xref: /petsc/src/tao/bound/impls/bnk/bnk.c (revision 4f4fdda464867e01e2f7b514a4f52c421e99e383)
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 = bnk->computehessian(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 exact 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   /* Prepare the reduced sub-matrices for the inactive set */
278   if (bnk->active_idx) {
279     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
280     ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
281     if (tao->hessian == tao->hessian_pre) {
282       bnk->Hpre_inactive = bnk->H_inactive;
283     } else {
284       ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
285       ierr = MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive);CHKERRQ(ierr);
286     }
287     if (bnk->bfgs_pre) {
288       ierr = PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx);CHKERRQ(ierr);
289     }
290   } else {
291     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
292     ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);CHKERRQ(ierr);
293     if (tao->hessian == tao->hessian_pre) {
294       bnk->Hpre_inactive = bnk->H_inactive;
295     } else {
296       ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
297       ierr = MatDuplicate(tao->hessian_pre, MAT_COPY_VALUES, &bnk->Hpre_inactive);CHKERRQ(ierr);
298     }
299     if (bnk->bfgs_pre) {
300       ierr = PCLMVMClearIS(bnk->bfgs_pre);CHKERRQ(ierr);
301     }
302   }
303   PetscFunctionReturn(0);
304 }
305 
306 /*------------------------------------------------------------*/
307 
308 /* Routine for estimating the active set */
309 
310 PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
311 {
312   PetscErrorCode               ierr;
313   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
314   PetscBool                    hessComputed, diagExists;
315 
316   PetscFunctionBegin;
317   switch (asType) {
318   case BNK_AS_NONE:
319     ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
320     ierr = VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);CHKERRQ(ierr);
321     ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
322     ierr = ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);CHKERRQ(ierr);
323     break;
324 
325   case BNK_AS_BERTSEKAS:
326     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
327     if (bnk->M) {
328       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
329       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
330     } else {
331       ierr = MatAssembled(tao->hessian, &hessComputed);CHKERRQ(ierr);
332       ierr = MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
333       if (hessComputed && diagExists) {
334         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
335         ierr = MatGetDiagonal(tao->hessian, bnk->Xwork);CHKERRQ(ierr);
336         ierr = VecAbs(bnk->Xwork);CHKERRQ(ierr);
337         ierr = VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);CHKERRQ(ierr);
338         ierr = VecReciprocal(bnk->Xwork);CHKERRQ(ierr);CHKERRQ(ierr);
339         ierr = VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);CHKERRQ(ierr);
340       } else {
341         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
342         ierr = VecCopy(bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
343       }
344     }
345     ierr = VecScale(bnk->W, -1.0);CHKERRQ(ierr);
346     ierr = TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol,
347                                    &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);CHKERRQ(ierr);
348     break;
349 
350   default:
351     break;
352   }
353   PetscFunctionReturn(0);
354 }
355 
356 /*------------------------------------------------------------*/
357 
358 /* Routine for bounding the step direction */
359 
360 PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
361 {
362   PetscErrorCode               ierr;
363   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
364 
365   PetscFunctionBegin;
366   switch (asType) {
367   case BNK_AS_NONE:
368     ierr = VecISSet(step, bnk->active_idx, 0.0);CHKERRQ(ierr);
369     break;
370 
371   case BNK_AS_BERTSEKAS:
372     ierr = TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step);CHKERRQ(ierr);
373     break;
374 
375   default:
376     break;
377   }
378   PetscFunctionReturn(0);
379 }
380 
381 /*------------------------------------------------------------*/
382 
383 /* Routine for taking a finite number of BNCG iterations to
384    accelerate Newton convergence.
385 
386    In practice, this approach simply trades off Hessian evaluations
387    for more gradient evaluations.
388 */
389 
390 PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
391 {
392   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
393   PetscErrorCode               ierr;
394 
395   PetscFunctionBegin;
396   *terminate = PETSC_FALSE;
397   if (bnk->max_cg_its > 0) {
398     /* Copy the current function value (important vectors are already shared) */
399     bnk->bncg_ctx->f = bnk->f;
400     /* Take some small finite number of BNCG iterations */
401     ierr = TaoSolve(bnk->bncg);CHKERRQ(ierr);
402     /* Add the number of gradient and function evaluations to the total */
403     tao->nfuncs += bnk->bncg->nfuncs;
404     tao->nfuncgrads += bnk->bncg->nfuncgrads;
405     tao->ngrads += bnk->bncg->ngrads;
406     tao->nhess += bnk->bncg->nhess;
407     bnk->tot_cg_its += bnk->bncg->niter;
408     /* Extract the BNCG function value out and save it into BNK */
409     bnk->f = bnk->bncg_ctx->f;
410     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) {
411       *terminate = PETSC_TRUE;
412     } else {
413       ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
414     }
415   }
416   PetscFunctionReturn(0);
417 }
418 
419 /*------------------------------------------------------------*/
420 
421 /* Routine for computing the Newton step. */
422 
423 PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason)
424 {
425   PetscErrorCode               ierr;
426   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
427   PetscInt                     bfgsUpdates = 0;
428   PetscInt                     kspits;
429 
430   PetscFunctionBegin;
431   /* If there are no inactive variables left, save some computation and return an adjusted zero step
432      that has (l-x) and (u-x) for lower and upper bounded variables. */
433   if (!bnk->inactive_idx) {
434     ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
435     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
436     PetscFunctionReturn(0);
437   }
438 
439   /* Shift the reduced Hessian matrix */
440   if ((shift) && (bnk->pert > 0)) {
441     ierr = MatShift(bnk->H_inactive, bnk->pert);CHKERRQ(ierr);
442     if (bnk->H_inactive != bnk->Hpre_inactive) {
443       ierr = MatShift(bnk->Hpre_inactive, bnk->pert);CHKERRQ(ierr);
444     }
445   }
446 
447   /* Solve the Newton system of equations */
448   tao->ksp_its = 0;
449   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
450   ierr = KSPReset(tao->ksp);CHKERRQ(ierr);
451   ierr = KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);CHKERRQ(ierr);
452   ierr = VecCopy(bnk->unprojected_gradient, bnk->Gwork);CHKERRQ(ierr);
453   if (bnk->active_idx) {
454     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
455     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
456   } else {
457     bnk->G_inactive = bnk->unprojected_gradient;
458     bnk->X_inactive = tao->stepdirection;
459   }
460   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
461     ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
462     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
463     ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
464     tao->ksp_its+=kspits;
465     tao->ksp_tot_its+=kspits;
466     ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
467 
468     if (0.0 == tao->trust) {
469       /* Radius was uninitialized; use the norm of the direction */
470       if (bnk->dnorm > 0.0) {
471         tao->trust = bnk->dnorm;
472 
473         /* Modify the radius if it is too large or small */
474         tao->trust = PetscMax(tao->trust, bnk->min_radius);
475         tao->trust = PetscMin(tao->trust, bnk->max_radius);
476       } else {
477         /* The direction was bad; set radius to default value and re-solve
478            the trust-region subproblem to get a direction */
479         tao->trust = tao->trust0;
480 
481         /* Modify the radius if it is too large or small */
482         tao->trust = PetscMax(tao->trust, bnk->min_radius);
483         tao->trust = PetscMin(tao->trust, bnk->max_radius);
484 
485         ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
486         ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
487         ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
488         tao->ksp_its+=kspits;
489         tao->ksp_tot_its+=kspits;
490         ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
491 
492         if (bnk->dnorm == 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial direction zero");
493       }
494     }
495   } else {
496     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
497     ierr = KSPGetIterationNumber(tao->ksp, &kspits);CHKERRQ(ierr);
498     tao->ksp_its += kspits;
499     tao->ksp_tot_its+=kspits;
500   }
501   /* Restore sub vectors back */
502   if (bnk->active_idx) {
503     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
504     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
505   }
506   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
507   ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
508   ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
509 
510   /* Record convergence reasons */
511   ierr = KSPGetConvergedReason(tao->ksp, ksp_reason);CHKERRQ(ierr);
512   if (KSP_CONVERGED_ATOL == *ksp_reason) {
513     ++bnk->ksp_atol;
514   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
515     ++bnk->ksp_rtol;
516   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
517     ++bnk->ksp_ctol;
518   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
519     ++bnk->ksp_negc;
520   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
521     ++bnk->ksp_dtol;
522   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
523     ++bnk->ksp_iter;
524   } else {
525     ++bnk->ksp_othr;
526   }
527 
528   /* Make sure the BFGS preconditioner is healthy */
529   if (bnk->M) {
530     ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
531     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
532       /* Preconditioner is numerically indefinite; reset the approximation. */
533       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
534       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
535     }
536   }
537   PetscFunctionReturn(0);
538 }
539 
540 /*------------------------------------------------------------*/
541 
542 /* Routine for recomputing the predicted reduction for a given step vector */
543 
544 PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
545 {
546   PetscErrorCode               ierr;
547   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
548 
549   PetscFunctionBegin;
550   /* Extract subvectors associated with the inactive set */
551   if (bnk->active_idx){
552     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
553     ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
554     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
555   } else {
556     bnk->X_inactive = tao->stepdirection;
557     bnk->inactive_work = bnk->Xwork;
558     bnk->G_inactive = bnk->Gwork;
559   }
560   /* Recompute the predicted decrease based on the quadratic model */
561   ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
562   ierr = VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive);CHKERRQ(ierr);
563   ierr = VecDot(bnk->inactive_work, bnk->X_inactive, prered);CHKERRQ(ierr);
564   /* Restore the sub vectors */
565   if (bnk->active_idx){
566     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
567     ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
568     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
569   }
570   PetscFunctionReturn(0);
571 }
572 
573 /*------------------------------------------------------------*/
574 
575 /* Routine for ensuring that the Newton step is a descent direction.
576 
577    The step direction falls back onto BFGS, scaled gradient and gradient steps
578    in the event that the Newton step fails the test.
579 */
580 
581 PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
582 {
583   PetscErrorCode               ierr;
584   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
585 
586   PetscReal                    gdx, e_min;
587   PetscInt                     bfgsUpdates;
588 
589   PetscFunctionBegin;
590   ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
591   if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
592     /* Newton step is not descent or direction produced Inf or NaN
593        Update the perturbation for next time */
594     if (bnk->pert <= 0.0) {
595       /* Initialize the perturbation */
596       bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
597       if (bnk->is_gltr) {
598         ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
599         bnk->pert = PetscMax(bnk->pert, -e_min);
600       }
601     } else {
602       /* Increase the perturbation */
603       bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
604     }
605 
606     if (!bnk->M) {
607       /* We don't have the bfgs matrix around and updated
608          Must use gradient direction in this case */
609       ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
610       *stepType = BNK_GRADIENT;
611     } else {
612       /* Attempt to use the BFGS direction */
613       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
614 
615       /* Check for success (descent direction)
616          NOTE: Negative gdx here means not a descent direction because
617          the fall-back step is missing a negative sign. */
618       ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
619       if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
620         /* BFGS direction is not descent or direction produced not a number
621            We can assert bfgsUpdates > 1 in this case because
622            the first solve produces the scaled gradient direction,
623            which is guaranteed to be descent */
624 
625         /* Use steepest descent direction (scaled) */
626         ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
627         ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
628         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
629 
630         *stepType = BNK_SCALED_GRADIENT;
631       } else {
632         ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
633         if (1 == bfgsUpdates) {
634           /* The first BFGS direction is always the scaled gradient */
635           *stepType = BNK_SCALED_GRADIENT;
636         } else {
637           *stepType = BNK_BFGS;
638         }
639       }
640     }
641     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
642     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
643     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
644   } else {
645     /* Computed Newton step is descent */
646     switch (ksp_reason) {
647     case KSP_DIVERGED_NANORINF:
648     case KSP_DIVERGED_BREAKDOWN:
649     case KSP_DIVERGED_INDEFINITE_MAT:
650     case KSP_DIVERGED_INDEFINITE_PC:
651     case KSP_CONVERGED_CG_NEG_CURVE:
652       /* Matrix or preconditioner is indefinite; increase perturbation */
653       if (bnk->pert <= 0.0) {
654         /* Initialize the perturbation */
655         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
656         if (bnk->is_gltr) {
657           ierr = KSPCGGLTRGetMinEig(tao->ksp, &e_min);CHKERRQ(ierr);
658           bnk->pert = PetscMax(bnk->pert, -e_min);
659         }
660       } else {
661         /* Increase the perturbation */
662         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
663       }
664       break;
665 
666     default:
667       /* Newton step computation is good; decrease perturbation */
668       bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
669       if (bnk->pert < bnk->pmin) {
670         bnk->pert = 0.0;
671       }
672       break;
673     }
674     *stepType = BNK_NEWTON;
675   }
676   PetscFunctionReturn(0);
677 }
678 
679 /*------------------------------------------------------------*/
680 
681 /* Routine for performing a bound-projected More-Thuente line search.
682 
683   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
684   Newton step does not produce a valid step length.
685 */
686 
687 PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
688 {
689   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
690   PetscErrorCode ierr;
691   TaoLineSearchConvergedReason ls_reason;
692 
693   PetscReal      e_min, gdx;
694   PetscInt       bfgsUpdates;
695 
696   PetscFunctionBegin;
697   /* Perform the linesearch */
698   ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
699   ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
700 
701   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_SCALED_GRADIENT && *stepType != BNK_GRADIENT) {
702     /* Linesearch failed, revert solution */
703     bnk->f = bnk->fold;
704     ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
705     ierr = VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);CHKERRQ(ierr);
706 
707     switch(*stepType) {
708     case BNK_NEWTON:
709       /* Failed to obtain acceptable iterate with Newton step
710          Update the perturbation for next time */
711       if (bnk->pert <= 0.0) {
712         /* Initialize the perturbation */
713         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
714         if (bnk->is_gltr) {
715           ierr = KSPCGGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
716           bnk->pert = PetscMax(bnk->pert, -e_min);
717         }
718       } else {
719         /* Increase the perturbation */
720         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
721       }
722 
723       if (!bnk->M) {
724         /* We don't have the bfgs matrix around and being updated
725            Must use gradient direction in this case */
726         ierr = VecCopy(bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
727         *stepType = BNK_GRADIENT;
728       } else {
729         /* Attempt to use the BFGS direction */
730         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
731         /* Check for success (descent direction)
732            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
733         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
734         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
735           /* BFGS direction is not descent or direction produced not a number
736              We can assert bfgsUpdates > 1 in this case
737              Use steepest descent direction (scaled) */
738           ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
739           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
740           ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
741 
742           bfgsUpdates = 1;
743           *stepType = BNK_SCALED_GRADIENT;
744         } else {
745           ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
746           if (1 == bfgsUpdates) {
747             /* The first BFGS direction is always the scaled gradient */
748             *stepType = BNK_SCALED_GRADIENT;
749           } else {
750             *stepType = BNK_BFGS;
751           }
752         }
753       }
754       break;
755 
756     case BNK_BFGS:
757       /* Can only enter if pc_type == BNK_PC_BFGS
758          Failed to obtain acceptable iterate with BFGS step
759          Attempt to use the scaled gradient direction */
760       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
761       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
762       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
763 
764       bfgsUpdates = 1;
765       *stepType = BNK_SCALED_GRADIENT;
766       break;
767     }
768     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
769     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
770     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
771 
772     /* Perform one last line search with the fall-back step */
773     ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
774     ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
775   }
776   *reason = ls_reason;
777   PetscFunctionReturn(0);
778 }
779 
780 /*------------------------------------------------------------*/
781 
782 /* Routine for updating the trust radius.
783 
784   Function features three different update methods:
785   1) Line-search step length based
786   2) Predicted decrease on the CG quadratic model
787   3) Interpolation
788 */
789 
790 PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
791 {
792   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
793   PetscErrorCode ierr;
794 
795   PetscReal      step, kappa;
796   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;
797 
798   PetscFunctionBegin;
799   /* Update trust region radius */
800   *accept = PETSC_FALSE;
801   switch(updateType) {
802   case BNK_UPDATE_STEP:
803     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
804     if (stepType == BNK_NEWTON) {
805       ierr = TaoLineSearchGetStepLength(tao->linesearch, &step);CHKERRQ(ierr);
806       if (step < bnk->nu1) {
807         /* Very bad step taken; reduce radius */
808         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
809       } else if (step < bnk->nu2) {
810         /* Reasonably bad step taken; reduce radius */
811         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
812       } else if (step < bnk->nu3) {
813         /*  Reasonable step was taken; leave radius alone */
814         if (bnk->omega3 < 1.0) {
815           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
816         } else if (bnk->omega3 > 1.0) {
817           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
818         }
819       } else if (step < bnk->nu4) {
820         /*  Full step taken; increase the radius */
821         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
822       } else {
823         /*  More than full step taken; increase the radius */
824         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
825       }
826     } else {
827       /*  Newton step was not good; reduce the radius */
828       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
829     }
830     break;
831 
832   case BNK_UPDATE_REDUCTION:
833     if (stepType == BNK_NEWTON) {
834       if ((prered < 0.0) || PetscIsInfOrNanReal(prered)) {
835         /* The predicted reduction has the wrong sign.  This cannot
836            happen in infinite precision arithmetic.  Step should
837            be rejected! */
838         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
839       } else {
840         if (PetscIsInfOrNanReal(actred)) {
841           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
842         } else {
843           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
844             kappa = 1.0;
845           } else {
846             kappa = actred / prered;
847           }
848           /* Accept or reject the step and update radius */
849           if (kappa < bnk->eta1) {
850             /* Reject the step */
851             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
852           } else {
853             /* Accept the step */
854             *accept = PETSC_TRUE;
855             /* Update the trust region radius only if the computed step is at the trust radius boundary */
856             if (bnk->dnorm == tao->trust) {
857               if (kappa < bnk->eta2) {
858                 /* Marginal bad step */
859                 tao->trust = bnk->alpha2 * tao->trust;
860               } else if (kappa < bnk->eta3) {
861                 /* Reasonable step */
862                 tao->trust = bnk->alpha3 * tao->trust;
863               } else if (kappa < bnk->eta4) {
864                 /* Good step */
865                 tao->trust = bnk->alpha4 * tao->trust;
866               } else {
867                 /* Very good step */
868                 tao->trust = bnk->alpha5 * tao->trust;
869               }
870             }
871           }
872         }
873       }
874     } else {
875       /*  Newton step was not good; reduce the radius */
876       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
877     }
878     break;
879 
880   default:
881     if (stepType == BNK_NEWTON) {
882       if (prered < 0.0) {
883         /*  The predicted reduction has the wrong sign.  This cannot */
884         /*  happen in infinite precision arithmetic.  Step should */
885         /*  be rejected! */
886         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
887       } else {
888         if (PetscIsInfOrNanReal(actred)) {
889           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
890         } else {
891           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
892             kappa = 1.0;
893           } else {
894             kappa = actred / prered;
895           }
896 
897           ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
898           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
899           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
900           tau_min = PetscMin(tau_1, tau_2);
901           tau_max = PetscMax(tau_1, tau_2);
902 
903           if (kappa >= 1.0 - bnk->mu1) {
904             /*  Great agreement */
905             *accept = PETSC_TRUE;
906             if (tau_max < 1.0) {
907               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
908             } else if (tau_max > bnk->gamma4) {
909               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
910             } else {
911               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
912             }
913           } else if (kappa >= 1.0 - bnk->mu2) {
914             /*  Good agreement */
915             *accept = PETSC_TRUE;
916             if (tau_max < bnk->gamma2) {
917               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
918             } else if (tau_max > bnk->gamma3) {
919               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
920             } else if (tau_max < 1.0) {
921               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
922             } else {
923               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
924             }
925           } else {
926             /*  Not good agreement */
927             if (tau_min > 1.0) {
928               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
929             } else if (tau_max < bnk->gamma1) {
930               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
931             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
932               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
933             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
934               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
935             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
936               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
937             } else {
938               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
939             }
940           }
941         }
942       }
943     } else {
944       /*  Newton step was not good; reduce the radius */
945       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
946     }
947     break;
948   }
949   /* Make sure the radius does not violate min and max settings */
950   tao->trust = PetscMin(tao->trust, bnk->max_radius);
951   tao->trust = PetscMax(tao->trust, bnk->min_radius);
952   PetscFunctionReturn(0);
953 }
954 
955 /* ---------------------------------------------------------- */
956 
957 PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
958 {
959   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
960 
961   PetscFunctionBegin;
962   switch (stepType) {
963   case BNK_NEWTON:
964     ++bnk->newt;
965     break;
966   case BNK_BFGS:
967     ++bnk->bfgs;
968     break;
969   case BNK_SCALED_GRADIENT:
970     ++bnk->sgrad;
971     break;
972   case BNK_GRADIENT:
973     ++bnk->grad;
974     break;
975   default:
976     break;
977   }
978   PetscFunctionReturn(0);
979 }
980 
981 /* ---------------------------------------------------------- */
982 
983 PetscErrorCode TaoSetUp_BNK(Tao tao)
984 {
985   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
986   PetscErrorCode ierr;
987   PetscInt       i;
988 
989   PetscFunctionBegin;
990   if (!tao->gradient) {
991     ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);
992   }
993   if (!tao->stepdirection) {
994     ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);
995   }
996   if (!bnk->W) {
997     ierr = VecDuplicate(tao->solution,&bnk->W);CHKERRQ(ierr);
998   }
999   if (!bnk->Xold) {
1000     ierr = VecDuplicate(tao->solution,&bnk->Xold);CHKERRQ(ierr);
1001   }
1002   if (!bnk->Gold) {
1003     ierr = VecDuplicate(tao->solution,&bnk->Gold);CHKERRQ(ierr);
1004   }
1005   if (!bnk->Xwork) {
1006     ierr = VecDuplicate(tao->solution,&bnk->Xwork);CHKERRQ(ierr);
1007   }
1008   if (!bnk->Gwork) {
1009     ierr = VecDuplicate(tao->solution,&bnk->Gwork);CHKERRQ(ierr);
1010   }
1011   if (!bnk->unprojected_gradient) {
1012     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient);CHKERRQ(ierr);
1013   }
1014   if (!bnk->unprojected_gradient_old) {
1015     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1016   }
1017   if (!bnk->Diag_min) {
1018     ierr = VecDuplicate(tao->solution,&bnk->Diag_min);CHKERRQ(ierr);
1019   }
1020   if (!bnk->Diag_max) {
1021     ierr = VecDuplicate(tao->solution,&bnk->Diag_max);CHKERRQ(ierr);
1022   }
1023   if (bnk->max_cg_its > 0) {
1024     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
1025     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
1026     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old));CHKERRQ(ierr);
1027     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old);CHKERRQ(ierr);
1028     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
1029     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient));CHKERRQ(ierr);
1030     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient);CHKERRQ(ierr);
1031     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
1032     ierr = PetscObjectReference((PetscObject)(bnk->Gold));CHKERRQ(ierr);
1033     ierr = VecDestroy(&bnk->bncg_ctx->G_old);CHKERRQ(ierr);
1034     bnk->bncg_ctx->G_old = bnk->Gold;
1035     ierr = PetscObjectReference((PetscObject)(tao->gradient));CHKERRQ(ierr);
1036     ierr = VecDestroy(&bnk->bncg->gradient);CHKERRQ(ierr);
1037     bnk->bncg->gradient = tao->gradient;
1038     ierr = PetscObjectReference((PetscObject)(tao->stepdirection));CHKERRQ(ierr);
1039     ierr = VecDestroy(&bnk->bncg->stepdirection);CHKERRQ(ierr);
1040     bnk->bncg->stepdirection = tao->stepdirection;
1041     ierr = TaoSetInitialVector(bnk->bncg, tao->solution);CHKERRQ(ierr);
1042     /* Copy over some settings from BNK into BNCG */
1043     ierr = TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its);CHKERRQ(ierr);
1044     ierr = TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol);CHKERRQ(ierr);
1045     ierr = TaoSetFunctionLowerBound(bnk->bncg, tao->fmin);CHKERRQ(ierr);
1046     ierr = TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP);CHKERRQ(ierr);
1047     ierr = TaoSetObjectiveRoutine(bnk->bncg, tao->ops->computeobjective, tao->user_objP);CHKERRQ(ierr);
1048     ierr = TaoSetGradientRoutine(bnk->bncg, tao->ops->computegradient, tao->user_gradP);CHKERRQ(ierr);
1049     ierr = TaoSetObjectiveAndGradientRoutine(bnk->bncg, tao->ops->computeobjectiveandgradient, tao->user_objgradP);CHKERRQ(ierr);
1050     ierr = PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg));CHKERRQ(ierr);
1051     for (i=0; i<tao->numbermonitors; ++i) {
1052       ierr = TaoSetMonitor(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);CHKERRQ(ierr);
1053       ierr = PetscObjectReference((PetscObject)(tao->monitorcontext[i]));CHKERRQ(ierr);
1054     }
1055   }
1056   bnk->X_inactive = 0;
1057   bnk->G_inactive = 0;
1058   bnk->inactive_work = 0;
1059   bnk->active_work = 0;
1060   bnk->inactive_idx = 0;
1061   bnk->active_idx = 0;
1062   bnk->active_lower = 0;
1063   bnk->active_upper = 0;
1064   bnk->active_fixed = 0;
1065   bnk->M = 0;
1066   bnk->H_inactive = 0;
1067   bnk->Hpre_inactive = 0;
1068   PetscFunctionReturn(0);
1069 }
1070 
1071 /*------------------------------------------------------------*/
1072 
1073 PetscErrorCode TaoDestroy_BNK(Tao tao)
1074 {
1075   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1076   PetscErrorCode ierr;
1077 
1078   PetscFunctionBegin;
1079   if (tao->setupcalled) {
1080     ierr = VecDestroy(&bnk->W);CHKERRQ(ierr);
1081     ierr = VecDestroy(&bnk->Xold);CHKERRQ(ierr);
1082     ierr = VecDestroy(&bnk->Gold);CHKERRQ(ierr);
1083     ierr = VecDestroy(&bnk->Xwork);CHKERRQ(ierr);
1084     ierr = VecDestroy(&bnk->Gwork);CHKERRQ(ierr);
1085     ierr = VecDestroy(&bnk->unprojected_gradient);CHKERRQ(ierr);
1086     ierr = VecDestroy(&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1087     ierr = VecDestroy(&bnk->Diag_min);CHKERRQ(ierr);
1088     ierr = VecDestroy(&bnk->Diag_max);CHKERRQ(ierr);
1089   }
1090   ierr = ISDestroy(&bnk->active_lower);CHKERRQ(ierr);
1091   ierr = ISDestroy(&bnk->active_upper);CHKERRQ(ierr);
1092   ierr = ISDestroy(&bnk->active_fixed);CHKERRQ(ierr);
1093   ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
1094   ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
1095   if (bnk->Hpre_inactive != tao->hessian_pre && bnk->Hpre_inactive != bnk->H_inactive) {
1096     ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
1097   }
1098   if (bnk->H_inactive != tao->hessian) {
1099     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
1100   }
1101   ierr = TaoDestroy(&bnk->bncg);CHKERRQ(ierr);
1102   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1103   PetscFunctionReturn(0);
1104 }
1105 
1106 /*------------------------------------------------------------*/
1107 
1108 PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
1109 {
1110   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1111   PetscErrorCode ierr;
1112   KSPType        ksp_type;
1113 
1114   PetscFunctionBegin;
1115   ierr = PetscOptionsHead(PetscOptionsObject,"Newton-Krylov method for bound constrained optimization");CHKERRQ(ierr);
1116   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);
1117   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);
1118   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);
1119   ierr = PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval,NULL);CHKERRQ(ierr);
1120   ierr = PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin,NULL);CHKERRQ(ierr);
1121   ierr = PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax,NULL);CHKERRQ(ierr);
1122   ierr = PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac,NULL);CHKERRQ(ierr);
1123   ierr = PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin,NULL);CHKERRQ(ierr);
1124   ierr = PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax,NULL);CHKERRQ(ierr);
1125   ierr = PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);CHKERRQ(ierr);
1126   ierr = PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac,NULL);CHKERRQ(ierr);
1127   ierr = PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac,NULL);CHKERRQ(ierr);
1128   ierr = PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac,NULL);CHKERRQ(ierr);
1129   ierr = PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1,NULL);CHKERRQ(ierr);
1130   ierr = PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2,NULL);CHKERRQ(ierr);
1131   ierr = PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3,NULL);CHKERRQ(ierr);
1132   ierr = PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4,NULL);CHKERRQ(ierr);
1133   ierr = PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1,NULL);CHKERRQ(ierr);
1134   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);
1135   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);
1136   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);
1137   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);
1138   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);
1139   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);
1140   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);
1141   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);
1142   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);
1143   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);
1144   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);
1145   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);
1146   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);
1147   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);
1148   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);
1149   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);
1150   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);
1151   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);
1152   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);
1153   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);
1154   ierr = PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1,NULL);CHKERRQ(ierr);
1155   ierr = PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2,NULL);CHKERRQ(ierr);
1156   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);
1157   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);
1158   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);
1159   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);
1160   ierr = PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta,NULL);CHKERRQ(ierr);
1161   ierr = PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);CHKERRQ(ierr);
1162   ierr = PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);CHKERRQ(ierr);
1163   ierr = PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);CHKERRQ(ierr);
1164   ierr = PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);CHKERRQ(ierr);
1165   ierr = PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);CHKERRQ(ierr);
1166   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);
1167   ierr = PetscOptionsTail();CHKERRQ(ierr);
1168   ierr = TaoSetFromOptions(bnk->bncg);CHKERRQ(ierr);
1169   ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr);
1170   ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr);
1171   ierr = KSPGetType(tao->ksp,&ksp_type);CHKERRQ(ierr);
1172   ierr = PetscStrcmp(ksp_type,KSPCGNASH,&bnk->is_nash);CHKERRQ(ierr);
1173   ierr = PetscStrcmp(ksp_type,KSPCGSTCG,&bnk->is_stcg);CHKERRQ(ierr);
1174   ierr = PetscStrcmp(ksp_type,KSPCGGLTR,&bnk->is_gltr);CHKERRQ(ierr);
1175   PetscFunctionReturn(0);
1176 }
1177 
1178 /*------------------------------------------------------------*/
1179 
1180 PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
1181 {
1182   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1183   PetscInt       nrejects;
1184   PetscBool      isascii;
1185   PetscErrorCode ierr;
1186 
1187   PetscFunctionBegin;
1188   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1189   if (isascii) {
1190     ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
1191     if (bnk->M) {
1192       ierr = MatLMVMGetRejectCount(bnk->M,&nrejects);CHKERRQ(ierr);
1193       ierr = PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %D\n",nrejects);CHKERRQ(ierr);
1194     }
1195     ierr = PetscViewerASCIIPrintf(viewer, "CG steps: %D\n", bnk->tot_cg_its);CHKERRQ(ierr);
1196     ierr = PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);CHKERRQ(ierr);
1197     if (bnk->M) {
1198       ierr = PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);CHKERRQ(ierr);
1199     }
1200     ierr = PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);CHKERRQ(ierr);
1201     ierr = PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);CHKERRQ(ierr);
1202     ierr = PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");CHKERRQ(ierr);
1203     ierr = PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);CHKERRQ(ierr);
1204     ierr = PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);CHKERRQ(ierr);
1205     ierr = PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);CHKERRQ(ierr);
1206     ierr = PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);CHKERRQ(ierr);
1207     ierr = PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);CHKERRQ(ierr);
1208     ierr = PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);CHKERRQ(ierr);
1209     ierr = PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);CHKERRQ(ierr);
1210     ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
1211   }
1212   PetscFunctionReturn(0);
1213 }
1214 
1215 /* ---------------------------------------------------------- */
1216 
1217 /*MC
1218   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
1219   At each iteration, the BNK methods solve the symmetric
1220   system of equations to obtain the step diretion dk:
1221               Hk dk = -gk
1222   for free variables only. The step can be globalized either through
1223   trust-region methods, or a line search, or a heuristic mixture of both.
1224 
1225     Options Database Keys:
1226 + -max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
1227 . -init_type - trust radius initialization method ("constant", "direction", "interpolation")
1228 . -update_type - trust radius update method ("step", "direction", "interpolation")
1229 . -as_type - active-set estimation method ("none", "bertsekas")
1230 . -as_tol - (developer) initial tolerance used in estimating bounded active variables (-as_type bertsekas)
1231 . -as_step - (developer) trial step length used in estimating bounded active variables (-as_type bertsekas)
1232 . -sval - (developer) Hessian perturbation starting value
1233 . -imin - (developer) minimum initial Hessian perturbation
1234 . -imax - (developer) maximum initial Hessian perturbation
1235 . -pmin - (developer) minimum Hessian perturbation
1236 . -pmax - (developer) aximum Hessian perturbation
1237 . -pgfac - (developer) Hessian perturbation growth factor
1238 . -psfac - (developer) Hessian perturbation shrink factor
1239 . -imfac - (developer) initial merit factor for Hessian perturbation
1240 . -pmgfac - (developer) merit growth factor for Hessian perturbation
1241 . -pmsfac - (developer) merit shrink factor for Hessian perturbation
1242 . -eta1 - (developer) threshold for rejecting step (-update_type reduction)
1243 . -eta2 - (developer) threshold for accepting marginal step (-update_type reduction)
1244 . -eta3 - (developer) threshold for accepting reasonable step (-update_type reduction)
1245 . -eta4 - (developer) threshold for accepting good step (-update_type reduction)
1246 . -alpha1 - (developer) radius reduction factor for rejected step (-update_type reduction)
1247 . -alpha2 - (developer) radius reduction factor for marginally accepted bad step (-update_type reduction)
1248 . -alpha3 - (developer) radius increase factor for reasonable accepted step (-update_type reduction)
1249 . -alpha4 - (developer) radius increase factor for good accepted step (-update_type reduction)
1250 . -alpha5 - (developer) radius increase factor for very good accepted step (-update_type reduction)
1251 . -epsilon - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-update_type reduction)
1252 . -mu1 - (developer) threshold for accepting very good step (-update_type interpolation)
1253 . -mu2 - (developer) threshold for accepting good step (-update_type interpolation)
1254 . -gamma1 - (developer) radius reduction factor for rejected very bad step (-update_type interpolation)
1255 . -gamma2 - (developer) radius reduction factor for rejected bad step (-update_type interpolation)
1256 . -gamma3 - (developer) radius increase factor for accepted good step (-update_type interpolation)
1257 . -gamma4 - (developer) radius increase factor for accepted very good step (-update_type interpolation)
1258 . -theta - (developer) trust region interpolation factor (-update_type interpolation)
1259 . -nu1 - (developer) threshold for small line-search step length (-update_type step)
1260 . -nu2 - (developer) threshold for reasonable line-search step length (-update_type step)
1261 . -nu3 - (developer) threshold for large line-search step length (-update_type step)
1262 . -nu4 - (developer) threshold for very large line-search step length (-update_type step)
1263 . -omega1 - (developer) radius reduction factor for very small line-search step length (-update_type step)
1264 . -omega2 - (developer) radius reduction factor for small line-search step length (-update_type step)
1265 . -omega3 - (developer) radius factor for decent line-search step length (-update_type step)
1266 . -omega4 - (developer) radius increase factor for large line-search step length (-update_type step)
1267 . -omega5 - (developer) radius increase factor for very large line-search step length (-update_type step)
1268 . -mu1_i -  (developer) threshold for accepting very good step (-init_type interpolation)
1269 . -mu2_i -  (developer) threshold for accepting good step (-init_type interpolation)
1270 . -gamma1_i - (developer) radius reduction factor for rejected very bad step (-init_type interpolation)
1271 . -gamma2_i - (developer) radius reduction factor for rejected bad step (-init_type interpolation)
1272 . -gamma3_i - (developer) radius increase factor for accepted good step (-init_type interpolation)
1273 . -gamma4_i - (developer) radius increase factor for accepted very good step (-init_type interpolation)
1274 - -theta_i - (developer) trust region interpolation factor (-init_type interpolation)
1275 
1276   Level: beginner
1277 M*/
1278 
1279 PetscErrorCode TaoCreate_BNK(Tao tao)
1280 {
1281   TAO_BNK        *bnk;
1282   const char     *morethuente_type = TAOLINESEARCHMT;
1283   PetscErrorCode ierr;
1284   PC             pc;
1285 
1286   PetscFunctionBegin;
1287   ierr = PetscNewLog(tao,&bnk);CHKERRQ(ierr);
1288 
1289   tao->ops->setup = TaoSetUp_BNK;
1290   tao->ops->view = TaoView_BNK;
1291   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
1292   tao->ops->destroy = TaoDestroy_BNK;
1293 
1294   /*  Override default settings (unless already changed) */
1295   if (!tao->max_it_changed) tao->max_it = 50;
1296   if (!tao->trust0_changed) tao->trust0 = 100.0;
1297 
1298   tao->data = (void*)bnk;
1299 
1300   /*  Hessian shifting parameters */
1301   bnk->computehessian = TaoBNKComputeHessian;
1302   bnk->computestep = TaoBNKComputeStep;
1303 
1304   bnk->sval   = 0.0;
1305   bnk->imin   = 1.0e-4;
1306   bnk->imax   = 1.0e+2;
1307   bnk->imfac  = 1.0e-1;
1308 
1309   bnk->pmin   = 1.0e-12;
1310   bnk->pmax   = 1.0e+2;
1311   bnk->pgfac  = 1.0e+1;
1312   bnk->psfac  = 4.0e-1;
1313   bnk->pmgfac = 1.0e-1;
1314   bnk->pmsfac = 1.0e-1;
1315 
1316   /*  Default values for trust-region radius update based on steplength */
1317   bnk->nu1 = 0.25;
1318   bnk->nu2 = 0.50;
1319   bnk->nu3 = 1.00;
1320   bnk->nu4 = 1.25;
1321 
1322   bnk->omega1 = 0.25;
1323   bnk->omega2 = 0.50;
1324   bnk->omega3 = 1.00;
1325   bnk->omega4 = 2.00;
1326   bnk->omega5 = 4.00;
1327 
1328   /*  Default values for trust-region radius update based on reduction */
1329   bnk->eta1 = 1.0e-4;
1330   bnk->eta2 = 0.25;
1331   bnk->eta3 = 0.50;
1332   bnk->eta4 = 0.90;
1333 
1334   bnk->alpha1 = 0.25;
1335   bnk->alpha2 = 0.50;
1336   bnk->alpha3 = 1.00;
1337   bnk->alpha4 = 2.00;
1338   bnk->alpha5 = 4.00;
1339 
1340   /*  Default values for trust-region radius update based on interpolation */
1341   bnk->mu1 = 0.10;
1342   bnk->mu2 = 0.50;
1343 
1344   bnk->gamma1 = 0.25;
1345   bnk->gamma2 = 0.50;
1346   bnk->gamma3 = 2.00;
1347   bnk->gamma4 = 4.00;
1348 
1349   bnk->theta = 0.05;
1350 
1351   /*  Default values for trust region initialization based on interpolation */
1352   bnk->mu1_i = 0.35;
1353   bnk->mu2_i = 0.50;
1354 
1355   bnk->gamma1_i = 0.0625;
1356   bnk->gamma2_i = 0.5;
1357   bnk->gamma3_i = 2.0;
1358   bnk->gamma4_i = 5.0;
1359 
1360   bnk->theta_i = 0.25;
1361 
1362   /*  Remaining parameters */
1363   bnk->max_cg_its = 0;
1364   bnk->min_radius = 1.0e-10;
1365   bnk->max_radius = 1.0e10;
1366   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
1367   bnk->as_tol = 1.0e-3;
1368   bnk->as_step = 1.0e-3;
1369   bnk->dmin = 1.0e-6;
1370   bnk->dmax = 1.0e6;
1371 
1372   bnk->M               = 0;
1373   bnk->bfgs_pre        = 0;
1374   bnk->init_type       = BNK_INIT_INTERPOLATION;
1375   bnk->update_type     = BNK_UPDATE_REDUCTION;
1376   bnk->as_type         = BNK_AS_BERTSEKAS;
1377 
1378   /* Create the embedded BNCG solver */
1379   ierr = TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg);CHKERRQ(ierr);
1380   ierr = PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1);CHKERRQ(ierr);
1381   ierr = TaoSetOptionsPrefix(bnk->bncg, "tao_bnk_");CHKERRQ(ierr);
1382   ierr = TaoSetType(bnk->bncg, TAOBNCG);CHKERRQ(ierr);
1383 
1384   /* Create the line search */
1385   ierr = TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);CHKERRQ(ierr);
1386   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr);
1387   ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr);
1388   ierr = TaoLineSearchSetType(tao->linesearch,morethuente_type);CHKERRQ(ierr);
1389   ierr = TaoLineSearchUseTaoRoutines(tao->linesearch,tao);CHKERRQ(ierr);
1390 
1391   /*  Set linear solver to default for symmetric matrices */
1392   ierr = KSPCreate(((PetscObject)tao)->comm,&tao->ksp);CHKERRQ(ierr);
1393   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr);
1394   ierr = KSPSetOptionsPrefix(tao->ksp,"tao_bnk_");CHKERRQ(ierr);
1395   ierr = KSPSetType(tao->ksp,KSPCGSTCG);CHKERRQ(ierr);
1396   ierr = KSPGetPC(tao->ksp, &pc);
1397   ierr = PCSetType(pc, PCLMVM);CHKERRQ(ierr);
1398   PetscFunctionReturn(0);
1399 }
1400