xref: /petsc/src/tao/bound/impls/bnk/bnk.c (revision 2f613bf53f46f9356e00a2ca2bd69453be72fc31)
1 #include <petsctaolinesearch.h>
2 #include <../src/tao/bound/impls/bnk/bnk.h>
3 #include <petscksp.h>
4 
5 static const char *BNK_INIT[64] = {"constant", "direction", "interpolation"};
6 static const char *BNK_UPDATE[64] = {"step", "reduction", "interpolation"};
7 static const char *BNK_AS[64] = {"none", "bertsekas"};
8 
9 /*------------------------------------------------------------*/
10 
11 /* Routine for initializing the KSP solver, the BFGS preconditioner, and the initial trust radius estimation */
12 
13 PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType, PetscBool *needH)
14 {
15   PetscErrorCode               ierr;
16   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
17   PC                           pc;
18 
19   PetscReal                    f_min, ftrial, prered, actred, kappa, sigma, resnorm;
20   PetscReal                    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
21   PetscBool                    is_bfgs, is_jacobi, is_symmetric, sym_set;
22   PetscInt                     n, N, nDiff;
23   PetscInt                     i_max = 5;
24   PetscInt                     j_max = 1;
25   PetscInt                     i, j;
26 
27   PetscFunctionBegin;
28   /* Project the current point onto the feasible set */
29   ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr);
30   ierr = TaoSetVariableBounds(bnk->bncg, tao->XL, tao->XU);CHKERRQ(ierr);
31   if (tao->bounded) {
32     ierr = TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);CHKERRQ(ierr);
33   }
34 
35   /* Project the initial point onto the feasible region */
36   ierr = TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);CHKERRQ(ierr);
37 
38   /* Check convergence criteria */
39   ierr = TaoComputeObjectiveAndGradient(tao, tao->solution, &bnk->f, bnk->unprojected_gradient);CHKERRQ(ierr);
40   ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
41   ierr = VecCopy(bnk->unprojected_gradient, tao->gradient);CHKERRQ(ierr);
42   ierr = VecISSet(tao->gradient, bnk->active_idx, 0.0);CHKERRQ(ierr);
43   ierr = TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm);CHKERRQ(ierr);
44 
45   /* Test the initial point for convergence */
46   ierr = VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);CHKERRQ(ierr);
47   ierr = VecNorm(bnk->W, NORM_2, &resnorm);CHKERRQ(ierr);
48   if (PetscIsInfOrNanReal(bnk->f) || PetscIsInfOrNanReal(resnorm)) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
49   ierr = TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
50   ierr = TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);CHKERRQ(ierr);
51   ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
52   if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
53 
54   /* Reset KSP stopping reason counters */
55   bnk->ksp_atol = 0;
56   bnk->ksp_rtol = 0;
57   bnk->ksp_dtol = 0;
58   bnk->ksp_ctol = 0;
59   bnk->ksp_negc = 0;
60   bnk->ksp_iter = 0;
61   bnk->ksp_othr = 0;
62 
63   /* Reset accepted step type counters */
64   bnk->tot_cg_its = 0;
65   bnk->newt = 0;
66   bnk->bfgs = 0;
67   bnk->sgrad = 0;
68   bnk->grad = 0;
69 
70   /* Initialize the Hessian perturbation */
71   bnk->pert = bnk->sval;
72 
73   /* Reset initial steplength to zero (this helps BNCG reset its direction internally) */
74   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
75 
76   /* Allocate the vectors needed for the BFGS approximation */
77   ierr = KSPGetPC(tao->ksp, &pc);CHKERRQ(ierr);
78   ierr = PetscObjectTypeCompare((PetscObject)pc, PCLMVM, &is_bfgs);CHKERRQ(ierr);
79   ierr = PetscObjectTypeCompare((PetscObject)pc, PCJACOBI, &is_jacobi);CHKERRQ(ierr);
80   if (is_bfgs) {
81     bnk->bfgs_pre = pc;
82     ierr = PCLMVMGetMatLMVM(bnk->bfgs_pre, &bnk->M);CHKERRQ(ierr);
83     ierr = VecGetLocalSize(tao->solution, &n);CHKERRQ(ierr);
84     ierr = VecGetSize(tao->solution, &N);CHKERRQ(ierr);
85     ierr = MatSetSizes(bnk->M, n, n, N, N);CHKERRQ(ierr);
86     ierr = MatLMVMAllocate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
87     ierr = MatIsSymmetricKnown(bnk->M, &sym_set, &is_symmetric);CHKERRQ(ierr);
88     if (!sym_set || !is_symmetric) SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_INCOMP, "LMVM matrix in the LMVM preconditioner must be symmetric.");
89   } else if (is_jacobi) {
90     ierr = PCJacobiSetUseAbs(pc,PETSC_TRUE);CHKERRQ(ierr);
91   }
92 
93   /* Prepare the min/max vectors for safeguarding diagonal scales */
94   ierr = VecSet(bnk->Diag_min, bnk->dmin);CHKERRQ(ierr);
95   ierr = VecSet(bnk->Diag_max, bnk->dmax);CHKERRQ(ierr);
96 
97   /* Initialize trust-region radius.  The initialization is only performed
98      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
99   *needH = PETSC_TRUE;
100   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
101     switch(initType) {
102     case BNK_INIT_CONSTANT:
103       /* Use the initial radius specified */
104       tao->trust = tao->trust0;
105       break;
106 
107     case BNK_INIT_INTERPOLATION:
108       /* Use interpolation based on the initial Hessian */
109       max_radius = 0.0;
110       tao->trust = tao->trust0;
111       for (j = 0; j < j_max; ++j) {
112         f_min = bnk->f;
113         sigma = 0.0;
114 
115         if (*needH) {
116           /* Compute the Hessian at the new step, and extract the inactive subsystem */
117           ierr = bnk->computehessian(tao);CHKERRQ(ierr);
118           ierr = TaoBNKEstimateActiveSet(tao, BNK_AS_NONE);CHKERRQ(ierr);
119           ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
120           if (bnk->active_idx) {
121             ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
122           } else {
123             ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);CHKERRQ(ierr);
124           }
125           *needH = PETSC_FALSE;
126         }
127 
128         for (i = 0; i < i_max; ++i) {
129           /* Take a steepest descent step and snap it to bounds */
130           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
131           ierr = VecAXPY(tao->solution, -tao->trust/bnk->gnorm, tao->gradient);CHKERRQ(ierr);
132           ierr = TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);CHKERRQ(ierr);
133           /* Compute the step we actually accepted */
134           ierr = VecCopy(tao->solution, bnk->W);CHKERRQ(ierr);
135           ierr = VecAXPY(bnk->W, -1.0, bnk->Xold);CHKERRQ(ierr);
136           /* Compute the objective at the trial */
137           ierr = TaoComputeObjective(tao, tao->solution, &ftrial);CHKERRQ(ierr);
138           if (PetscIsInfOrNanReal(bnk->f)) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
139           ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
140           if (PetscIsInfOrNanReal(ftrial)) {
141             tau = bnk->gamma1_i;
142           } else {
143             if (ftrial < f_min) {
144               f_min = ftrial;
145               sigma = -tao->trust / bnk->gnorm;
146             }
147 
148             /* Compute the predicted and actual reduction */
149             if (bnk->active_idx) {
150               ierr = VecGetSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
151               ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
152             } else {
153               bnk->X_inactive = bnk->W;
154               bnk->inactive_work = bnk->Xwork;
155             }
156             ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
157             ierr = VecDot(bnk->X_inactive, bnk->inactive_work, &prered);CHKERRQ(ierr);
158             if (bnk->active_idx) {
159               ierr = VecRestoreSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
160               ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
161             }
162             prered = tao->trust * (bnk->gnorm - 0.5 * tao->trust * prered / (bnk->gnorm * bnk->gnorm));
163             actred = bnk->f - ftrial;
164             if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
165               kappa = 1.0;
166             } else {
167               kappa = actred / prered;
168             }
169 
170             tau_1 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust + (1.0 - bnk->theta_i) * prered - actred);
171             tau_2 = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust - (1.0 + bnk->theta_i) * prered + actred);
172             tau_min = PetscMin(tau_1, tau_2);
173             tau_max = PetscMax(tau_1, tau_2);
174 
175             if (PetscAbsScalar(kappa - (PetscReal)1.0) <= bnk->mu1_i) {
176               /*  Great agreement */
177               max_radius = PetscMax(max_radius, tao->trust);
178 
179               if (tau_max < 1.0) {
180                 tau = bnk->gamma3_i;
181               } else if (tau_max > bnk->gamma4_i) {
182                 tau = bnk->gamma4_i;
183               } else {
184                 tau = tau_max;
185               }
186             } else if (PetscAbsScalar(kappa - (PetscReal)1.0) <= bnk->mu2_i) {
187               /*  Good agreement */
188               max_radius = PetscMax(max_radius, tao->trust);
189 
190               if (tau_max < bnk->gamma2_i) {
191                 tau = bnk->gamma2_i;
192               } else if (tau_max > bnk->gamma3_i) {
193                 tau = bnk->gamma3_i;
194               } else {
195                 tau = tau_max;
196               }
197             } else {
198               /*  Not good agreement */
199               if (tau_min > 1.0) {
200                 tau = bnk->gamma2_i;
201               } else if (tau_max < bnk->gamma1_i) {
202                 tau = bnk->gamma1_i;
203               } else if ((tau_min < bnk->gamma1_i) && (tau_max >= 1.0)) {
204                 tau = bnk->gamma1_i;
205               } else if ((tau_1 >= bnk->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
206                 tau = tau_1;
207               } else if ((tau_2 >= bnk->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
208                 tau = tau_2;
209               } else {
210                 tau = tau_max;
211               }
212             }
213           }
214           tao->trust = tau * tao->trust;
215         }
216 
217         if (f_min < bnk->f) {
218           /* We accidentally found a solution better than the initial, so accept it */
219           bnk->f = f_min;
220           ierr = VecCopy(tao->solution, bnk->Xold);CHKERRQ(ierr);
221           ierr = VecAXPY(tao->solution,sigma,tao->gradient);CHKERRQ(ierr);
222           ierr = TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);CHKERRQ(ierr);
223           ierr = VecCopy(tao->solution, tao->stepdirection);CHKERRQ(ierr);
224           ierr = VecAXPY(tao->stepdirection, -1.0, bnk->Xold);CHKERRQ(ierr);
225           ierr = TaoComputeGradient(tao,tao->solution,bnk->unprojected_gradient);CHKERRQ(ierr);
226           ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
227           ierr = VecCopy(bnk->unprojected_gradient, tao->gradient);CHKERRQ(ierr);
228           ierr = VecISSet(tao->gradient, bnk->active_idx, 0.0);CHKERRQ(ierr);
229           /* Compute gradient at the new iterate and flip switch to compute the Hessian later */
230           ierr = TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm);CHKERRQ(ierr);
231           *needH = PETSC_TRUE;
232           /* Test the new step for convergence */
233           ierr = VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);CHKERRQ(ierr);
234           ierr = VecNorm(bnk->W, NORM_2, &resnorm);CHKERRQ(ierr);
235           if (PetscIsInfOrNanReal(resnorm)) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
236           ierr = TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
237           ierr = TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);CHKERRQ(ierr);
238           ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
239           if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(0);
240           /* active BNCG recycling early because we have a stepdirection computed */
241           ierr = TaoSetRecycleHistory(bnk->bncg, PETSC_TRUE);CHKERRQ(ierr);
242         }
243       }
244       tao->trust = PetscMax(tao->trust, max_radius);
245 
246       /* Ensure that the trust radius is within the limits */
247       tao->trust = PetscMax(tao->trust, bnk->min_radius);
248       tao->trust = PetscMin(tao->trust, bnk->max_radius);
249       break;
250 
251     default:
252       /* Norm of the first direction will initialize radius */
253       tao->trust = 0.0;
254       break;
255     }
256   }
257   PetscFunctionReturn(0);
258 }
259 
260 /*------------------------------------------------------------*/
261 
262 /* Routine for computing the exact Hessian and preparing the preconditioner at the new iterate */
263 
264 PetscErrorCode TaoBNKComputeHessian(Tao tao)
265 {
266   PetscErrorCode               ierr;
267   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
268 
269   PetscFunctionBegin;
270   /* Compute the Hessian */
271   ierr = TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);CHKERRQ(ierr);
272   /* Add a correction to the BFGS preconditioner */
273   if (bnk->M) {
274     ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
275   }
276   /* Prepare the reduced sub-matrices for the inactive set */
277   if (bnk->Hpre_inactive) {
278     ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
279   }
280   if (bnk->H_inactive) {
281     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
282   }
283   if (bnk->active_idx) {
284     ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
285     if (tao->hessian == tao->hessian_pre) {
286       ierr = PetscObjectReference((PetscObject)bnk->H_inactive);CHKERRQ(ierr);
287       bnk->Hpre_inactive = bnk->H_inactive;
288     } else {
289       ierr = MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive);CHKERRQ(ierr);
290     }
291     if (bnk->bfgs_pre) {
292       ierr = PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx);CHKERRQ(ierr);
293     }
294   } else {
295     ierr = MatDuplicate(tao->hessian, MAT_COPY_VALUES, &bnk->H_inactive);CHKERRQ(ierr);
296     if (tao->hessian == tao->hessian_pre) {
297       ierr = PetscObjectReference((PetscObject)bnk->H_inactive);CHKERRQ(ierr);
298       bnk->Hpre_inactive = bnk->H_inactive;
299     } else {
300       ierr = MatDuplicate(tao->hessian_pre, MAT_COPY_VALUES, &bnk->Hpre_inactive);CHKERRQ(ierr);
301     }
302     if (bnk->bfgs_pre) {
303       ierr = PCLMVMClearIS(bnk->bfgs_pre);CHKERRQ(ierr);
304     }
305   }
306   PetscFunctionReturn(0);
307 }
308 
309 /*------------------------------------------------------------*/
310 
311 /* Routine for estimating the active set */
312 
313 PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
314 {
315   PetscErrorCode               ierr;
316   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
317   PetscBool                    hessComputed, diagExists;
318 
319   PetscFunctionBegin;
320   switch (asType) {
321   case BNK_AS_NONE:
322     ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
323     ierr = VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);CHKERRQ(ierr);
324     ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
325     ierr = ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);CHKERRQ(ierr);
326     break;
327 
328   case BNK_AS_BERTSEKAS:
329     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
330     if (bnk->M) {
331       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
332       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
333     } else {
334       hessComputed = diagExists = PETSC_FALSE;
335       if (tao->hessian) {
336         ierr = MatAssembled(tao->hessian, &hessComputed);CHKERRQ(ierr);
337       }
338       if (hessComputed) {
339         ierr = MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
340       }
341       if (diagExists) {
342         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
343         ierr = MatGetDiagonal(tao->hessian, bnk->Xwork);CHKERRQ(ierr);
344         ierr = VecAbs(bnk->Xwork);CHKERRQ(ierr);
345         ierr = VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);CHKERRQ(ierr);
346         ierr = VecReciprocal(bnk->Xwork);CHKERRQ(ierr);
347         ierr = VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);CHKERRQ(ierr);
348       } else {
349         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
350         ierr = VecCopy(bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
351       }
352     }
353     ierr = VecScale(bnk->W, -1.0);CHKERRQ(ierr);
354     ierr = TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol,
355                                    &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);CHKERRQ(ierr);
356     break;
357 
358   default:
359     break;
360   }
361   PetscFunctionReturn(0);
362 }
363 
364 /*------------------------------------------------------------*/
365 
366 /* Routine for bounding the step direction */
367 
368 PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
369 {
370   PetscErrorCode               ierr;
371   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
372 
373   PetscFunctionBegin;
374   switch (asType) {
375   case BNK_AS_NONE:
376     ierr = VecISSet(step, bnk->active_idx, 0.0);CHKERRQ(ierr);
377     break;
378 
379   case BNK_AS_BERTSEKAS:
380     ierr = TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step);CHKERRQ(ierr);
381     break;
382 
383   default:
384     break;
385   }
386   PetscFunctionReturn(0);
387 }
388 
389 /*------------------------------------------------------------*/
390 
391 /* Routine for taking a finite number of BNCG iterations to
392    accelerate Newton convergence.
393 
394    In practice, this approach simply trades off Hessian evaluations
395    for more gradient evaluations.
396 */
397 
398 PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
399 {
400   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
401   PetscErrorCode               ierr;
402 
403   PetscFunctionBegin;
404   *terminate = PETSC_FALSE;
405   if (bnk->max_cg_its > 0) {
406     /* Copy the current function value (important vectors are already shared) */
407     bnk->bncg_ctx->f = bnk->f;
408     /* Take some small finite number of BNCG iterations */
409     ierr = TaoSolve(bnk->bncg);CHKERRQ(ierr);
410     /* Add the number of gradient and function evaluations to the total */
411     tao->nfuncs += bnk->bncg->nfuncs;
412     tao->nfuncgrads += bnk->bncg->nfuncgrads;
413     tao->ngrads += bnk->bncg->ngrads;
414     tao->nhess += bnk->bncg->nhess;
415     bnk->tot_cg_its += bnk->bncg->niter;
416     /* Extract the BNCG function value out and save it into BNK */
417     bnk->f = bnk->bncg_ctx->f;
418     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) {
419       *terminate = PETSC_TRUE;
420     } else {
421       ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
422     }
423   }
424   PetscFunctionReturn(0);
425 }
426 
427 /*------------------------------------------------------------*/
428 
429 /* Routine for computing the Newton step. */
430 
431 PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason, PetscInt *step_type)
432 {
433   PetscErrorCode               ierr;
434   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
435   PetscInt                     bfgsUpdates = 0;
436   PetscInt                     kspits;
437   PetscBool                    is_lmvm;
438 
439   PetscFunctionBegin;
440   /* If there are no inactive variables left, save some computation and return an adjusted zero step
441      that has (l-x) and (u-x) for lower and upper bounded variables. */
442   if (!bnk->inactive_idx) {
443     ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
444     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
445     PetscFunctionReturn(0);
446   }
447 
448   /* Shift the reduced Hessian matrix */
449   if ((shift) && (bnk->pert > 0)) {
450     ierr = PetscObjectTypeCompare((PetscObject)tao->hessian, MATLMVM, &is_lmvm);CHKERRQ(ierr);
451     if (is_lmvm) {
452       ierr = MatShift(tao->hessian, bnk->pert);CHKERRQ(ierr);
453     } else {
454       ierr = MatShift(bnk->H_inactive, bnk->pert);CHKERRQ(ierr);
455       if (bnk->H_inactive != bnk->Hpre_inactive) {
456         ierr = MatShift(bnk->Hpre_inactive, bnk->pert);CHKERRQ(ierr);
457       }
458     }
459   }
460 
461   /* Solve the Newton system of equations */
462   tao->ksp_its = 0;
463   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
464   ierr = KSPReset(tao->ksp);CHKERRQ(ierr);
465   ierr = KSPResetFromOptions(tao->ksp);CHKERRQ(ierr);
466   ierr = KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);CHKERRQ(ierr);
467   ierr = VecCopy(bnk->unprojected_gradient, bnk->Gwork);CHKERRQ(ierr);
468   if (bnk->active_idx) {
469     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
470     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
471   } else {
472     bnk->G_inactive = bnk->unprojected_gradient;
473     bnk->X_inactive = tao->stepdirection;
474   }
475   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
476     ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
477     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
478     ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
479     tao->ksp_its+=kspits;
480     tao->ksp_tot_its+=kspits;
481     ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
482 
483     if (0.0 == tao->trust) {
484       /* Radius was uninitialized; use the norm of the direction */
485       if (bnk->dnorm > 0.0) {
486         tao->trust = bnk->dnorm;
487 
488         /* Modify the radius if it is too large or small */
489         tao->trust = PetscMax(tao->trust, bnk->min_radius);
490         tao->trust = PetscMin(tao->trust, bnk->max_radius);
491       } else {
492         /* The direction was bad; set radius to default value and re-solve
493            the trust-region subproblem to get a direction */
494         tao->trust = tao->trust0;
495 
496         /* Modify the radius if it is too large or small */
497         tao->trust = PetscMax(tao->trust, bnk->min_radius);
498         tao->trust = PetscMin(tao->trust, bnk->max_radius);
499 
500         ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
501         ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
502         ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
503         tao->ksp_its+=kspits;
504         tao->ksp_tot_its+=kspits;
505         ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
506 
507         if (bnk->dnorm == 0.0) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_PLIB, "Initial direction zero");
508       }
509     }
510   } else {
511     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
512     ierr = KSPGetIterationNumber(tao->ksp, &kspits);CHKERRQ(ierr);
513     tao->ksp_its += kspits;
514     tao->ksp_tot_its+=kspits;
515   }
516   /* Restore sub vectors back */
517   if (bnk->active_idx) {
518     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
519     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
520   }
521   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
522   ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
523   ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
524 
525   /* Record convergence reasons */
526   ierr = KSPGetConvergedReason(tao->ksp, ksp_reason);CHKERRQ(ierr);
527   if (KSP_CONVERGED_ATOL == *ksp_reason) {
528     ++bnk->ksp_atol;
529   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
530     ++bnk->ksp_rtol;
531   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
532     ++bnk->ksp_ctol;
533   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
534     ++bnk->ksp_negc;
535   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
536     ++bnk->ksp_dtol;
537   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
538     ++bnk->ksp_iter;
539   } else {
540     ++bnk->ksp_othr;
541   }
542 
543   /* Make sure the BFGS preconditioner is healthy */
544   if (bnk->M) {
545     ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
546     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
547       /* Preconditioner is numerically indefinite; reset the approximation. */
548       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
549       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
550     }
551   }
552   *step_type = BNK_NEWTON;
553   PetscFunctionReturn(0);
554 }
555 
556 /*------------------------------------------------------------*/
557 
558 /* Routine for recomputing the predicted reduction for a given step vector */
559 
560 PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
561 {
562   PetscErrorCode               ierr;
563   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
564 
565   PetscFunctionBegin;
566   /* Extract subvectors associated with the inactive set */
567   if (bnk->active_idx) {
568     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
569     ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
570     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
571   } else {
572     bnk->X_inactive = tao->stepdirection;
573     bnk->inactive_work = bnk->Xwork;
574     bnk->G_inactive = bnk->Gwork;
575   }
576   /* Recompute the predicted decrease based on the quadratic model */
577   ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
578   ierr = VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive);CHKERRQ(ierr);
579   ierr = VecDot(bnk->inactive_work, bnk->X_inactive, prered);CHKERRQ(ierr);
580   /* Restore the sub vectors */
581   if (bnk->active_idx) {
582     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
583     ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
584     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
585   }
586   PetscFunctionReturn(0);
587 }
588 
589 /*------------------------------------------------------------*/
590 
591 /* Routine for ensuring that the Newton step is a descent direction.
592 
593    The step direction falls back onto BFGS, scaled gradient and gradient steps
594    in the event that the Newton step fails the test.
595 */
596 
597 PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
598 {
599   PetscErrorCode               ierr;
600   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
601 
602   PetscReal                    gdx, e_min;
603   PetscInt                     bfgsUpdates;
604 
605   PetscFunctionBegin;
606   switch (*stepType) {
607   case BNK_NEWTON:
608     ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
609     if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
610       /* Newton step is not descent or direction produced Inf or NaN
611         Update the perturbation for next time */
612       if (bnk->pert <= 0.0) {
613         /* Initialize the perturbation */
614         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
615         if (bnk->is_gltr) {
616           ierr = KSPGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
617           bnk->pert = PetscMax(bnk->pert, -e_min);
618         }
619       } else {
620         /* Increase the perturbation */
621         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
622       }
623 
624       if (!bnk->M) {
625         /* We don't have the bfgs matrix around and updated
626           Must use gradient direction in this case */
627         ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
628         *stepType = BNK_GRADIENT;
629       } else {
630         /* Attempt to use the BFGS direction */
631         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
632 
633         /* Check for success (descent direction)
634           NOTE: Negative gdx here means not a descent direction because
635           the fall-back step is missing a negative sign. */
636         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
637         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
638           /* BFGS direction is not descent or direction produced not a number
639             We can assert bfgsUpdates > 1 in this case because
640             the first solve produces the scaled gradient direction,
641             which is guaranteed to be descent */
642 
643           /* Use steepest descent direction (scaled) */
644           ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
645           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
646           ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
647 
648           *stepType = BNK_SCALED_GRADIENT;
649         } else {
650           ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
651           if (1 == bfgsUpdates) {
652             /* The first BFGS direction is always the scaled gradient */
653             *stepType = BNK_SCALED_GRADIENT;
654           } else {
655             *stepType = BNK_BFGS;
656           }
657         }
658       }
659       /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
660       ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
661       ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
662     } else {
663       /* Computed Newton step is descent */
664       switch (ksp_reason) {
665       case KSP_DIVERGED_NANORINF:
666       case KSP_DIVERGED_BREAKDOWN:
667       case KSP_DIVERGED_INDEFINITE_MAT:
668       case KSP_DIVERGED_INDEFINITE_PC:
669       case KSP_CONVERGED_CG_NEG_CURVE:
670         /* Matrix or preconditioner is indefinite; increase perturbation */
671         if (bnk->pert <= 0.0) {
672           /* Initialize the perturbation */
673           bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
674           if (bnk->is_gltr) {
675             ierr = KSPGLTRGetMinEig(tao->ksp, &e_min);CHKERRQ(ierr);
676             bnk->pert = PetscMax(bnk->pert, -e_min);
677           }
678         } else {
679           /* Increase the perturbation */
680           bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
681         }
682         break;
683 
684       default:
685         /* Newton step computation is good; decrease perturbation */
686         bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
687         if (bnk->pert < bnk->pmin) {
688           bnk->pert = 0.0;
689         }
690         break;
691       }
692       *stepType = BNK_NEWTON;
693     }
694     break;
695 
696   case BNK_BFGS:
697     /* Check for success (descent direction) */
698     ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
699     if (gdx >= 0 || PetscIsInfOrNanReal(gdx)) {
700       /* Step is not descent or solve was not successful
701          Use steepest descent direction (scaled) */
702       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
703       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
704       ierr = MatSolve(bnk->M, tao->gradient, tao->stepdirection);CHKERRQ(ierr);
705       ierr = VecScale(tao->stepdirection,-1.0);CHKERRQ(ierr);
706       ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
707       *stepType = BNK_SCALED_GRADIENT;
708     } else {
709       *stepType = BNK_BFGS;
710     }
711     break;
712 
713   case BNK_SCALED_GRADIENT:
714     break;
715 
716   default:
717     break;
718   }
719 
720   PetscFunctionReturn(0);
721 }
722 
723 /*------------------------------------------------------------*/
724 
725 /* Routine for performing a bound-projected More-Thuente line search.
726 
727   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
728   Newton step does not produce a valid step length.
729 */
730 
731 PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
732 {
733   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
734   PetscErrorCode ierr;
735   TaoLineSearchConvergedReason ls_reason;
736 
737   PetscReal      e_min, gdx;
738   PetscInt       bfgsUpdates;
739 
740   PetscFunctionBegin;
741   /* Perform the linesearch */
742   ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
743   ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
744 
745   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_SCALED_GRADIENT && *stepType != BNK_GRADIENT) {
746     /* Linesearch failed, revert solution */
747     bnk->f = bnk->fold;
748     ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
749     ierr = VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);CHKERRQ(ierr);
750 
751     switch(*stepType) {
752     case BNK_NEWTON:
753       /* Failed to obtain acceptable iterate with Newton step
754          Update the perturbation for next time */
755       if (bnk->pert <= 0.0) {
756         /* Initialize the perturbation */
757         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
758         if (bnk->is_gltr) {
759           ierr = KSPGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
760           bnk->pert = PetscMax(bnk->pert, -e_min);
761         }
762       } else {
763         /* Increase the perturbation */
764         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
765       }
766 
767       if (!bnk->M) {
768         /* We don't have the bfgs matrix around and being updated
769            Must use gradient direction in this case */
770         ierr = VecCopy(bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
771         *stepType = BNK_GRADIENT;
772       } else {
773         /* Attempt to use the BFGS direction */
774         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
775         /* Check for success (descent direction)
776            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
777         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
778         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
779           /* BFGS direction is not descent or direction produced not a number
780              We can assert bfgsUpdates > 1 in this case
781              Use steepest descent direction (scaled) */
782           ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
783           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
784           ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
785 
786           bfgsUpdates = 1;
787           *stepType = BNK_SCALED_GRADIENT;
788         } else {
789           ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
790           if (1 == bfgsUpdates) {
791             /* The first BFGS direction is always the scaled gradient */
792             *stepType = BNK_SCALED_GRADIENT;
793           } else {
794             *stepType = BNK_BFGS;
795           }
796         }
797       }
798       break;
799 
800     case BNK_BFGS:
801       /* Can only enter if pc_type == BNK_PC_BFGS
802          Failed to obtain acceptable iterate with BFGS step
803          Attempt to use the scaled gradient direction */
804       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
805       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
806       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
807 
808       bfgsUpdates = 1;
809       *stepType = BNK_SCALED_GRADIENT;
810       break;
811     }
812     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
813     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
814     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
815 
816     /* Perform one last line search with the fall-back step */
817     ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
818     ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
819   }
820   *reason = ls_reason;
821   PetscFunctionReturn(0);
822 }
823 
824 /*------------------------------------------------------------*/
825 
826 /* Routine for updating the trust radius.
827 
828   Function features three different update methods:
829   1) Line-search step length based
830   2) Predicted decrease on the CG quadratic model
831   3) Interpolation
832 */
833 
834 PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
835 {
836   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
837   PetscErrorCode ierr;
838 
839   PetscReal      step, kappa;
840   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;
841 
842   PetscFunctionBegin;
843   /* Update trust region radius */
844   *accept = PETSC_FALSE;
845   switch(updateType) {
846   case BNK_UPDATE_STEP:
847     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
848     if (stepType == BNK_NEWTON) {
849       ierr = TaoLineSearchGetStepLength(tao->linesearch, &step);CHKERRQ(ierr);
850       if (step < bnk->nu1) {
851         /* Very bad step taken; reduce radius */
852         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
853       } else if (step < bnk->nu2) {
854         /* Reasonably bad step taken; reduce radius */
855         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
856       } else if (step < bnk->nu3) {
857         /*  Reasonable step was taken; leave radius alone */
858         if (bnk->omega3 < 1.0) {
859           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
860         } else if (bnk->omega3 > 1.0) {
861           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
862         }
863       } else if (step < bnk->nu4) {
864         /*  Full step taken; increase the radius */
865         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
866       } else {
867         /*  More than full step taken; increase the radius */
868         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
869       }
870     } else {
871       /*  Newton step was not good; reduce the radius */
872       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
873     }
874     break;
875 
876   case BNK_UPDATE_REDUCTION:
877     if (stepType == BNK_NEWTON) {
878       if ((prered < 0.0) || PetscIsInfOrNanReal(prered)) {
879         /* The predicted reduction has the wrong sign.  This cannot
880            happen in infinite precision arithmetic.  Step should
881            be rejected! */
882         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
883       } else {
884         if (PetscIsInfOrNanReal(actred)) {
885           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
886         } else {
887           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
888             kappa = 1.0;
889           } else {
890             kappa = actred / prered;
891           }
892           /* Accept or reject the step and update radius */
893           if (kappa < bnk->eta1) {
894             /* Reject the step */
895             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
896           } else {
897             /* Accept the step */
898             *accept = PETSC_TRUE;
899             /* Update the trust region radius only if the computed step is at the trust radius boundary */
900             if (bnk->dnorm == tao->trust) {
901               if (kappa < bnk->eta2) {
902                 /* Marginal bad step */
903                 tao->trust = bnk->alpha2 * tao->trust;
904               } else if (kappa < bnk->eta3) {
905                 /* Reasonable step */
906                 tao->trust = bnk->alpha3 * tao->trust;
907               } else if (kappa < bnk->eta4) {
908                 /* Good step */
909                 tao->trust = bnk->alpha4 * tao->trust;
910               } else {
911                 /* Very good step */
912                 tao->trust = bnk->alpha5 * tao->trust;
913               }
914             }
915           }
916         }
917       }
918     } else {
919       /*  Newton step was not good; reduce the radius */
920       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
921     }
922     break;
923 
924   default:
925     if (stepType == BNK_NEWTON) {
926       if (prered < 0.0) {
927         /*  The predicted reduction has the wrong sign.  This cannot */
928         /*  happen in infinite precision arithmetic.  Step should */
929         /*  be rejected! */
930         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
931       } else {
932         if (PetscIsInfOrNanReal(actred)) {
933           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
934         } else {
935           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
936             kappa = 1.0;
937           } else {
938             kappa = actred / prered;
939           }
940 
941           ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
942           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
943           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
944           tau_min = PetscMin(tau_1, tau_2);
945           tau_max = PetscMax(tau_1, tau_2);
946 
947           if (kappa >= 1.0 - bnk->mu1) {
948             /*  Great agreement */
949             *accept = PETSC_TRUE;
950             if (tau_max < 1.0) {
951               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
952             } else if (tau_max > bnk->gamma4) {
953               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
954             } else {
955               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
956             }
957           } else if (kappa >= 1.0 - bnk->mu2) {
958             /*  Good agreement */
959             *accept = PETSC_TRUE;
960             if (tau_max < bnk->gamma2) {
961               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
962             } else if (tau_max > bnk->gamma3) {
963               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
964             } else if (tau_max < 1.0) {
965               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
966             } else {
967               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
968             }
969           } else {
970             /*  Not good agreement */
971             if (tau_min > 1.0) {
972               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
973             } else if (tau_max < bnk->gamma1) {
974               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
975             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
976               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
977             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
978               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
979             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
980               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
981             } else {
982               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
983             }
984           }
985         }
986       }
987     } else {
988       /*  Newton step was not good; reduce the radius */
989       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
990     }
991     break;
992   }
993   /* Make sure the radius does not violate min and max settings */
994   tao->trust = PetscMin(tao->trust, bnk->max_radius);
995   tao->trust = PetscMax(tao->trust, bnk->min_radius);
996   PetscFunctionReturn(0);
997 }
998 
999 /* ---------------------------------------------------------- */
1000 
1001 PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
1002 {
1003   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1004 
1005   PetscFunctionBegin;
1006   switch (stepType) {
1007   case BNK_NEWTON:
1008     ++bnk->newt;
1009     break;
1010   case BNK_BFGS:
1011     ++bnk->bfgs;
1012     break;
1013   case BNK_SCALED_GRADIENT:
1014     ++bnk->sgrad;
1015     break;
1016   case BNK_GRADIENT:
1017     ++bnk->grad;
1018     break;
1019   default:
1020     break;
1021   }
1022   PetscFunctionReturn(0);
1023 }
1024 
1025 /* ---------------------------------------------------------- */
1026 
1027 PetscErrorCode TaoSetUp_BNK(Tao tao)
1028 {
1029   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1030   PetscErrorCode ierr;
1031   PetscInt       i;
1032   KSPType        ksp_type;
1033 
1034   PetscFunctionBegin;
1035   if (!tao->gradient) {
1036     ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);
1037   }
1038   if (!tao->stepdirection) {
1039     ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);
1040   }
1041   if (!bnk->W) {
1042     ierr = VecDuplicate(tao->solution,&bnk->W);CHKERRQ(ierr);
1043   }
1044   if (!bnk->Xold) {
1045     ierr = VecDuplicate(tao->solution,&bnk->Xold);CHKERRQ(ierr);
1046   }
1047   if (!bnk->Gold) {
1048     ierr = VecDuplicate(tao->solution,&bnk->Gold);CHKERRQ(ierr);
1049   }
1050   if (!bnk->Xwork) {
1051     ierr = VecDuplicate(tao->solution,&bnk->Xwork);CHKERRQ(ierr);
1052   }
1053   if (!bnk->Gwork) {
1054     ierr = VecDuplicate(tao->solution,&bnk->Gwork);CHKERRQ(ierr);
1055   }
1056   if (!bnk->unprojected_gradient) {
1057     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient);CHKERRQ(ierr);
1058   }
1059   if (!bnk->unprojected_gradient_old) {
1060     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1061   }
1062   if (!bnk->Diag_min) {
1063     ierr = VecDuplicate(tao->solution,&bnk->Diag_min);CHKERRQ(ierr);
1064   }
1065   if (!bnk->Diag_max) {
1066     ierr = VecDuplicate(tao->solution,&bnk->Diag_max);CHKERRQ(ierr);
1067   }
1068   if (bnk->max_cg_its > 0) {
1069     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
1070     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
1071     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old));CHKERRQ(ierr);
1072     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old);CHKERRQ(ierr);
1073     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
1074     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient));CHKERRQ(ierr);
1075     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient);CHKERRQ(ierr);
1076     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
1077     ierr = PetscObjectReference((PetscObject)(bnk->Gold));CHKERRQ(ierr);
1078     ierr = VecDestroy(&bnk->bncg_ctx->G_old);CHKERRQ(ierr);
1079     bnk->bncg_ctx->G_old = bnk->Gold;
1080     ierr = PetscObjectReference((PetscObject)(tao->gradient));CHKERRQ(ierr);
1081     ierr = VecDestroy(&bnk->bncg->gradient);CHKERRQ(ierr);
1082     bnk->bncg->gradient = tao->gradient;
1083     ierr = PetscObjectReference((PetscObject)(tao->stepdirection));CHKERRQ(ierr);
1084     ierr = VecDestroy(&bnk->bncg->stepdirection);CHKERRQ(ierr);
1085     bnk->bncg->stepdirection = tao->stepdirection;
1086     ierr = TaoSetInitialVector(bnk->bncg, tao->solution);CHKERRQ(ierr);
1087     /* Copy over some settings from BNK into BNCG */
1088     ierr = TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its);CHKERRQ(ierr);
1089     ierr = TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol);CHKERRQ(ierr);
1090     ierr = TaoSetFunctionLowerBound(bnk->bncg, tao->fmin);CHKERRQ(ierr);
1091     ierr = TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP);CHKERRQ(ierr);
1092     ierr = TaoSetObjectiveRoutine(bnk->bncg, tao->ops->computeobjective, tao->user_objP);CHKERRQ(ierr);
1093     ierr = TaoSetGradientRoutine(bnk->bncg, tao->ops->computegradient, tao->user_gradP);CHKERRQ(ierr);
1094     ierr = TaoSetObjectiveAndGradientRoutine(bnk->bncg, tao->ops->computeobjectiveandgradient, tao->user_objgradP);CHKERRQ(ierr);
1095     ierr = PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg));CHKERRQ(ierr);
1096     for (i=0; i<tao->numbermonitors; ++i) {
1097       ierr = TaoSetMonitor(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);CHKERRQ(ierr);
1098       ierr = PetscObjectReference((PetscObject)(tao->monitorcontext[i]));CHKERRQ(ierr);
1099     }
1100   }
1101   ierr = KSPGetType(tao->ksp,&ksp_type);CHKERRQ(ierr);
1102   ierr = PetscStrcmp(ksp_type,KSPNASH,&bnk->is_nash);CHKERRQ(ierr);
1103   ierr = PetscStrcmp(ksp_type,KSPSTCG,&bnk->is_stcg);CHKERRQ(ierr);
1104   ierr = PetscStrcmp(ksp_type,KSPGLTR,&bnk->is_gltr);CHKERRQ(ierr);
1105   bnk->X_inactive = NULL;
1106   bnk->G_inactive = NULL;
1107   bnk->inactive_work = NULL;
1108   bnk->active_work = NULL;
1109   bnk->inactive_idx = NULL;
1110   bnk->active_idx = NULL;
1111   bnk->active_lower = NULL;
1112   bnk->active_upper = NULL;
1113   bnk->active_fixed = NULL;
1114   bnk->M = NULL;
1115   bnk->H_inactive = NULL;
1116   bnk->Hpre_inactive = NULL;
1117   PetscFunctionReturn(0);
1118 }
1119 
1120 /*------------------------------------------------------------*/
1121 
1122 PetscErrorCode TaoDestroy_BNK(Tao tao)
1123 {
1124   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1125   PetscErrorCode ierr;
1126 
1127   PetscFunctionBegin;
1128   if (tao->setupcalled) {
1129     ierr = VecDestroy(&bnk->W);CHKERRQ(ierr);
1130     ierr = VecDestroy(&bnk->Xold);CHKERRQ(ierr);
1131     ierr = VecDestroy(&bnk->Gold);CHKERRQ(ierr);
1132     ierr = VecDestroy(&bnk->Xwork);CHKERRQ(ierr);
1133     ierr = VecDestroy(&bnk->Gwork);CHKERRQ(ierr);
1134     ierr = VecDestroy(&bnk->unprojected_gradient);CHKERRQ(ierr);
1135     ierr = VecDestroy(&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1136     ierr = VecDestroy(&bnk->Diag_min);CHKERRQ(ierr);
1137     ierr = VecDestroy(&bnk->Diag_max);CHKERRQ(ierr);
1138   }
1139   ierr = ISDestroy(&bnk->active_lower);CHKERRQ(ierr);
1140   ierr = ISDestroy(&bnk->active_upper);CHKERRQ(ierr);
1141   ierr = ISDestroy(&bnk->active_fixed);CHKERRQ(ierr);
1142   ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
1143   ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
1144   ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
1145   ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
1146   ierr = TaoDestroy(&bnk->bncg);CHKERRQ(ierr);
1147   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1148   PetscFunctionReturn(0);
1149 }
1150 
1151 /*------------------------------------------------------------*/
1152 
1153 PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
1154 {
1155   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1156   PetscErrorCode ierr;
1157 
1158   PetscFunctionBegin;
1159   ierr = PetscOptionsHead(PetscOptionsObject,"Newton-Krylov method for bound constrained optimization");CHKERRQ(ierr);
1160   ierr = PetscOptionsEList("-tao_bnk_init_type", "radius initialization type", "", BNK_INIT, BNK_INIT_TYPES, BNK_INIT[bnk->init_type], &bnk->init_type, NULL);CHKERRQ(ierr);
1161   ierr = PetscOptionsEList("-tao_bnk_update_type", "radius update type", "", BNK_UPDATE, BNK_UPDATE_TYPES, BNK_UPDATE[bnk->update_type], &bnk->update_type, NULL);CHKERRQ(ierr);
1162   ierr = PetscOptionsEList("-tao_bnk_as_type", "active set estimation method", "", BNK_AS, BNK_AS_TYPES, BNK_AS[bnk->as_type], &bnk->as_type, NULL);CHKERRQ(ierr);
1163   ierr = PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval,NULL);CHKERRQ(ierr);
1164   ierr = PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin,NULL);CHKERRQ(ierr);
1165   ierr = PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax,NULL);CHKERRQ(ierr);
1166   ierr = PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac,NULL);CHKERRQ(ierr);
1167   ierr = PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin,NULL);CHKERRQ(ierr);
1168   ierr = PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax,NULL);CHKERRQ(ierr);
1169   ierr = PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);CHKERRQ(ierr);
1170   ierr = PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac,NULL);CHKERRQ(ierr);
1171   ierr = PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac,NULL);CHKERRQ(ierr);
1172   ierr = PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac,NULL);CHKERRQ(ierr);
1173   ierr = PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1,NULL);CHKERRQ(ierr);
1174   ierr = PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2,NULL);CHKERRQ(ierr);
1175   ierr = PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3,NULL);CHKERRQ(ierr);
1176   ierr = PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4,NULL);CHKERRQ(ierr);
1177   ierr = PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1,NULL);CHKERRQ(ierr);
1178   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);
1179   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);
1180   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);
1181   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);
1182   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);
1183   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);
1184   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);
1185   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);
1186   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);
1187   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);
1188   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);
1189   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);
1190   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);
1191   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);
1192   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);
1193   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);
1194   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);
1195   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);
1196   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);
1197   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);
1198   ierr = PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1,NULL);CHKERRQ(ierr);
1199   ierr = PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2,NULL);CHKERRQ(ierr);
1200   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);
1201   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);
1202   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);
1203   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);
1204   ierr = PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta,NULL);CHKERRQ(ierr);
1205   ierr = PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);CHKERRQ(ierr);
1206   ierr = PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);CHKERRQ(ierr);
1207   ierr = PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);CHKERRQ(ierr);
1208   ierr = PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);CHKERRQ(ierr);
1209   ierr = PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);CHKERRQ(ierr);
1210   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);
1211   ierr = PetscOptionsTail();CHKERRQ(ierr);
1212   ierr = TaoSetFromOptions(bnk->bncg);CHKERRQ(ierr);
1213   ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr);
1214   ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr);
1215   PetscFunctionReturn(0);
1216 }
1217 
1218 /*------------------------------------------------------------*/
1219 
1220 PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
1221 {
1222   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1223   PetscInt       nrejects;
1224   PetscBool      isascii;
1225   PetscErrorCode ierr;
1226 
1227   PetscFunctionBegin;
1228   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1229   if (isascii) {
1230     ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
1231     if (bnk->M) {
1232       ierr = MatLMVMGetRejectCount(bnk->M,&nrejects);CHKERRQ(ierr);
1233       ierr = PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %D\n",nrejects);CHKERRQ(ierr);
1234     }
1235     ierr = PetscViewerASCIIPrintf(viewer, "CG steps: %D\n", bnk->tot_cg_its);CHKERRQ(ierr);
1236     ierr = PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);CHKERRQ(ierr);
1237     if (bnk->M) {
1238       ierr = PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);CHKERRQ(ierr);
1239     }
1240     ierr = PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);CHKERRQ(ierr);
1241     ierr = PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);CHKERRQ(ierr);
1242     ierr = PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");CHKERRQ(ierr);
1243     ierr = PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);CHKERRQ(ierr);
1244     ierr = PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);CHKERRQ(ierr);
1245     ierr = PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);CHKERRQ(ierr);
1246     ierr = PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);CHKERRQ(ierr);
1247     ierr = PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);CHKERRQ(ierr);
1248     ierr = PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);CHKERRQ(ierr);
1249     ierr = PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);CHKERRQ(ierr);
1250     ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
1251   }
1252   PetscFunctionReturn(0);
1253 }
1254 
1255 /* ---------------------------------------------------------- */
1256 
1257 /*MC
1258   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
1259   At each iteration, the BNK methods solve the symmetric
1260   system of equations to obtain the step diretion dk:
1261               Hk dk = -gk
1262   for free variables only. The step can be globalized either through
1263   trust-region methods, or a line search, or a heuristic mixture of both.
1264 
1265     Options Database Keys:
1266 + -max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
1267 . -init_type - trust radius initialization method ("constant", "direction", "interpolation")
1268 . -update_type - trust radius update method ("step", "direction", "interpolation")
1269 . -as_type - active-set estimation method ("none", "bertsekas")
1270 . -as_tol - (developer) initial tolerance used in estimating bounded active variables (-as_type bertsekas)
1271 . -as_step - (developer) trial step length used in estimating bounded active variables (-as_type bertsekas)
1272 . -sval - (developer) Hessian perturbation starting value
1273 . -imin - (developer) minimum initial Hessian perturbation
1274 . -imax - (developer) maximum initial Hessian perturbation
1275 . -pmin - (developer) minimum Hessian perturbation
1276 . -pmax - (developer) aximum Hessian perturbation
1277 . -pgfac - (developer) Hessian perturbation growth factor
1278 . -psfac - (developer) Hessian perturbation shrink factor
1279 . -imfac - (developer) initial merit factor for Hessian perturbation
1280 . -pmgfac - (developer) merit growth factor for Hessian perturbation
1281 . -pmsfac - (developer) merit shrink factor for Hessian perturbation
1282 . -eta1 - (developer) threshold for rejecting step (-update_type reduction)
1283 . -eta2 - (developer) threshold for accepting marginal step (-update_type reduction)
1284 . -eta3 - (developer) threshold for accepting reasonable step (-update_type reduction)
1285 . -eta4 - (developer) threshold for accepting good step (-update_type reduction)
1286 . -alpha1 - (developer) radius reduction factor for rejected step (-update_type reduction)
1287 . -alpha2 - (developer) radius reduction factor for marginally accepted bad step (-update_type reduction)
1288 . -alpha3 - (developer) radius increase factor for reasonable accepted step (-update_type reduction)
1289 . -alpha4 - (developer) radius increase factor for good accepted step (-update_type reduction)
1290 . -alpha5 - (developer) radius increase factor for very good accepted step (-update_type reduction)
1291 . -epsilon - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-update_type reduction)
1292 . -mu1 - (developer) threshold for accepting very good step (-update_type interpolation)
1293 . -mu2 - (developer) threshold for accepting good step (-update_type interpolation)
1294 . -gamma1 - (developer) radius reduction factor for rejected very bad step (-update_type interpolation)
1295 . -gamma2 - (developer) radius reduction factor for rejected bad step (-update_type interpolation)
1296 . -gamma3 - (developer) radius increase factor for accepted good step (-update_type interpolation)
1297 . -gamma4 - (developer) radius increase factor for accepted very good step (-update_type interpolation)
1298 . -theta - (developer) trust region interpolation factor (-update_type interpolation)
1299 . -nu1 - (developer) threshold for small line-search step length (-update_type step)
1300 . -nu2 - (developer) threshold for reasonable line-search step length (-update_type step)
1301 . -nu3 - (developer) threshold for large line-search step length (-update_type step)
1302 . -nu4 - (developer) threshold for very large line-search step length (-update_type step)
1303 . -omega1 - (developer) radius reduction factor for very small line-search step length (-update_type step)
1304 . -omega2 - (developer) radius reduction factor for small line-search step length (-update_type step)
1305 . -omega3 - (developer) radius factor for decent line-search step length (-update_type step)
1306 . -omega4 - (developer) radius increase factor for large line-search step length (-update_type step)
1307 . -omega5 - (developer) radius increase factor for very large line-search step length (-update_type step)
1308 . -mu1_i -  (developer) threshold for accepting very good step (-init_type interpolation)
1309 . -mu2_i -  (developer) threshold for accepting good step (-init_type interpolation)
1310 . -gamma1_i - (developer) radius reduction factor for rejected very bad step (-init_type interpolation)
1311 . -gamma2_i - (developer) radius reduction factor for rejected bad step (-init_type interpolation)
1312 . -gamma3_i - (developer) radius increase factor for accepted good step (-init_type interpolation)
1313 . -gamma4_i - (developer) radius increase factor for accepted very good step (-init_type interpolation)
1314 - -theta_i - (developer) trust region interpolation factor (-init_type interpolation)
1315 
1316   Level: beginner
1317 M*/
1318 
1319 PetscErrorCode TaoCreate_BNK(Tao tao)
1320 {
1321   TAO_BNK        *bnk;
1322   const char     *morethuente_type = TAOLINESEARCHMT;
1323   PetscErrorCode ierr;
1324   PC             pc;
1325 
1326   PetscFunctionBegin;
1327   ierr = PetscNewLog(tao,&bnk);CHKERRQ(ierr);
1328 
1329   tao->ops->setup = TaoSetUp_BNK;
1330   tao->ops->view = TaoView_BNK;
1331   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
1332   tao->ops->destroy = TaoDestroy_BNK;
1333 
1334   /*  Override default settings (unless already changed) */
1335   if (!tao->max_it_changed) tao->max_it = 50;
1336   if (!tao->trust0_changed) tao->trust0 = 100.0;
1337 
1338   tao->data = (void*)bnk;
1339 
1340   /*  Hessian shifting parameters */
1341   bnk->computehessian = TaoBNKComputeHessian;
1342   bnk->computestep = TaoBNKComputeStep;
1343 
1344   bnk->sval   = 0.0;
1345   bnk->imin   = 1.0e-4;
1346   bnk->imax   = 1.0e+2;
1347   bnk->imfac  = 1.0e-1;
1348 
1349   bnk->pmin   = 1.0e-12;
1350   bnk->pmax   = 1.0e+2;
1351   bnk->pgfac  = 1.0e+1;
1352   bnk->psfac  = 4.0e-1;
1353   bnk->pmgfac = 1.0e-1;
1354   bnk->pmsfac = 1.0e-1;
1355 
1356   /*  Default values for trust-region radius update based on steplength */
1357   bnk->nu1 = 0.25;
1358   bnk->nu2 = 0.50;
1359   bnk->nu3 = 1.00;
1360   bnk->nu4 = 1.25;
1361 
1362   bnk->omega1 = 0.25;
1363   bnk->omega2 = 0.50;
1364   bnk->omega3 = 1.00;
1365   bnk->omega4 = 2.00;
1366   bnk->omega5 = 4.00;
1367 
1368   /*  Default values for trust-region radius update based on reduction */
1369   bnk->eta1 = 1.0e-4;
1370   bnk->eta2 = 0.25;
1371   bnk->eta3 = 0.50;
1372   bnk->eta4 = 0.90;
1373 
1374   bnk->alpha1 = 0.25;
1375   bnk->alpha2 = 0.50;
1376   bnk->alpha3 = 1.00;
1377   bnk->alpha4 = 2.00;
1378   bnk->alpha5 = 4.00;
1379 
1380   /*  Default values for trust-region radius update based on interpolation */
1381   bnk->mu1 = 0.10;
1382   bnk->mu2 = 0.50;
1383 
1384   bnk->gamma1 = 0.25;
1385   bnk->gamma2 = 0.50;
1386   bnk->gamma3 = 2.00;
1387   bnk->gamma4 = 4.00;
1388 
1389   bnk->theta = 0.05;
1390 
1391   /*  Default values for trust region initialization based on interpolation */
1392   bnk->mu1_i = 0.35;
1393   bnk->mu2_i = 0.50;
1394 
1395   bnk->gamma1_i = 0.0625;
1396   bnk->gamma2_i = 0.5;
1397   bnk->gamma3_i = 2.0;
1398   bnk->gamma4_i = 5.0;
1399 
1400   bnk->theta_i = 0.25;
1401 
1402   /*  Remaining parameters */
1403   bnk->max_cg_its = 0;
1404   bnk->min_radius = 1.0e-10;
1405   bnk->max_radius = 1.0e10;
1406   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
1407   bnk->as_tol = 1.0e-3;
1408   bnk->as_step = 1.0e-3;
1409   bnk->dmin = 1.0e-6;
1410   bnk->dmax = 1.0e6;
1411 
1412   bnk->M               = NULL;
1413   bnk->bfgs_pre        = NULL;
1414   bnk->init_type       = BNK_INIT_INTERPOLATION;
1415   bnk->update_type     = BNK_UPDATE_REDUCTION;
1416   bnk->as_type         = BNK_AS_BERTSEKAS;
1417 
1418   bnk->is_stcg         = PETSC_FALSE;
1419   bnk->is_gltr         = PETSC_FALSE;
1420   bnk->is_nash         = PETSC_FALSE;
1421 
1422   /* Create the embedded BNCG solver */
1423   ierr = TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg);CHKERRQ(ierr);
1424   ierr = PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1);CHKERRQ(ierr);
1425   ierr = TaoSetOptionsPrefix(bnk->bncg, "tao_bnk_");CHKERRQ(ierr);
1426   ierr = TaoSetType(bnk->bncg, TAOBNCG);CHKERRQ(ierr);
1427 
1428   /* Create the line search */
1429   ierr = TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);CHKERRQ(ierr);
1430   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr);
1431   ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr);
1432   ierr = TaoLineSearchSetType(tao->linesearch,morethuente_type);CHKERRQ(ierr);
1433   ierr = TaoLineSearchUseTaoRoutines(tao->linesearch,tao);CHKERRQ(ierr);
1434 
1435   /*  Set linear solver to default for symmetric matrices */
1436   ierr = KSPCreate(((PetscObject)tao)->comm,&tao->ksp);CHKERRQ(ierr);
1437   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr);
1438   ierr = KSPSetOptionsPrefix(tao->ksp,"tao_bnk_");CHKERRQ(ierr);
1439   ierr = KSPSetType(tao->ksp,KSPSTCG);CHKERRQ(ierr);
1440   ierr = KSPGetPC(tao->ksp, &pc);CHKERRQ(ierr);
1441   ierr = PCSetType(pc, PCLMVM);CHKERRQ(ierr);
1442   PetscFunctionReturn(0);
1443 }
1444