5 * tao->trust * prered / (bnk->gnorm * bnk->gnorm));
164             actred = bnk->f - ftrial;
165             if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
166               kappa = 1.0;
167             } else {
168               kappa = actred / prered;
169             }
170 
171             tau_1 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust + (1.0 - bnk->theta_i) * prered - actred);
172             tau_2 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust - (1.0 + bnk->theta_i) * prered + actred);
173             tau_min = PetscMin(tau_1, tau_2);
174             tau_max = PetscMax(tau_1, tau_2);
175 
176             if (PetscAbsScalar(kappa - 1.0) <= bnk->mu1_i) {
177               /*  Great agreement */
178               max_radius = PetscMax(max_radius, tao->trust);
179 
180               if (tau_max < 1.0) {
181                 tau = bnk->gamma3_i;
182               } else if (tau_max > bnk->gamma4_i) {
183                 tau = bnk->gamma4_i;
184               } else {
185                 tau = tau_max;
186               }
187             } else if (PetscAbsScalar(kappa - 1.0) <= bnk->mu2_i) {
188               /*  Good agreement */
189               max_radius = PetscMax(max_radius, tao->trust);
190 
191               if (tau_max < bnk->gamma2_i) {
192                 tau = bnk->gamma2_i;
193               } else if (tau_max > bnk->gamma3_i) {
194                 tau = bnk->gamma3_i;
195               } else {
196                 tau = tau_max;
197               }
198             } else {
199               /*  Not good agreement */
200               if (tau_min > 1.0) {
201                 tau = bnk->gamma2_i;
202               } else if (tau_max < bnk->gamma1_i) {
203                 tau = bnk->gamma1_i;
204               } else if ((tau_min < bnk->gamma1_i) && (tau_max >= 1.0)) {
205                 tau = bnk->gamma1_i;
206               } else if ((tau_1 >= bnk->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
207                 tau = tau_1;
208               } else if ((tau_2 >= bnk->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
209                 tau = tau_2;
210               } else {
211                 tau = tau_max;
212               }
213             }
214           }
215           tao->trust = tau * tao->trust;
216         }
217 
218         if (f_min < bnk->f) {
219           /* We accidentally found a solution better than the initial, so accept it */
220           bnk->f = f_min;
221           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
222           ierr = VecAXPY(tao->solution,sigma,tao->gradient);CHKERRQ(ierr);
223           ierr = TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);CHKERRQ(ierr);
224           ierr = VecCopy(tao->solution, tao->stepdirection);CHKERRQ(ierr);
225           ierr = VecAXPY(tao->stepdirection, -1.0, bnk->Xold);CHKERRQ(ierr);
226           ierr = TaoComputeGradient(tao,tao->solution,bnk->unprojected_gradient);CHKERRQ(ierr);
227           ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
228           ierr = VecCopy(bnk->unprojected_gradient, tao->gradient);CHKERRQ(ierr);
229           ierr = VecISSet(tao->gradient, bnk->active_idx, 0.0);CHKERRQ(ierr);
230           /* Compute gradient at the new iterate and flip switch to compute the Hessian later */
231           ierr = VecNorm(tao->gradient, NORM_2, &bnk->gnorm);CHKERRQ(ierr);
232           *needH = PETSC_TRUE;
233           /* Test the new step for convergence */
234           ierr = VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);CHKERRQ(ierr);
235           ierr = VecNorm(bnk->W, NORM_2, &resnorm);CHKERRQ(ierr);
236           if (PetscIsInfOrNanReal(resnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
237           ierr = TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
238           ierr = TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);CHKERRQ(ierr);
239           ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
240           if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
241           /* active BNCG recycling early because we have a stepdirection computed */
242           ierr = TaoBNCGSetRecycleFlag(bnk->bncg, PETSC_TRUE);CHKERRQ(ierr);
243         }
244       }
245       tao->trust = PetscMax(tao->trust, max_radius);
246 
247       /* Ensure that the trust radius is within the limits */
248       tao->trust = PetscMax(tao->trust, bnk->min_radius);
249       tao->trust = PetscMin(tao->trust, bnk->max_radius);
250       break;
251 
252     default:
253       /* Norm of the first direction will initialize radius */
254       tao->trust = 0.0;
255       break;
256     }
257   }
258   PetscFunctionReturn(0);
259 }
260 
261 /*------------------------------------------------------------*/
262 
263 /* Routine for computing the Hessian and preparing the preconditioner at the new iterate */
264 
265 PetscErrorCode TaoBNKComputeHessian(Tao tao)
266 {
267   PetscErrorCode               ierr;
268   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
269 
270   PetscFunctionBegin;
271   /* Compute the Hessian */
272   ierr = TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);CHKERRQ(ierr);
273   /* Add a correction to the BFGS preconditioner */
274   if (bnk->M) {
275     ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
276   }
277   PetscFunctionReturn(0);
278 }
279 
280 /*------------------------------------------------------------*/
281 
282 /* Routine for estimating the active set */
283 
284 PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
285 {
286   PetscErrorCode               ierr;
287   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
288   PetscBool                    hessComputed, diagExists;
289 
290   PetscFunctionBegin;
291   switch (asType) {
292   case BNK_AS_NONE:
293     ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
294     ierr = VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);CHKERRQ(ierr);
295     ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
296     ierr = ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);CHKERRQ(ierr);
297     break;
298 
299   case BNK_AS_BERTSEKAS:
300     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
301     if (bnk->M) {
302       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
303       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
304     } else {
305       ierr = MatAssembled(tao->hessian, &hessComputed);CHKERRQ(ierr);
306       ierr = MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
307       if (hessComputed && diagExists) {
308         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
309         ierr = MatGetDiagonal(tao->hessian, bnk->Xwork);CHKERRQ(ierr);
310         ierr = VecAbs(bnk->Xwork);CHKERRQ(ierr);
311         ierr = VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);CHKERRQ(ierr);
312         ierr = VecReciprocal(bnk->Xwork);CHKERRQ(ierr);CHKERRQ(ierr);
313         ierr = VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);CHKERRQ(ierr);
314       } else {
315         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
316         ierr = VecCopy(bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
317       }
318     }
319     ierr = VecScale(bnk->W, -1.0);CHKERRQ(ierr);
320     ierr = TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol,
321                                    &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);CHKERRQ(ierr);
322     break;
323 
324   default:
325     break;
326   }
327   PetscFunctionReturn(0);
328 }
329 
330 /*------------------------------------------------------------*/
331 
332 /* Routine for bounding the step direction */
333 
334 PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
335 {
336   PetscErrorCode               ierr;
337   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
338 
339   PetscFunctionBegin;
340   switch (asType) {
341   case BNK_AS_NONE:
342     ierr = VecISSet(step, bnk->active_idx, 0.0);CHKERRQ(ierr);
343     break;
344 
345   case BNK_AS_BERTSEKAS:
346     ierr = TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step);CHKERRQ(ierr);
347     break;
348 
349   default:
350     break;
351   }
352   PetscFunctionReturn(0);
353 }
354 
355 /*------------------------------------------------------------*/
356 
357 /* Routine for taking a finite number of BNCG iterations to
358    accelerate Newton convergence.
359 
360    In practice, this approach simply trades off Hessian evaluations
361    for more gradient evaluations.
362 */
363 
364 PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
365 {
366   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
367   PetscErrorCode               ierr;
368 
369   PetscFunctionBegin;
370   *terminate = PETSC_FALSE;
371   if (bnk->max_cg_its > 0) {
372     /* Copy the current function value (important vectors are already shared) */
373     bnk->bncg_ctx->f = bnk->f;
374     /* Take some small finite number of BNCG iterations */
375     ierr = TaoSolve(bnk->bncg);CHKERRQ(ierr);
376     /* Add the number of gradient and function evaluations to the total */
377     tao->nfuncs += bnk->bncg->nfuncs;
378     tao->nfuncgrads += bnk->bncg->nfuncgrads;
379     tao->ngrads += bnk->bncg->ngrads;
380     tao->nhess += bnk->bncg->nhess;
381     bnk->tot_cg_its += bnk->bncg->niter;
382     /* Extract the BNCG function value out and save it into BNK */
383     bnk->f = bnk->bncg_ctx->f;
384     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) {
385       *terminate = PETSC_TRUE;
386     } else {
387       ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
388     }
389   }
390   PetscFunctionReturn(0);
391 }
392 
393 /*------------------------------------------------------------*/
394 
395 /* Routine for computing the Newton step. */
396 
397 PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason)
398 {
399   PetscErrorCode               ierr;
400   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
401   PetscInt                     bfgsUpdates = 0;
402   PetscInt                     kspits;
403 
404   PetscFunctionBegin;
405   /* If there are no inactive variables left, save some computation and return an adjusted zero step
406      that has (l-x) and (u-x) for lower and upper bounded variables. */
407   if (!bnk->inactive_idx) {
408     ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
409     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
410     PetscFunctionReturn(0);
411   }
412 
413   /* Prepare the reduced sub-matrices for the inactive set */
414   if (bnk->active_idx) {
415     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
416     ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
417     if (tao->hessian == tao->hessian_pre) {
418       bnk->Hpre_inactive = bnk->H_inactive;
419     } else {
420       ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
421       ierr = MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive);CHKERRQ(ierr);
422     }
423     if (bnk->bfgs_pre) {
424       ierr = PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx);CHKERRQ(ierr);
425     }
426   } else {
427     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
428     ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);CHKERRQ(ierr);
429     if (tao->hessian == tao->hessian_pre) {
430       bnk->Hpre_inactive = bnk->H_inactive;
431     } else {
432       ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
433       ierr = MatDuplicate(tao->hessian_pre, MAT_COPY_VALUES, &bnk->Hpre_inactive);CHKERRQ(ierr);
434     }
435     if (bnk->bfgs_pre) {
436       ierr = PCLMVMClearIS(bnk->bfgs_pre);CHKERRQ(ierr);
437     }
438   }
439 
440   /* Shift the reduced Hessian matrix */
441   if ((shift) && (bnk->pert > 0)) {
442     ierr = MatShift(bnk->H_inactive, bnk->pert);CHKERRQ(ierr);
443     if (bnk->H_inactive != bnk->Hpre_inactive) {
444       ierr = MatShift(bnk->Hpre_inactive, bnk->pert);CHKERRQ(ierr);
445     }
446   }
447 
448   /* Solve the Newton system of equations */
449   tao->ksp_its = 0;
450   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
451   ierr = KSPReset(tao->ksp);CHKERRQ(ierr);
452   ierr = KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);CHKERRQ(ierr);
453   ierr = VecCopy(bnk->unprojected_gradient, bnk->Gwork);CHKERRQ(ierr);
454   if (bnk->active_idx) {
455     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
456     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
457   } else {
458     bnk->G_inactive = bnk->unprojected_gradient;
459     bnk->X_inactive = tao->stepdirection;
460   }
461   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
462     ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
463     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
464     ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
465     tao->ksp_its+=kspits;
466     tao->ksp_tot_its+=kspits;
467     ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
468 
469     if (0.0 == tao->trust) {
470       /* Radius was uninitialized; use the norm of the direction */
471       if (bnk->dnorm > 0.0) {
472         tao->trust = bnk->dnorm;
473 
474         /* Modify the radius if it is too large or small */
475         tao->trust = PetscMax(tao->trust, bnk->min_radius);
476         tao->trust = PetscMin(tao->trust, bnk->max_radius);
477       } else {
478         /* The direction was bad; set radius to default value and re-solve
479            the trust-region subproblem to get a direction */
480         tao->trust = tao->trust0;
481 
482         /* Modify the radius if it is too large or small */
483         tao->trust = PetscMax(tao->trust, bnk->min_radius);
484         tao->trust = PetscMin(tao->trust, bnk->max_radius);
485 
486         ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
487         ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
488         ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
489         tao->ksp_its+=kspits;
490         tao->ksp_tot_its+=kspits;
491         ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
492 
493         if (bnk->dnorm == 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial direction zero");
494       }
495     }
496   } else {
497     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
498     ierr = KSPGetIterationNumber(tao->ksp, &kspits);CHKERRQ(ierr);
499     tao->ksp_its += kspits;
500     tao->ksp_tot_its+=kspits;
501   }
502   /* Restore sub vectors back */
503   if (bnk->active_idx) {
504     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
505     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
506   }
507   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
508   ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
509   ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
510 
511   /* Record convergence reasons */
512   ierr = KSPGetConvergedReason(tao->ksp, ksp_reason);CHKERRQ(ierr);
513   if (KSP_CONVERGED_ATOL == *ksp_reason) {
514     ++bnk->ksp_atol;
515   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
516     ++bnk->ksp_rtol;
517   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
518     ++bnk->ksp_ctol;
519   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
520     ++bnk->ksp_negc;
521   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
522     ++bnk->ksp_dtol;
523   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
524     ++bnk->ksp_iter;
525   } else {
526     ++bnk->ksp_othr;
527   }
528 
529   /* Make sure the BFGS preconditioner is healthy */
530   if (bnk->M) {
531     ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
532     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
533       /* Preconditioner is numerically indefinite; reset the approximation. */
534       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
535       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
536     }
537   }
538   PetscFunctionReturn(0);
539 }
540 
541 /*------------------------------------------------------------*/
542 
543 /* Routine for recomputing the predicted reduction for a given step vector */
544 
545 PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
546 {
547   PetscErrorCode               ierr;
548   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
549 
550   PetscFunctionBegin;
551   /* Extract subvectors associated with the inactive set */
552   if (bnk->active_idx){
553     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
554     ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
555     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
556   } else {
557     bnk->X_inactive = tao->stepdirection;
558     bnk->inactive_work = bnk->Xwork;
559     bnk->G_inactive = bnk->Gwork;
560   }
561   /* Recompute the predicted decrease based on the quadratic model */
562   ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
563   ierr = VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive);CHKERRQ(ierr);
564   ierr = VecDot(bnk->inactive_work, bnk->X_inactive, prered);CHKERRQ(ierr);
565   /* Restore the sub vectors */
566   if (bnk->active_idx){
567     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
568     ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
569     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
570   }
571   PetscFunctionReturn(0);
572 }
573 
574 /*------------------------------------------------------------*/
575 
576 /* Routine for ensuring that the Newton step is a descent direction.
577 
578    The step direction falls back onto BFGS, scaled gradient and gradient steps
579    in the event that the Newton step fails the test.
580 */
581 
582 PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
583 {
584   PetscErrorCode               ierr;
585   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
586 
587   PetscReal                    gdx, e_min;
588   PetscInt                     bfgsUpdates;
589 
590   PetscFunctionBegin;
591   ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
592   if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
593     /* Newton step is not descent or direction produced Inf or NaN
594        Update the perturbation for next time */
595     if (bnk->pert <= 0.0) {
596       /* Initialize the perturbation */
597       bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
598       if (bnk->is_gltr) {
599         ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
600         bnk->pert = PetscMax(bnk->pert, -e_min);
601       }
602     } else {
603       /* Increase the perturbation */
604       bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
605     }
606 
607     if (!bnk->M) {
608       /* We don't have the bfgs matrix around and updated
609          Must use gradient direction in this case */
610       ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
611       *stepType = BNK_GRADIENT;
612     } else {
613       /* Attempt to use the BFGS direction */
614       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
615 
616       /* Check for success (descent direction)
617          NOTE: Negative gdx here means not a descent direction because
618          the fall-back step is missing a negative sign. */
619       ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
620       if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
621         /* BFGS direction is not descent or direction produced not a number
622            We can assert bfgsUpdates > 1 in this case because
623            the first solve produces the scaled gradient direction,
624            which is guaranteed to be descent */
625 
626         /* Use steepest descent direction (scaled) */
627         ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
628         ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
629         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
630 
631         *stepType = BNK_SCALED_GRADIENT;
632       } else {
633         ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
634         if (1 == bfgsUpdates) {
635           /* The first BFGS direction is always the scaled gradient */
636           *stepType = BNK_SCALED_GRADIENT;
637         } else {
638           *stepType = BNK_BFGS;
639         }
640       }
641     }
642     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
643     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
644     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
645   } else {
646     /* Computed Newton step is descent */
647     switch (ksp_reason) {
648     case KSP_DIVERGED_NANORINF:
649     case KSP_DIVERGED_BREAKDOWN:
650     case KSP_DIVERGED_INDEFINITE_MAT:
651     case KSP_DIVERGED_INDEFINITE_PC:
652     case KSP_CONVERGED_CG_NEG_CURVE:
653       /* Matrix or preconditioner is indefinite; increase perturbation */
654       if (bnk->pert <= 0.0) {
655         /* Initialize the perturbation */
656         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
657         if (bnk->is_gltr) {
658           ierr = KSPCGGLTRGetMinEig(tao->ksp, &e_min);CHKERRQ(ierr);
659           bnk->pert = PetscMax(bnk->pert, -e_min);
660         }
661       } else {
662         /* Increase the perturbation */
663         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
664       }
665       break;
666 
667     default:
668       /* Newton step computation is good; decrease perturbation */
669       bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
670       if (bnk->pert < bnk->pmin) {
671         bnk->pert = 0.0;
672       }
673       break;
674     }
675     *stepType = BNK_NEWTON;
676   }
677   PetscFunctionReturn(0);
678 }
679 
680 /*------------------------------------------------------------*/
681 
682 /* Routine for performing a bound-projected More-Thuente line search.
683 
684   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
685   Newton step does not produce a valid step length.
686 */
687 
688 PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
689 {
690   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
691   PetscErrorCode ierr;
692   TaoLineSearchConvergedReason ls_reason;
693 
694   PetscReal      e_min, gdx;
695   PetscInt       bfgsUpdates;
696 
697   PetscFunctionBegin;
698   /* Perform the linesearch */
699   ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
700   ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
701 
702   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_SCALED_GRADIENT && *stepType != BNK_GRADIENT) {
703     /* Linesearch failed, revert solution */
704     bnk->f = bnk->fold;
705     ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
706     ierr = VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);CHKERRQ(ierr);
707 
708     switch(*stepType) {
709     case BNK_NEWTON:
710       /* Failed to obtain acceptable iterate with Newton step
711          Update the perturbation for next time */
712       if (bnk->pert <= 0.0) {
713         /* Initialize the perturbation */
714         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
715         if (bnk->is_gltr) {
716           ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
717           bnk->pert = PetscMax(bnk->pert, -e_min);
718         }
719       } else {
720         /* Increase the perturbation */
721         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
722       }
723 
724       if (!bnk->M) {
725         /* We don't have the bfgs matrix around and being updated
726            Must use gradient direction in this case */
727         ierr = VecCopy(bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
728         *stepType = BNK_GRADIENT;
729       } else {
730         /* Attempt to use the BFGS direction */
731         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
732         /* Check for success (descent direction)
733            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
734         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
735         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
736           /* BFGS direction is not descent or direction produced not a number
737              We can assert bfgsUpdates > 1 in this case
738              Use steepest descent direction (scaled) */
739           ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
740           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
741           ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
742 
743           bfgsUpdates = 1;
744           *stepType = BNK_SCALED_GRADIENT;
745         } else {
746           ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
747           if (1 == bfgsUpdates) {
748             /* The first BFGS direction is always the scaled gradient */
749             *stepType = BNK_SCALED_GRADIENT;
750           } else {
751             *stepType = BNK_BFGS;
752           }
753         }
754       }
755       break;
756 
757     case BNK_BFGS:
758       /* Can only enter if pc_type == BNK_PC_BFGS
759          Failed to obtain acceptable iterate with BFGS step
760          Attempt to use the scaled gradient direction */
761       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
762       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
763       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
764 
765       bfgsUpdates = 1;
766       *stepType = BNK_SCALED_GRADIENT;
767       break;
768     }
769     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
770     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
771     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
772 
773     /* Perform one last line search with the fall-back step */
774     ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
775     ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
776   }
777   *reason = ls_reason;
778   PetscFunctionReturn(0);
779 }
780 
781 /*------------------------------------------------------------*/
782 
783 /* Routine for updating the trust radius.
784 
785   Function features three different update methods:
786   1) Line-search step length based
787   2) Predicted decrease on the CG quadratic model
788   3) Interpolation
789 */
790 
791 PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
792 {
793   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
794   PetscErrorCode ierr;
795 
796   PetscReal      step, kappa;
797   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;
798 
799   PetscFunctionBegin;
800   /* Update trust region radius */
801   *accept = PETSC_FALSE;
802   switch(updateType) {
803   case BNK_UPDATE_STEP:
804     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
805     if (stepType == BNK_NEWTON) {
806       ierr = TaoLineSearchGetStepLength(tao->linesearch, &step);CHKERRQ(ierr);
807       if (step < bnk->nu1) {
808         /* Very bad step taken; reduce radius */
809         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
810       } else if (step < bnk->nu2) {
811         /* Reasonably bad step taken; reduce radius */
812         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
813       } else if (step < bnk->nu3) {
814         /*  Reasonable step was taken; leave radius alone */
815         if (bnk->omega3 < 1.0) {
816           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
817         } else if (bnk->omega3 > 1.0) {
818           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
819         }
820       } else if (step < bnk->nu4) {
821         /*  Full step taken; increase the radius */
822         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
823       } else {
824         /*  More than full step taken; increase the radius */
825         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
826       }
827     } else {
828       /*  Newton step was not good; reduce the radius */
829       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
830     }
831     break;
832 
833   case BNK_UPDATE_REDUCTION:
834     if (stepType == BNK_NEWTON) {
835       if (prered < 0.0) {
836         /* The predicted reduction has the wrong sign.  This cannot
837            happen in infinite precision arithmetic.  Step should
838            be rejected! */
839         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
840       } else {
841         if (PetscIsInfOrNanReal(actred)) {
842           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
843         } else {
844           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
845             kappa = 1.0;
846           } else {
847             kappa = actred / prered;
848           }
849 
850           /* Accept or reject the step and update radius */
851           if (kappa < bnk->eta1) {
852             /* Reject the step */
853             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
854           } else {
855             /* Accept the step */
856             *accept = PETSC_TRUE;
857             /* Update the trust region radius only if the computed step is at the trust radius boundary */
858             if (bnk->dnorm == tao->trust) {
859               if (kappa < bnk->eta2) {
860                 /* Marginal bad step */
861                 tao->trust = bnk->alpha2 * tao->trust;
862               } else if (kappa < bnk->eta3) {
863                 /* Reasonable step */
864                 tao->trust = bnk->alpha3 * tao->trust;
865               } else if (kappa < bnk->eta4) {
866                 /* Good step */
867                 tao->trust = bnk->alpha4 * tao->trust;
868               } else {
869                 /* Very good step */
870                 tao->trust = bnk->alpha5 * tao->trust;
871               }
872             }
873           }
874         }
875       }
876     } else {
877       /*  Newton step was not good; reduce the radius */
878       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
879     }
880     break;
881 
882   default:
883     if (stepType == BNK_NEWTON) {
884       if (prered < 0.0) {
885         /*  The predicted reduction has the wrong sign.  This cannot */
886         /*  happen in infinite precision arithmetic.  Step should */
887         /*  be rejected! */
888         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
889       } else {
890         if (PetscIsInfOrNanReal(actred)) {
891           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
892         } else {
893           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
894             kappa = 1.0;
895           } else {
896             kappa = actred / prered;
897           }
898 
899           ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
900           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
901           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
902           tau_min = PetscMin(tau_1, tau_2);
903           tau_max = PetscMax(tau_1, tau_2);
904 
905           if (kappa >= 1.0 - bnk->mu1) {
906             /*  Great agreement */
907             *accept = PETSC_TRUE;
908             if (tau_max < 1.0) {
909               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
910             } else if (tau_max > bnk->gamma4) {
911               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
912             } else {
913               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
914             }
915           } else if (kappa >= 1.0 - bnk->mu2) {
916             /*  Good agreement */
917             *accept = PETSC_TRUE;
918             if (tau_max < bnk->gamma2) {
919               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
920             } else if (tau_max > bnk->gamma3) {
921               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
922             } else if (tau_max < 1.0) {
923               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
924             } else {
925               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
926             }
927           } else {
928             /*  Not good agreement */
929             if (tau_min > 1.0) {
930               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
931             } else if (tau_max < bnk->gamma1) {
932               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
933             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
934               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
935             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
936               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
937             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
938               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
939             } else {
940               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
941             }
942           }
943         }
944       }
945     } else {
946       /*  Newton step was not good; reduce the radius */
947       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
948     }
949     break;
950   }
951   /* Make sure the radius does not violate min and max settings */
952   tao->trust = PetscMin(tao->trust, bnk->max_radius);
953   tao->trust = PetscMax(tao->trust, bnk->min_radius);
954   PetscFunctionReturn(0);
955 }
956 
957 /* ---------------------------------------------------------- */
958 
959 PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
960 {
961   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
962 
963   PetscFunctionBegin;
964   switch (stepType) {
965   case BNK_NEWTON:
966     ++bnk->newt;
967     break;
968   case BNK_BFGS:
969     ++bnk->bfgs;
970     break;
971   case BNK_SCALED_GRADIENT:
972     ++bnk->sgrad;
973     break;
974   case BNK_GRADIENT:
975     ++bnk->grad;
976     break;
977   default:
978     break;
979   }
980   PetscFunctionReturn(0);
981 }
982 
983 /* ---------------------------------------------------------- */
984 
985 PetscErrorCode TaoSetUp_BNK(Tao tao)
986 {
987   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
988   PetscErrorCode ierr;
989   KSPType        ksp_type;
990   PetscInt       i;
991 
992   PetscFunctionBegin;
993   if (!tao->gradient) {
994     ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);
995   }
996   if (!tao->stepdirection) {
997     ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);
998   }
999   if (!bnk->W) {
1000     ierr = VecDuplicate(tao->solution,&bnk->W);CHKERRQ(ierr);
1001   }
1002   if (!bnk->Xold) {
1003     ierr = VecDuplicate(tao->solution,&bnk->Xold);CHKERRQ(ierr);
1004   }
1005   if (!bnk->Gold) {
1006     ierr = VecDuplicate(tao->solution,&bnk->Gold);CHKERRQ(ierr);
1007   }
1008   if (!bnk->Xwork) {
1009     ierr = VecDuplicate(tao->solution,&bnk->Xwork);CHKERRQ(ierr);
1010   }
1011   if (!bnk->Gwork) {
1012     ierr = VecDuplicate(tao->solution,&bnk->Gwork);CHKERRQ(ierr);
1013   }
1014   if (!bnk->unprojected_gradient) {
1015     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient);CHKERRQ(ierr);
1016   }
1017   if (!bnk->unprojected_gradient_old) {
1018     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1019   }
1020   if (!bnk->Diag_min) {
1021     ierr = VecDuplicate(tao->solution,&bnk->Diag_min);CHKERRQ(ierr);
1022   }
1023   if (!bnk->Diag_max) {
1024     ierr = VecDuplicate(tao->solution,&bnk->Diag_max);CHKERRQ(ierr);
1025   }
1026   if (bnk->max_cg_its > 0) {
1027     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
1028     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
1029     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old));CHKERRQ(ierr);
1030     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old);CHKERRQ(ierr);
1031     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
1032     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient));CHKERRQ(ierr);
1033     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient);CHKERRQ(ierr);
1034     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
1035     ierr = PetscObjectReference((PetscObject)(bnk->Gold));CHKERRQ(ierr);
1036     ierr = VecDestroy(&bnk->bncg_ctx->G_old);CHKERRQ(ierr);
1037     bnk->bncg_ctx->G_old = bnk->Gold;
1038     ierr = PetscObjectReference((PetscObject)(tao->gradient));CHKERRQ(ierr);
1039     ierr = VecDestroy(&bnk->bncg->gradient);CHKERRQ(ierr);
1040     bnk->bncg->gradient = tao->gradient;
1041     ierr = PetscObjectReference((PetscObject)(tao->stepdirection));CHKERRQ(ierr);
1042     ierr = VecDestroy(&bnk->bncg->stepdirection);CHKERRQ(ierr);
1043     bnk->bncg->stepdirection = tao->stepdirection;
1044     ierr = TaoSetInitialVector(bnk->bncg, tao->solution);CHKERRQ(ierr);
1045     /* Copy over some settings from BNK into BNCG */
1046     ierr = TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its);CHKERRQ(ierr);
1047     ierr = TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol);CHKERRQ(ierr);
1048     ierr = TaoSetFunctionLowerBound(bnk->bncg, tao->fmin);CHKERRQ(ierr);
1049     ierr = TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP);CHKERRQ(ierr);
1050     ierr = TaoSetObjectiveRoutine(bnk->bncg, tao->ops->computeobjective, tao->user_objP);CHKERRQ(ierr);
1051     ierr = TaoSetGradientRoutine(bnk->bncg, tao->ops->computegradient, tao->user_gradP);CHKERRQ(ierr);
1052     ierr = TaoSetObjectiveAndGradientRoutine(bnk->bncg, tao->ops->computeobjectiveandgradient, tao->user_objgradP);CHKERRQ(ierr);
1053     ierr = PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg));CHKERRQ(ierr);
1054     for (i=0; i<tao->numbermonitors; ++i) {
1055       ierr = TaoSetMonitor(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);CHKERRQ(ierr);
1056       ierr = PetscObjectReference((PetscObject)(tao->monitorcontext[i]));CHKERRQ(ierr);
1057     }
1058   }
1059   bnk->X_inactive = 0;
1060   bnk->G_inactive = 0;
1061   bnk->inactive_work = 0;
1062   bnk->active_work = 0;
1063   bnk->inactive_idx = 0;
1064   bnk->active_idx = 0;
1065   bnk->active_lower = 0;
1066   bnk->active_upper = 0;
1067   bnk->active_fixed = 0;
1068   bnk->M = 0;
1069   bnk->H_inactive = 0;
1070   bnk->Hpre_inactive = 0;
1071   ierr = KSPGetType(tao->ksp,&ksp_type);CHKERRQ(ierr);
1072   ierr = PetscStrcmp(ksp_type,KSPCGNASH,&bnk->is_nash);CHKERRQ(ierr);
1073   ierr = PetscStrcmp(ksp_type,KSPCGSTCG,&bnk->is_stcg);CHKERRQ(ierr);
1074   ierr = PetscStrcmp(ksp_type,KSPCGGLTR,&bnk->is_gltr);CHKERRQ(ierr);
1075   PetscFunctionReturn(0);
1076 }
1077 
1078 /*------------------------------------------------------------*/
1079 
1080 static PetscErrorCode TaoDestroy_BNK(Tao tao)
1081 {
1082   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1083   PetscErrorCode ierr;
1084 
1085   PetscFunctionBegin;
1086   if (tao->setupcalled) {
1087     ierr = VecDestroy(&bnk->W);CHKERRQ(ierr);
1088     ierr = VecDestroy(&bnk->Xold);CHKERRQ(ierr);
1089     ierr = VecDestroy(&bnk->Gold);CHKERRQ(ierr);
1090     ierr = VecDestroy(&bnk->Xwork);CHKERRQ(ierr);
1091     ierr = VecDestroy(&bnk->Gwork);CHKERRQ(ierr);
1092     ierr = VecDestroy(&bnk->unprojected_gradient);CHKERRQ(ierr);
1093     ierr = VecDestroy(&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1094     ierr = VecDestroy(&bnk->Diag_min);CHKERRQ(ierr);
1095     ierr = VecDestroy(&bnk->Diag_max);CHKERRQ(ierr);
1096   }
1097   ierr = ISDestroy(&bnk->active_lower);CHKERRQ(ierr);
1098   ierr = ISDestroy(&bnk->active_upper);CHKERRQ(ierr);
1099   ierr = ISDestroy(&bnk->active_fixed);CHKERRQ(ierr);
1100   ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
1101   ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
1102   if (bnk->Hpre_inactive != tao->hessian_pre && bnk->Hpre_inactive != bnk->H_inactive) {
1103     ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
1104   }
1105   if (bnk->H_inactive != tao->hessian) {
1106     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
1107   }
1108   ierr = TaoDestroy(&bnk->bncg);CHKERRQ(ierr);
1109   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1110   PetscFunctionReturn(0);
1111 }
1112 
1113 /*------------------------------------------------------------*/
1114 
1115 static PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
1116 {
1117   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1118   PetscErrorCode ierr;
1119 
1120   PetscFunctionBegin;
1121   ierr = PetscOptionsHead(PetscOptionsObject,"Newton line search method for unconstrained optimization");CHKERRQ(ierr);
1122   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);
1123   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);
1124   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);
1125   ierr = PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval,NULL);CHKERRQ(ierr);
1126   ierr = PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin,NULL);CHKERRQ(ierr);
1127   ierr = PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax,NULL);CHKERRQ(ierr);
1128   ierr = PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac,NULL);CHKERRQ(ierr);
1129   ierr = PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin,NULL);CHKERRQ(ierr);
1130   ierr = PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax,NULL);CHKERRQ(ierr);
1131   ierr = PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);CHKERRQ(ierr);
1132   ierr = PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac,NULL);CHKERRQ(ierr);
1133   ierr = PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac,NULL);CHKERRQ(ierr);
1134   ierr = PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac,NULL);CHKERRQ(ierr);
1135   ierr = PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1,NULL);CHKERRQ(ierr);
1136   ierr = PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2,NULL);CHKERRQ(ierr);
1137   ierr = PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3,NULL);CHKERRQ(ierr);
1138   ierr = PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4,NULL);CHKERRQ(ierr);
1139   ierr = PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1,NULL);CHKERRQ(ierr);
1140   ierr = PetscOptionsReal("-tao_bnk_alpha2", "(developer) radius reduction factor for marginally accepted bad step (-tao_bnk_update_type reduction)", "", bnk->alpha2, &bnk->alpha2,NULL);CHKERRQ(ierr);
1141   ierr = PetscOptionsReal("-tao_bnk_alpha3", "(developer) radius increase factor for reasonable accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha3, &bnk->alpha3,NULL);CHKERRQ(ierr);
1142   ierr = PetscOptionsReal("-tao_bnk_alpha4", "(developer) radius increase factor for good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha4, &bnk->alpha4,NULL);CHKERRQ(ierr);
1143   ierr = PetscOptionsReal("-tao_bnk_alpha5", "(developer) radius increase factor for very good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha5, &bnk->alpha5,NULL);CHKERRQ(ierr);
1144   ierr = PetscOptionsReal("-tao_bnk_nu1", "(developer) threshold for small line-search step length (-tao_bnk_update_type step)", "", bnk->nu1, &bnk->nu1,NULL);CHKERRQ(ierr);
1145   ierr = PetscOptionsReal("-tao_bnk_nu2", "(developer) threshold for reasonable line-search step length (-tao_bnk_update_type step)", "", bnk->nu2, &bnk->nu2,NULL);CHKERRQ(ierr);
1146   ierr = PetscOptionsReal("-tao_bnk_nu3", "(developer) threshold for large line-search step length (-tao_bnk_update_type step)", "", bnk->nu3, &bnk->nu3,NULL);CHKERRQ(ierr);
1147   ierr = PetscOptionsReal("-tao_bnk_nu4", "(developer) threshold for very large line-search step length (-tao_bnk_update_type step)", "", bnk->nu4, &bnk->nu4,NULL);CHKERRQ(ierr);
1148   ierr = PetscOptionsReal("-tao_bnk_omega1", "(developer) radius reduction factor for very small line-search step length (-tao_bnk_update_type step)", "", bnk->omega1, &bnk->omega1,NULL);CHKERRQ(ierr);
1149   ierr = PetscOptionsReal("-tao_bnk_omega2", "(developer) radius reduction factor for small line-search step length (-tao_bnk_update_type step)", "", bnk->omega2, &bnk->omega2,NULL);CHKERRQ(ierr);
1150   ierr = PetscOptionsReal("-tao_bnk_omega3", "(developer) radius factor for decent line-search step length (-tao_bnk_update_type step)", "", bnk->omega3, &bnk->omega3,NULL);CHKERRQ(ierr);
1151   ierr = PetscOptionsReal("-tao_bnk_omega4", "(developer) radius increase factor for large line-search step length (-tao_bnk_update_type step)", "", bnk->omega4, &bnk->omega4,NULL);CHKERRQ(ierr);
1152   ierr = PetscOptionsReal("-tao_bnk_omega5", "(developer) radius increase factor for very large line-search step length (-tao_bnk_update_type step)", "", bnk->omega5, &bnk->omega5,NULL);CHKERRQ(ierr);
1153   ierr = PetscOptionsReal("-tao_bnk_mu1_i", "(developer) threshold for accepting very good step (-tao_bnk_init_type interpolation)", "", bnk->mu1_i, &bnk->mu1_i,NULL);CHKERRQ(ierr);
1154   ierr = PetscOptionsReal("-tao_bnk_mu2_i", "(developer) threshold for accepting good step (-tao_bnk_init_type interpolation)", "", bnk->mu2_i, &bnk->mu2_i,NULL);CHKERRQ(ierr);
1155   ierr = PetscOptionsReal("-tao_bnk_gamma1_i", "(developer) radius reduction factor for rejected very bad step (-tao_bnk_init_type interpolation)", "", bnk->gamma1_i, &bnk->gamma1_i,NULL);CHKERRQ(ierr);
1156   ierr = PetscOptionsReal("-tao_bnk_gamma2_i", "(developer) radius reduction factor for rejected bad step (-tao_bnk_init_type interpolation)", "", bnk->gamma2_i, &bnk->gamma2_i,NULL);CHKERRQ(ierr);
1157   ierr = PetscOptionsReal("-tao_bnk_gamma3_i", "(developer) radius increase factor for accepted good step (-tao_bnk_init_type interpolation)", "", bnk->gamma3_i, &bnk->gamma3_i,NULL);CHKERRQ(ierr);
1158   ierr = PetscOptionsReal("-tao_bnk_gamma4_i", "(developer) radius increase factor for accepted very good step (-tao_bnk_init_type interpolation)", "", bnk->gamma4_i, &bnk->gamma4_i,NULL);CHKERRQ(ierr);
1159   ierr = PetscOptionsReal("-tao_bnk_theta_i", "(developer) trust region interpolation factor (-tao_bnk_init_type interpolation)", "", bnk->theta_i, &bnk->theta_i,NULL);CHKERRQ(ierr);
1160   ierr = PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1,NULL);CHKERRQ(ierr);
1161   ierr = PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2,NULL);CHKERRQ(ierr);
1162   ierr = PetscOptionsReal("-tao_bnk_gamma1", "(developer) radius reduction factor for rejected very bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma1, &bnk->gamma1,NULL);CHKERRQ(ierr);
1163   ierr = PetscOptionsReal("-tao_bnk_gamma2", "(developer) radius reduction factor for rejected bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma2, &bnk->gamma2,NULL);CHKERRQ(ierr);
1164   ierr = PetscOptionsReal("-tao_bnk_gamma3", "(developer) radius increase factor for accepted good step (-tao_bnk_update_type interpolation)", "", bnk->gamma3, &bnk->gamma3,NULL);CHKERRQ(ierr);
1165   ierr = PetscOptionsReal("-tao_bnk_gamma4", "(developer) radius increase factor for accepted very good step (-tao_bnk_update_type interpolation)", "", bnk->gamma4, &bnk->gamma4,NULL);CHKERRQ(ierr);
1166   ierr = PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta,NULL);CHKERRQ(ierr);
1167   ierr = PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);CHKERRQ(ierr);
1168   ierr = PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);CHKERRQ(ierr);
1169   ierr = PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);CHKERRQ(ierr);
1170   ierr = PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);CHKERRQ(ierr);
1171   ierr = PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);CHKERRQ(ierr);
1172   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);
1173   ierr = PetscOptionsTail();CHKERRQ(ierr);
1174   ierr = TaoSetFromOptions(bnk->bncg);CHKERRQ(ierr);
1175   ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr);
1176   ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr);
1177   PetscFunctionReturn(0);
1178 }
1179 
1180 /*------------------------------------------------------------*/
1181 
1182 static PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
1183 {
1184   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1185   PetscInt       nrejects;
1186   PetscBool      isascii;
1187   PetscErrorCode ierr;
1188 
1189   PetscFunctionBegin;
1190   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1191   if (isascii) {
1192     ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
1193     if (bnk->M) {
1194       ierr = MatLMVMGetRejectCount(bnk->M,&nrejects);CHKERRQ(ierr);
1195       ierr = PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %D\n",nrejects);CHKERRQ(ierr);
1196     }
1197     ierr = PetscViewerASCIIPrintf(viewer, "CG steps: %D\n", bnk->tot_cg_its);CHKERRQ(ierr);
1198     ierr = PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);CHKERRQ(ierr);
1199     if (bnk->M) {
1200       ierr = PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);CHKERRQ(ierr);
1201     }
1202     ierr = PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);CHKERRQ(ierr);
1203     ierr = PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);CHKERRQ(ierr);
1204     ierr = PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");CHKERRQ(ierr);
1205     ierr = PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);CHKERRQ(ierr);
1206     ierr = PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);CHKERRQ(ierr);
1207     ierr = PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);CHKERRQ(ierr);
1208     ierr = PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);CHKERRQ(ierr);
1209     ierr = PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);CHKERRQ(ierr);
1210     ierr = PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);CHKERRQ(ierr);
1211     ierr = PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);CHKERRQ(ierr);
1212     ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
1213   }
1214   PetscFunctionReturn(0);
1215 }
1216 
1217 /* ---------------------------------------------------------- */
1218 
1219 /*MC
1220   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
1221   At each iteration, the BNK methods solve the symmetric
1222   system of equations to obtain the step diretion dk:
1223               Hk dk = -gk
1224   for free variables only. The step can be globalized either through
1225   trust-region methods, or a line search, or a heuristic mixture of both.
1226 
1227     Options Database Keys:
1228 + -tao_bnk_max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
1229 . -tao_bnk_init_type - trust radius initialization method ("constant", "direction", "interpolation")
1230 . -tao_bnk_update_type - trust radius update method ("step", "direction", "interpolation")
1231 . -tao_bnk_as_type - active-set estimation method ("none", "bertsekas")
1232 . -tao_bnk_as_tol - (developer) initial tolerance used in estimating bounded active variables (-tao_bnk_as_type bertsekas)
1233 . -tao_bnk_as_step - (developer) trial step length used in estimating bounded active variables (-tao_bnk_as_type bertsekas)
1234 . -tao_bnk_sval - (developer) Hessian perturbation starting value
1235 . -tao_bnk_imin - (developer) minimum initial Hessian perturbation
1236 . -tao_bnk_imax - (developer) maximum initial Hessian perturbation
1237 . -tao_bnk_pmin - (developer) minimum Hessian perturbation
1238 . -tao_bnk_pmax - (developer) aximum Hessian perturbation
1239 . -tao_bnk_pgfac - (developer) Hessian perturbation growth factor
1240 . -tao_bnk_psfac - (developer) Hessian perturbation shrink factor
1241 . -tao_bnk_imfac - (developer) initial merit factor for Hessian perturbation
1242 . -tao_bnk_pmgfac - (developer) merit growth factor for Hessian perturbation
1243 . -tao_bnk_pmsfac - (developer) merit shrink factor for Hessian perturbation
1244 . -tao_bnk_eta1 - (developer) threshold for rejecting step (-tao_bnk_update_type reduction)
1245 . -tao_bnk_eta2 - (developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)
1246 . -tao_bnk_eta3 - (developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)
1247 . -tao_bnk_eta4 - (developer) threshold for accepting good step (-tao_bnk_update_type reduction)
1248 . -tao_bnk_alpha1 - (developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)
1249 . -tao_bnk_alpha2 - (developer) radius reduction factor for marginally accepted bad step (-tao_bnk_update_type reduction)
1250 . -tao_bnk_alpha3 - (developer) radius increase factor for reasonable accepted step (-tao_bnk_update_type reduction)
1251 . -tao_bnk_alpha4 - (developer) radius increase factor for good accepted step (-tao_bnk_update_type reduction)
1252 . -tao_bnk_alpha5 - (developer) radius increase factor for very good accepted step (-tao_bnk_update_type reduction)
1253 . -tao_bnk_epsilon - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-tao_bnk_update_type reduction)
1254 . -tao_bnk_mu1 - (developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)
1255 . -tao_bnk_mu2 - (developer) threshold for accepting good step (-tao_bnk_update_type interpolation)
1256 . -tao_bnk_gamma1 - (developer) radius reduction factor for rejected very bad step (-tao_bnk_update_type interpolation)
1257 . -tao_bnk_gamma2 - (developer) radius reduction factor for rejected bad step (-tao_bnk_update_type interpolation)
1258 . -tao_bnk_gamma3 - (developer) radius increase factor for accepted good step (-tao_bnk_update_type interpolation)
1259 . -tao_bnk_gamma4 - (developer) radius increase factor for accepted very good step (-tao_bnk_update_type interpolation)
1260 . -tao_bnk_theta - (developer) trust region interpolation factor (-tao_bnk_update_type interpolation)
1261 . -tao_bnk_nu1 - (developer) threshold for small line-search step length (-tao_bnk_update_type step)
1262 . -tao_bnk_nu2 - (developer) threshold for reasonable line-search step length (-tao_bnk_update_type step)
1263 . -tao_bnk_nu3 - (developer) threshold for large line-search step length (-tao_bnk_update_type step)
1264 . -tao_bnk_nu4 - (developer) threshold for very large line-search step length (-tao_bnk_update_type step)
1265 . -tao_bnk_omega1 - (developer) radius reduction factor for very small line-search step length (-tao_bnk_update_type step)
1266 . -tao_bnk_omega2 - (developer) radius reduction factor for small line-search step length (-tao_bnk_update_type step)
1267 . -tao_bnk_omega3 - (developer) radius factor for decent line-search step length (-tao_bnk_update_type step)
1268 . -tao_bnk_omega4 - (developer) radius increase factor for large line-search step length (-tao_bnk_update_type step)
1269 . -tao_bnk_omega5 - (developer) radius increase factor for very large line-search step length (-tao_bnk_update_type step)
1270 . -tao_bnk_mu1_i -  (developer) threshold for accepting very good step (-tao_bnk_init_type interpolation)
1271 . -tao_bnk_mu2_i -  (developer) threshold for accepting good step (-tao_bnk_init_type interpolation)
1272 . -tao_bnk_gamma1_i - (developer) radius reduction factor for rejected very bad step (-tao_bnk_init_type interpolation)
1273 . -tao_bnk_gamma2_i - (developer) radius reduction factor for rejected bad step (-tao_bnk_init_type interpolation)
1274 . -tao_bnk_gamma3_i - (developer) radius increase factor for accepted good step (-tao_bnk_init_type interpolation)
1275 . -tao_bnk_gamma4_i - (developer) radius increase factor for accepted very good step (-tao_bnk_init_type interpolation)
1276 - -tao_bnk_theta_i - (developer) trust region interpolation factor (-tao_bnk_init_type interpolation)
1277 
1278   Level: beginner
1279 M*/
1280 
1281 PetscErrorCode TaoCreate_BNK(Tao tao)
1282 {
1283   TAO_BNK        *bnk;
1284   const char     *morethuente_type = TAOLINESEARCHMT;
1285   PetscErrorCode ierr;
1286   PC             pc;
1287 
1288   PetscFunctionBegin;
1289   ierr = PetscNewLog(tao,&bnk);CHKERRQ(ierr);
1290 
1291   tao->ops->setup = TaoSetUp_BNK;
1292   tao->ops->view = TaoView_BNK;
1293   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
1294   tao->ops->destroy = TaoDestroy_BNK;
1295 
1296   /*  Override default settings (unless already changed) */
1297   if (!tao->max_it_changed) tao->max_it = 50;
1298   if (!tao->trust0_changed) tao->trust0 = 100.0;
1299 
1300   tao->data = (void*)bnk;
1301 
1302   /*  Hessian shifting parameters */
1303   bnk->sval   = 0.0;
1304   bnk->imin   = 1.0e-4;
1305   bnk->imax   = 1.0e+2;
1306   bnk->imfac  = 1.0e-1;
1307 
1308   bnk->pmin   = 1.0e-12;
1309   bnk->pmax   = 1.0e+2;
1310   bnk->pgfac  = 1.0e+1;
1311   bnk->psfac  = 4.0e-1;
1312   bnk->pmgfac = 1.0e-1;
1313   bnk->pmsfac = 1.0e-1;
1314 
1315   /*  Default values for trust-region radius update based on steplength */
1316   bnk->nu1 = 0.25;
1317   bnk->nu2 = 0.50;
1318   bnk->nu3 = 1.00;
1319   bnk->nu4 = 1.25;
1320 
1321   bnk->omega1 = 0.25;
1322   bnk->omega2 = 0.50;
1323   bnk->omega3 = 1.00;
1324   bnk->omega4 = 2.00;
1325   bnk->omega5 = 4.00;
1326 
1327   /*  Default values for trust-region radius update based on reduction */
1328   bnk->eta1 = 1.0e-4;
1329   bnk->eta2 = 0.25;
1330   bnk->eta3 = 0.50;
1331   bnk->eta4 = 0.90;
1332 
1333   bnk->alpha1 = 0.25;
1334   bnk->alpha2 = 0.50;
1335   bnk->alpha3 = 1.00;
1336   bnk->alpha4 = 2.00;
1337   bnk->alpha5 = 4.00;
1338 
1339   /*  Default values for trust-region radius update based on interpolation */
1340   bnk->mu1 = 0.10;
1341   bnk->mu2 = 0.50;
1342 
1343   bnk->gamma1 = 0.25;
1344   bnk->gamma2 = 0.50;
1345   bnk->gamma3 = 2.00;
1346   bnk->gamma4 = 4.00;
1347 
1348   bnk->theta = 0.05;
1349 
1350   /*  Default values for trust region initialization based on interpolation */
1351   bnk->mu1_i = 0.35;
1352   bnk->mu2_i = 0.50;
1353 
1354   bnk->gamma1_i = 0.0625;
1355   bnk->gamma2_i = 0.5;
1356   bnk->gamma3_i = 2.0;
1357   bnk->gamma4_i = 5.0;
1358 
1359   bnk->theta_i = 0.25;
1360 
1361   /*  Remaining parameters */
1362   bnk->max_cg_its = 0;
1363   bnk->min_radius = 1.0e-10;
1364   bnk->max_radius = 1.0e10;
1365   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
1366   bnk->as_tol = 1.0e-3;
1367   bnk->as_step = 1.0e-3;
1368   bnk->dmin = 1.0e-6;
1369   bnk->dmax = 1.0e6;
1370 
1371   bnk->M               = 0;
1372   bnk->bfgs_pre        = 0;
1373   bnk->init_type       = BNK_INIT_INTERPOLATION;
1374   bnk->update_type     = BNK_UPDATE_INTERPOLATION;
1375   bnk->as_type         = BNK_AS_BERTSEKAS;
1376 
1377   /* Create the embedded BNCG solver */
1378   ierr = TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg);CHKERRQ(ierr);
1379   ierr = PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1);CHKERRQ(ierr);
1380   ierr = TaoSetOptionsPrefix(bnk->bncg, "tao_bnk_");CHKERRQ(ierr);
1381   ierr = TaoSetType(bnk->bncg, TAOBNCG);CHKERRQ(ierr);
1382 
1383   /* Create the line search */
1384   ierr = TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);CHKERRQ(ierr);
1385   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr);
1386   ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr);
1387   ierr = TaoLineSearchSetType(tao->linesearch,morethuente_type);CHKERRQ(ierr);
1388   ierr = TaoLineSearchUseTaoRoutines(tao->linesearch,tao);CHKERRQ(ierr);
1389 
1390   /*  Set linear solver to default for symmetric matrices */
1391   ierr = KSPCreate(((PetscObject)tao)->comm,&tao->ksp);CHKERRQ(ierr);
1392   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr);
1393   ierr = KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);CHKERRQ(ierr);
1394   ierr = KSPSetType(tao->ksp,KSPCGSTCG);CHKERRQ(ierr);
1395   ierr = KSPGetPC(tao->ksp, &pc);
1396   ierr = PCSetType(pc, PCLMVM);CHKERRQ(ierr);
1397   PetscFunctionReturn(0);
1398 }
1399