xref: /petsc/src/tao/bound/impls/bnk/bnk.c (revision ccb4e88a40f0b86eaeca07ff64c64e4de2fae686)
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 = PetscObjectReference((PetscObject)tao->hessian);CHKERRQ(ierr);
124             bnk->H_inactive = tao->hessian;
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(PetscObjectComm((PetscObject)tao),PETSC_ERR_USER, "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 - (PetscReal)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 - (PetscReal)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 = TaoGradientNorm(tao, 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(PetscObjectComm((PetscObject)tao),PETSC_ERR_USER, "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 = TaoSetRecycleHistory(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->Hpre_inactive) {
279     ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
280   }
281   if (bnk->H_inactive) {
282     ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
283   }
284   if (bnk->active_idx) {
285     ierr = MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);CHKERRQ(ierr);
286     if (tao->hessian == tao->hessian_pre) {
287       ierr = PetscObjectReference((PetscObject)bnk->H_inactive);CHKERRQ(ierr);
288       bnk->Hpre_inactive = bnk->H_inactive;
289     } else {
290       ierr = MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive);CHKERRQ(ierr);
291     }
292     if (bnk->bfgs_pre) {
293       ierr = PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx);CHKERRQ(ierr);
294     }
295   } else {
296     ierr = PetscObjectReference((PetscObject)tao->hessian);CHKERRQ(ierr);
297     bnk->H_inactive = tao->hessian;
298     if (tao->hessian == tao->hessian_pre) {
299       ierr = PetscObjectReference((PetscObject)bnk->H_inactive);CHKERRQ(ierr);
300       bnk->Hpre_inactive = bnk->H_inactive;
301     } else {
302       ierr = PetscObjectReference((PetscObject)tao->hessian_pre);
303       bnk->Hpre_inactive = tao->hessian_pre;
304     }
305     if (bnk->bfgs_pre) {
306       ierr = PCLMVMClearIS(bnk->bfgs_pre);CHKERRQ(ierr);
307     }
308   }
309   PetscFunctionReturn(0);
310 }
311 
312 /*------------------------------------------------------------*/
313 
314 /* Routine for estimating the active set */
315 
316 PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
317 {
318   PetscErrorCode               ierr;
319   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
320   PetscBool                    hessComputed, diagExists;
321 
322   PetscFunctionBegin;
323   switch (asType) {
324   case BNK_AS_NONE:
325     ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
326     ierr = VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);CHKERRQ(ierr);
327     ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
328     ierr = ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);CHKERRQ(ierr);
329     break;
330 
331   case BNK_AS_BERTSEKAS:
332     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
333     if (bnk->M) {
334       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
335       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
336     } else {
337       hessComputed = diagExists = PETSC_FALSE;
338       if (tao->hessian) {
339         ierr = MatAssembled(tao->hessian, &hessComputed);CHKERRQ(ierr);
340       }
341       if (hessComputed) {
342         ierr = MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);CHKERRQ(ierr);
343       }
344       if (diagExists) {
345         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
346         ierr = MatGetDiagonal(tao->hessian, bnk->Xwork);CHKERRQ(ierr);
347         ierr = VecAbs(bnk->Xwork);CHKERRQ(ierr);
348         ierr = VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);CHKERRQ(ierr);
349         ierr = VecReciprocal(bnk->Xwork);CHKERRQ(ierr);
350         ierr = VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);CHKERRQ(ierr);
351       } else {
352         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
353         ierr = VecCopy(bnk->unprojected_gradient, bnk->W);CHKERRQ(ierr);
354       }
355     }
356     ierr = VecScale(bnk->W, -1.0);CHKERRQ(ierr);
357     ierr = TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol,
358                                    &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);CHKERRQ(ierr);
359     break;
360 
361   default:
362     break;
363   }
364   PetscFunctionReturn(0);
365 }
366 
367 /*------------------------------------------------------------*/
368 
369 /* Routine for bounding the step direction */
370 
371 PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
372 {
373   PetscErrorCode               ierr;
374   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
375 
376   PetscFunctionBegin;
377   switch (asType) {
378   case BNK_AS_NONE:
379     ierr = VecISSet(step, bnk->active_idx, 0.0);CHKERRQ(ierr);
380     break;
381 
382   case BNK_AS_BERTSEKAS:
383     ierr = TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step);CHKERRQ(ierr);
384     break;
385 
386   default:
387     break;
388   }
389   PetscFunctionReturn(0);
390 }
391 
392 /*------------------------------------------------------------*/
393 
394 /* Routine for taking a finite number of BNCG iterations to
395    accelerate Newton convergence.
396 
397    In practice, this approach simply trades off Hessian evaluations
398    for more gradient evaluations.
399 */
400 
401 PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
402 {
403   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
404   PetscErrorCode               ierr;
405 
406   PetscFunctionBegin;
407   *terminate = PETSC_FALSE;
408   if (bnk->max_cg_its > 0) {
409     /* Copy the current function value (important vectors are already shared) */
410     bnk->bncg_ctx->f = bnk->f;
411     /* Take some small finite number of BNCG iterations */
412     ierr = TaoSolve(bnk->bncg);CHKERRQ(ierr);
413     /* Add the number of gradient and function evaluations to the total */
414     tao->nfuncs += bnk->bncg->nfuncs;
415     tao->nfuncgrads += bnk->bncg->nfuncgrads;
416     tao->ngrads += bnk->bncg->ngrads;
417     tao->nhess += bnk->bncg->nhess;
418     bnk->tot_cg_its += bnk->bncg->niter;
419     /* Extract the BNCG function value out and save it into BNK */
420     bnk->f = bnk->bncg_ctx->f;
421     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) {
422       *terminate = PETSC_TRUE;
423     } else {
424       ierr = TaoBNKEstimateActiveSet(tao, bnk->as_type);CHKERRQ(ierr);
425     }
426   }
427   PetscFunctionReturn(0);
428 }
429 
430 /*------------------------------------------------------------*/
431 
432 /* Routine for computing the Newton step. */
433 
434 PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason, PetscInt *step_type)
435 {
436   PetscErrorCode               ierr;
437   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
438   PetscInt                     bfgsUpdates = 0;
439   PetscInt                     kspits;
440   PetscBool                    is_lmvm;
441 
442   PetscFunctionBegin;
443   /* If there are no inactive variables left, save some computation and return an adjusted zero step
444      that has (l-x) and (u-x) for lower and upper bounded variables. */
445   if (!bnk->inactive_idx) {
446     ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
447     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
448     PetscFunctionReturn(0);
449   }
450 
451   /* Shift the reduced Hessian matrix */
452   if ((shift) && (bnk->pert > 0)) {
453     ierr = PetscObjectTypeCompare((PetscObject)tao->hessian, MATLMVM, &is_lmvm);CHKERRQ(ierr);
454     if (is_lmvm) {
455       ierr = MatShift(tao->hessian, bnk->pert);CHKERRQ(ierr);
456     } else {
457       ierr = MatShift(bnk->H_inactive, bnk->pert);CHKERRQ(ierr);
458       if (bnk->H_inactive != bnk->Hpre_inactive) {
459         ierr = MatShift(bnk->Hpre_inactive, bnk->pert);CHKERRQ(ierr);
460       }
461     }
462   }
463 
464   /* Solve the Newton system of equations */
465   tao->ksp_its = 0;
466   ierr = VecSet(tao->stepdirection, 0.0);CHKERRQ(ierr);
467   ierr = KSPReset(tao->ksp);CHKERRQ(ierr);
468   ierr = KSPResetFromOptions(tao->ksp);CHKERRQ(ierr);
469   ierr = KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);CHKERRQ(ierr);
470   ierr = VecCopy(bnk->unprojected_gradient, bnk->Gwork);CHKERRQ(ierr);
471   if (bnk->active_idx) {
472     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
473     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
474   } else {
475     bnk->G_inactive = bnk->unprojected_gradient;
476     bnk->X_inactive = tao->stepdirection;
477   }
478   if (bnk->is_nash || bnk->is_stcg || bnk->is_gltr) {
479     ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
480     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
481     ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
482     tao->ksp_its+=kspits;
483     tao->ksp_tot_its+=kspits;
484     ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
485 
486     if (0.0 == tao->trust) {
487       /* Radius was uninitialized; use the norm of the direction */
488       if (bnk->dnorm > 0.0) {
489         tao->trust = bnk->dnorm;
490 
491         /* Modify the radius if it is too large or small */
492         tao->trust = PetscMax(tao->trust, bnk->min_radius);
493         tao->trust = PetscMin(tao->trust, bnk->max_radius);
494       } else {
495         /* The direction was bad; set radius to default value and re-solve
496            the trust-region subproblem to get a direction */
497         tao->trust = tao->trust0;
498 
499         /* Modify the radius if it is too large or small */
500         tao->trust = PetscMax(tao->trust, bnk->min_radius);
501         tao->trust = PetscMin(tao->trust, bnk->max_radius);
502 
503         ierr = KSPCGSetRadius(tao->ksp,tao->trust);CHKERRQ(ierr);
504         ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
505         ierr = KSPGetIterationNumber(tao->ksp,&kspits);CHKERRQ(ierr);
506         tao->ksp_its+=kspits;
507         tao->ksp_tot_its+=kspits;
508         ierr = KSPCGGetNormD(tao->ksp,&bnk->dnorm);CHKERRQ(ierr);
509 
510         if (bnk->dnorm == 0.0) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_PLIB, "Initial direction zero");
511       }
512     }
513   } else {
514     ierr = KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);CHKERRQ(ierr);
515     ierr = KSPGetIterationNumber(tao->ksp, &kspits);CHKERRQ(ierr);
516     tao->ksp_its += kspits;
517     tao->ksp_tot_its+=kspits;
518   }
519   /* Restore sub vectors back */
520   if (bnk->active_idx) {
521     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
522     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
523   }
524   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
525   ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
526   ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
527 
528   /* Record convergence reasons */
529   ierr = KSPGetConvergedReason(tao->ksp, ksp_reason);CHKERRQ(ierr);
530   if (KSP_CONVERGED_ATOL == *ksp_reason) {
531     ++bnk->ksp_atol;
532   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
533     ++bnk->ksp_rtol;
534   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
535     ++bnk->ksp_ctol;
536   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
537     ++bnk->ksp_negc;
538   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
539     ++bnk->ksp_dtol;
540   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
541     ++bnk->ksp_iter;
542   } else {
543     ++bnk->ksp_othr;
544   }
545 
546   /* Make sure the BFGS preconditioner is healthy */
547   if (bnk->M) {
548     ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
549     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
550       /* Preconditioner is numerically indefinite; reset the approximation. */
551       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
552       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
553     }
554   }
555   *step_type = BNK_NEWTON;
556   PetscFunctionReturn(0);
557 }
558 
559 /*------------------------------------------------------------*/
560 
561 /* Routine for recomputing the predicted reduction for a given step vector */
562 
563 PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
564 {
565   PetscErrorCode               ierr;
566   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
567 
568   PetscFunctionBegin;
569   /* Extract subvectors associated with the inactive set */
570   if (bnk->active_idx) {
571     ierr = VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
572     ierr = VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
573     ierr = VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
574   } else {
575     bnk->X_inactive = tao->stepdirection;
576     bnk->inactive_work = bnk->Xwork;
577     bnk->G_inactive = bnk->Gwork;
578   }
579   /* Recompute the predicted decrease based on the quadratic model */
580   ierr = MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);CHKERRQ(ierr);
581   ierr = VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive);CHKERRQ(ierr);
582   ierr = VecDot(bnk->inactive_work, bnk->X_inactive, prered);CHKERRQ(ierr);
583   /* Restore the sub vectors */
584   if (bnk->active_idx) {
585     ierr = VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);CHKERRQ(ierr);
586     ierr = VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);CHKERRQ(ierr);
587     ierr = VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);CHKERRQ(ierr);
588   }
589   PetscFunctionReturn(0);
590 }
591 
592 /*------------------------------------------------------------*/
593 
594 /* Routine for ensuring that the Newton step is a descent direction.
595 
596    The step direction falls back onto BFGS, scaled gradient and gradient steps
597    in the event that the Newton step fails the test.
598 */
599 
600 PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
601 {
602   PetscErrorCode               ierr;
603   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
604 
605   PetscReal                    gdx, e_min;
606   PetscInt                     bfgsUpdates;
607 
608   PetscFunctionBegin;
609   switch (*stepType) {
610   case BNK_NEWTON:
611     ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
612     if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
613       /* Newton step is not descent or direction produced Inf or NaN
614         Update the perturbation for next time */
615       if (bnk->pert <= 0.0) {
616         /* Initialize the perturbation */
617         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
618         if (bnk->is_gltr) {
619           ierr = KSPGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
620           bnk->pert = PetscMax(bnk->pert, -e_min);
621         }
622       } else {
623         /* Increase the perturbation */
624         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
625       }
626 
627       if (!bnk->M) {
628         /* We don't have the bfgs matrix around and updated
629           Must use gradient direction in this case */
630         ierr = VecCopy(tao->gradient, tao->stepdirection);CHKERRQ(ierr);
631         *stepType = BNK_GRADIENT;
632       } else {
633         /* Attempt to use the BFGS direction */
634         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
635 
636         /* Check for success (descent direction)
637           NOTE: Negative gdx here means not a descent direction because
638           the fall-back step is missing a negative sign. */
639         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
640         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
641           /* BFGS direction is not descent or direction produced not a number
642             We can assert bfgsUpdates > 1 in this case because
643             the first solve produces the scaled gradient direction,
644             which is guaranteed to be descent */
645 
646           /* Use steepest descent direction (scaled) */
647           ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
648           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
649           ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
650 
651           *stepType = BNK_SCALED_GRADIENT;
652         } else {
653           ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
654           if (1 == bfgsUpdates) {
655             /* The first BFGS direction is always the scaled gradient */
656             *stepType = BNK_SCALED_GRADIENT;
657           } else {
658             *stepType = BNK_BFGS;
659           }
660         }
661       }
662       /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
663       ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
664       ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
665     } else {
666       /* Computed Newton step is descent */
667       switch (ksp_reason) {
668       case KSP_DIVERGED_NANORINF:
669       case KSP_DIVERGED_BREAKDOWN:
670       case KSP_DIVERGED_INDEFINITE_MAT:
671       case KSP_DIVERGED_INDEFINITE_PC:
672       case KSP_CONVERGED_CG_NEG_CURVE:
673         /* Matrix or preconditioner is indefinite; increase perturbation */
674         if (bnk->pert <= 0.0) {
675           /* Initialize the perturbation */
676           bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
677           if (bnk->is_gltr) {
678             ierr = KSPGLTRGetMinEig(tao->ksp, &e_min);CHKERRQ(ierr);
679             bnk->pert = PetscMax(bnk->pert, -e_min);
680           }
681         } else {
682           /* Increase the perturbation */
683           bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
684         }
685         break;
686 
687       default:
688         /* Newton step computation is good; decrease perturbation */
689         bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
690         if (bnk->pert < bnk->pmin) {
691           bnk->pert = 0.0;
692         }
693         break;
694       }
695       *stepType = BNK_NEWTON;
696     }
697     break;
698 
699   case BNK_BFGS:
700     /* Check for success (descent direction) */
701     ierr = VecDot(tao->stepdirection, tao->gradient, &gdx);CHKERRQ(ierr);
702     if (gdx >= 0 || PetscIsInfOrNanReal(gdx)) {
703       /* Step is not descent or solve was not successful
704          Use steepest descent direction (scaled) */
705       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
706       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
707       ierr = MatSolve(bnk->M, tao->gradient, tao->stepdirection);CHKERRQ(ierr);
708       ierr = VecScale(tao->stepdirection,-1.0);CHKERRQ(ierr);
709       ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
710       *stepType = BNK_SCALED_GRADIENT;
711     } else {
712       *stepType = BNK_BFGS;
713     }
714     break;
715 
716   case BNK_SCALED_GRADIENT:
717     break;
718 
719   default:
720     break;
721   }
722 
723   PetscFunctionReturn(0);
724 }
725 
726 /*------------------------------------------------------------*/
727 
728 /* Routine for performing a bound-projected More-Thuente line search.
729 
730   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
731   Newton step does not produce a valid step length.
732 */
733 
734 PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
735 {
736   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
737   PetscErrorCode ierr;
738   TaoLineSearchConvergedReason ls_reason;
739 
740   PetscReal      e_min, gdx;
741   PetscInt       bfgsUpdates;
742 
743   PetscFunctionBegin;
744   /* Perform the linesearch */
745   ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
746   ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
747 
748   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_SCALED_GRADIENT && *stepType != BNK_GRADIENT) {
749     /* Linesearch failed, revert solution */
750     bnk->f = bnk->fold;
751     ierr = VecCopy(bnk->Xold, tao->solution);CHKERRQ(ierr);
752     ierr = VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);CHKERRQ(ierr);
753 
754     switch(*stepType) {
755     case BNK_NEWTON:
756       /* Failed to obtain acceptable iterate with Newton step
757          Update the perturbation for next time */
758       if (bnk->pert <= 0.0) {
759         /* Initialize the perturbation */
760         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
761         if (bnk->is_gltr) {
762           ierr = KSPGLTRGetMinEig(tao->ksp,&e_min);CHKERRQ(ierr);
763           bnk->pert = PetscMax(bnk->pert, -e_min);
764         }
765       } else {
766         /* Increase the perturbation */
767         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
768       }
769 
770       if (!bnk->M) {
771         /* We don't have the bfgs matrix around and being updated
772            Must use gradient direction in this case */
773         ierr = VecCopy(bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
774         *stepType = BNK_GRADIENT;
775       } else {
776         /* Attempt to use the BFGS direction */
777         ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
778         /* Check for success (descent direction)
779            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
780         ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
781         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
782           /* BFGS direction is not descent or direction produced not a number
783              We can assert bfgsUpdates > 1 in this case
784              Use steepest descent direction (scaled) */
785           ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
786           ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
787           ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
788 
789           bfgsUpdates = 1;
790           *stepType = BNK_SCALED_GRADIENT;
791         } else {
792           ierr = MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);CHKERRQ(ierr);
793           if (1 == bfgsUpdates) {
794             /* The first BFGS direction is always the scaled gradient */
795             *stepType = BNK_SCALED_GRADIENT;
796           } else {
797             *stepType = BNK_BFGS;
798           }
799         }
800       }
801       break;
802 
803     case BNK_BFGS:
804       /* Can only enter if pc_type == BNK_PC_BFGS
805          Failed to obtain acceptable iterate with BFGS step
806          Attempt to use the scaled gradient direction */
807       ierr = MatLMVMReset(bnk->M, PETSC_FALSE);CHKERRQ(ierr);
808       ierr = MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);CHKERRQ(ierr);
809       ierr = MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);CHKERRQ(ierr);
810 
811       bfgsUpdates = 1;
812       *stepType = BNK_SCALED_GRADIENT;
813       break;
814     }
815     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
816     ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr);
817     ierr = TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);CHKERRQ(ierr);
818 
819     /* Perform one last line search with the fall-back step */
820     ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);CHKERRQ(ierr);
821     ierr = TaoAddLineSearchCounts(tao);CHKERRQ(ierr);
822   }
823   *reason = ls_reason;
824   PetscFunctionReturn(0);
825 }
826 
827 /*------------------------------------------------------------*/
828 
829 /* Routine for updating the trust radius.
830 
831   Function features three different update methods:
832   1) Line-search step length based
833   2) Predicted decrease on the CG quadratic model
834   3) Interpolation
835 */
836 
837 PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
838 {
839   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
840   PetscErrorCode ierr;
841 
842   PetscReal      step, kappa;
843   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;
844 
845   PetscFunctionBegin;
846   /* Update trust region radius */
847   *accept = PETSC_FALSE;
848   switch(updateType) {
849   case BNK_UPDATE_STEP:
850     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
851     if (stepType == BNK_NEWTON) {
852       ierr = TaoLineSearchGetStepLength(tao->linesearch, &step);CHKERRQ(ierr);
853       if (step < bnk->nu1) {
854         /* Very bad step taken; reduce radius */
855         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
856       } else if (step < bnk->nu2) {
857         /* Reasonably bad step taken; reduce radius */
858         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
859       } else if (step < bnk->nu3) {
860         /*  Reasonable step was taken; leave radius alone */
861         if (bnk->omega3 < 1.0) {
862           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
863         } else if (bnk->omega3 > 1.0) {
864           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
865         }
866       } else if (step < bnk->nu4) {
867         /*  Full step taken; increase the radius */
868         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
869       } else {
870         /*  More than full step taken; increase the radius */
871         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
872       }
873     } else {
874       /*  Newton step was not good; reduce the radius */
875       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
876     }
877     break;
878 
879   case BNK_UPDATE_REDUCTION:
880     if (stepType == BNK_NEWTON) {
881       if ((prered < 0.0) || PetscIsInfOrNanReal(prered)) {
882         /* The predicted reduction has the wrong sign.  This cannot
883            happen in infinite precision arithmetic.  Step should
884            be rejected! */
885         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
886       } else {
887         if (PetscIsInfOrNanReal(actred)) {
888           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
889         } else {
890           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
891             kappa = 1.0;
892           } else {
893             kappa = actred / prered;
894           }
895           /* Accept or reject the step and update radius */
896           if (kappa < bnk->eta1) {
897             /* Reject the step */
898             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
899           } else {
900             /* Accept the step */
901             *accept = PETSC_TRUE;
902             /* Update the trust region radius only if the computed step is at the trust radius boundary */
903             if (bnk->dnorm == tao->trust) {
904               if (kappa < bnk->eta2) {
905                 /* Marginal bad step */
906                 tao->trust = bnk->alpha2 * tao->trust;
907               } else if (kappa < bnk->eta3) {
908                 /* Reasonable step */
909                 tao->trust = bnk->alpha3 * tao->trust;
910               } else if (kappa < bnk->eta4) {
911                 /* Good step */
912                 tao->trust = bnk->alpha4 * tao->trust;
913               } else {
914                 /* Very good step */
915                 tao->trust = bnk->alpha5 * tao->trust;
916               }
917             }
918           }
919         }
920       }
921     } else {
922       /*  Newton step was not good; reduce the radius */
923       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
924     }
925     break;
926 
927   default:
928     if (stepType == BNK_NEWTON) {
929       if (prered < 0.0) {
930         /*  The predicted reduction has the wrong sign.  This cannot */
931         /*  happen in infinite precision arithmetic.  Step should */
932         /*  be rejected! */
933         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
934       } else {
935         if (PetscIsInfOrNanReal(actred)) {
936           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
937         } else {
938           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
939             kappa = 1.0;
940           } else {
941             kappa = actred / prered;
942           }
943 
944           ierr = VecDot(tao->gradient, tao->stepdirection, &gdx);CHKERRQ(ierr);
945           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
946           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
947           tau_min = PetscMin(tau_1, tau_2);
948           tau_max = PetscMax(tau_1, tau_2);
949 
950           if (kappa >= 1.0 - bnk->mu1) {
951             /*  Great agreement */
952             *accept = PETSC_TRUE;
953             if (tau_max < 1.0) {
954               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
955             } else if (tau_max > bnk->gamma4) {
956               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
957             } else {
958               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
959             }
960           } else if (kappa >= 1.0 - bnk->mu2) {
961             /*  Good agreement */
962             *accept = PETSC_TRUE;
963             if (tau_max < bnk->gamma2) {
964               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
965             } else if (tau_max > bnk->gamma3) {
966               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
967             } else if (tau_max < 1.0) {
968               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
969             } else {
970               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
971             }
972           } else {
973             /*  Not good agreement */
974             if (tau_min > 1.0) {
975               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
976             } else if (tau_max < bnk->gamma1) {
977               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
978             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
979               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
980             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
981               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
982             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
983               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
984             } else {
985               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
986             }
987           }
988         }
989       }
990     } else {
991       /*  Newton step was not good; reduce the radius */
992       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
993     }
994     break;
995   }
996   /* Make sure the radius does not violate min and max settings */
997   tao->trust = PetscMin(tao->trust, bnk->max_radius);
998   tao->trust = PetscMax(tao->trust, bnk->min_radius);
999   PetscFunctionReturn(0);
1000 }
1001 
1002 /* ---------------------------------------------------------- */
1003 
1004 PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
1005 {
1006   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1007 
1008   PetscFunctionBegin;
1009   switch (stepType) {
1010   case BNK_NEWTON:
1011     ++bnk->newt;
1012     break;
1013   case BNK_BFGS:
1014     ++bnk->bfgs;
1015     break;
1016   case BNK_SCALED_GRADIENT:
1017     ++bnk->sgrad;
1018     break;
1019   case BNK_GRADIENT:
1020     ++bnk->grad;
1021     break;
1022   default:
1023     break;
1024   }
1025   PetscFunctionReturn(0);
1026 }
1027 
1028 /* ---------------------------------------------------------- */
1029 
1030 PetscErrorCode TaoSetUp_BNK(Tao tao)
1031 {
1032   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1033   PetscErrorCode ierr;
1034   PetscInt       i;
1035   KSPType        ksp_type;
1036 
1037   PetscFunctionBegin;
1038   if (!tao->gradient) {
1039     ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);
1040   }
1041   if (!tao->stepdirection) {
1042     ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);
1043   }
1044   if (!bnk->W) {
1045     ierr = VecDuplicate(tao->solution,&bnk->W);CHKERRQ(ierr);
1046   }
1047   if (!bnk->Xold) {
1048     ierr = VecDuplicate(tao->solution,&bnk->Xold);CHKERRQ(ierr);
1049   }
1050   if (!bnk->Gold) {
1051     ierr = VecDuplicate(tao->solution,&bnk->Gold);CHKERRQ(ierr);
1052   }
1053   if (!bnk->Xwork) {
1054     ierr = VecDuplicate(tao->solution,&bnk->Xwork);CHKERRQ(ierr);
1055   }
1056   if (!bnk->Gwork) {
1057     ierr = VecDuplicate(tao->solution,&bnk->Gwork);CHKERRQ(ierr);
1058   }
1059   if (!bnk->unprojected_gradient) {
1060     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient);CHKERRQ(ierr);
1061   }
1062   if (!bnk->unprojected_gradient_old) {
1063     ierr = VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1064   }
1065   if (!bnk->Diag_min) {
1066     ierr = VecDuplicate(tao->solution,&bnk->Diag_min);CHKERRQ(ierr);
1067   }
1068   if (!bnk->Diag_max) {
1069     ierr = VecDuplicate(tao->solution,&bnk->Diag_max);CHKERRQ(ierr);
1070   }
1071   if (bnk->max_cg_its > 0) {
1072     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
1073     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
1074     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old));CHKERRQ(ierr);
1075     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old);CHKERRQ(ierr);
1076     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
1077     ierr = PetscObjectReference((PetscObject)(bnk->unprojected_gradient));CHKERRQ(ierr);
1078     ierr = VecDestroy(&bnk->bncg_ctx->unprojected_gradient);CHKERRQ(ierr);
1079     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
1080     ierr = PetscObjectReference((PetscObject)(bnk->Gold));CHKERRQ(ierr);
1081     ierr = VecDestroy(&bnk->bncg_ctx->G_old);CHKERRQ(ierr);
1082     bnk->bncg_ctx->G_old = bnk->Gold;
1083     ierr = PetscObjectReference((PetscObject)(tao->gradient));CHKERRQ(ierr);
1084     ierr = VecDestroy(&bnk->bncg->gradient);CHKERRQ(ierr);
1085     bnk->bncg->gradient = tao->gradient;
1086     ierr = PetscObjectReference((PetscObject)(tao->stepdirection));CHKERRQ(ierr);
1087     ierr = VecDestroy(&bnk->bncg->stepdirection);CHKERRQ(ierr);
1088     bnk->bncg->stepdirection = tao->stepdirection;
1089     ierr = TaoSetInitialVector(bnk->bncg, tao->solution);CHKERRQ(ierr);
1090     /* Copy over some settings from BNK into BNCG */
1091     ierr = TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its);CHKERRQ(ierr);
1092     ierr = TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol);CHKERRQ(ierr);
1093     ierr = TaoSetFunctionLowerBound(bnk->bncg, tao->fmin);CHKERRQ(ierr);
1094     ierr = TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP);CHKERRQ(ierr);
1095     ierr = TaoSetObjectiveRoutine(bnk->bncg, tao->ops->computeobjective, tao->user_objP);CHKERRQ(ierr);
1096     ierr = TaoSetGradientRoutine(bnk->bncg, tao->ops->computegradient, tao->user_gradP);CHKERRQ(ierr);
1097     ierr = TaoSetObjectiveAndGradientRoutine(bnk->bncg, tao->ops->computeobjectiveandgradient, tao->user_objgradP);CHKERRQ(ierr);
1098     ierr = PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg));CHKERRQ(ierr);
1099     for (i=0; i<tao->numbermonitors; ++i) {
1100       ierr = TaoSetMonitor(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);CHKERRQ(ierr);
1101       ierr = PetscObjectReference((PetscObject)(tao->monitorcontext[i]));CHKERRQ(ierr);
1102     }
1103   }
1104   ierr = KSPGetType(tao->ksp,&ksp_type);CHKERRQ(ierr);
1105   ierr = PetscStrcmp(ksp_type,KSPNASH,&bnk->is_nash);CHKERRQ(ierr);
1106   ierr = PetscStrcmp(ksp_type,KSPSTCG,&bnk->is_stcg);CHKERRQ(ierr);
1107   ierr = PetscStrcmp(ksp_type,KSPGLTR,&bnk->is_gltr);CHKERRQ(ierr);
1108   bnk->X_inactive = NULL;
1109   bnk->G_inactive = NULL;
1110   bnk->inactive_work = NULL;
1111   bnk->active_work = NULL;
1112   bnk->inactive_idx = NULL;
1113   bnk->active_idx = NULL;
1114   bnk->active_lower = NULL;
1115   bnk->active_upper = NULL;
1116   bnk->active_fixed = NULL;
1117   bnk->M = NULL;
1118   bnk->H_inactive = NULL;
1119   bnk->Hpre_inactive = NULL;
1120   PetscFunctionReturn(0);
1121 }
1122 
1123 /*------------------------------------------------------------*/
1124 
1125 PetscErrorCode TaoDestroy_BNK(Tao tao)
1126 {
1127   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1128   PetscErrorCode ierr;
1129 
1130   PetscFunctionBegin;
1131   if (tao->setupcalled) {
1132     ierr = VecDestroy(&bnk->W);CHKERRQ(ierr);
1133     ierr = VecDestroy(&bnk->Xold);CHKERRQ(ierr);
1134     ierr = VecDestroy(&bnk->Gold);CHKERRQ(ierr);
1135     ierr = VecDestroy(&bnk->Xwork);CHKERRQ(ierr);
1136     ierr = VecDestroy(&bnk->Gwork);CHKERRQ(ierr);
1137     ierr = VecDestroy(&bnk->unprojected_gradient);CHKERRQ(ierr);
1138     ierr = VecDestroy(&bnk->unprojected_gradient_old);CHKERRQ(ierr);
1139     ierr = VecDestroy(&bnk->Diag_min);CHKERRQ(ierr);
1140     ierr = VecDestroy(&bnk->Diag_max);CHKERRQ(ierr);
1141   }
1142   ierr = ISDestroy(&bnk->active_lower);CHKERRQ(ierr);
1143   ierr = ISDestroy(&bnk->active_upper);CHKERRQ(ierr);
1144   ierr = ISDestroy(&bnk->active_fixed);CHKERRQ(ierr);
1145   ierr = ISDestroy(&bnk->active_idx);CHKERRQ(ierr);
1146   ierr = ISDestroy(&bnk->inactive_idx);CHKERRQ(ierr);
1147   ierr = MatDestroy(&bnk->Hpre_inactive);CHKERRQ(ierr);
1148   ierr = MatDestroy(&bnk->H_inactive);CHKERRQ(ierr);
1149   ierr = TaoDestroy(&bnk->bncg);CHKERRQ(ierr);
1150   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1151   PetscFunctionReturn(0);
1152 }
1153 
1154 /*------------------------------------------------------------*/
1155 
1156 PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
1157 {
1158   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1159   PetscErrorCode ierr;
1160 
1161   PetscFunctionBegin;
1162   ierr = PetscOptionsHead(PetscOptionsObject,"Newton-Krylov method for bound constrained optimization");CHKERRQ(ierr);
1163   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);
1164   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);
1165   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);
1166   ierr = PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval,NULL);CHKERRQ(ierr);
1167   ierr = PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin,NULL);CHKERRQ(ierr);
1168   ierr = PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax,NULL);CHKERRQ(ierr);
1169   ierr = PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac,NULL);CHKERRQ(ierr);
1170   ierr = PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin,NULL);CHKERRQ(ierr);
1171   ierr = PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax,NULL);CHKERRQ(ierr);
1172   ierr = PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);CHKERRQ(ierr);
1173   ierr = PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac,NULL);CHKERRQ(ierr);
1174   ierr = PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac,NULL);CHKERRQ(ierr);
1175   ierr = PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac,NULL);CHKERRQ(ierr);
1176   ierr = PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1,NULL);CHKERRQ(ierr);
1177   ierr = PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2,NULL);CHKERRQ(ierr);
1178   ierr = PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3,NULL);CHKERRQ(ierr);
1179   ierr = PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4,NULL);CHKERRQ(ierr);
1180   ierr = PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1,NULL);CHKERRQ(ierr);
1181   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);
1182   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);
1183   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);
1184   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);
1185   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);
1186   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);
1187   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);
1188   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);
1189   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);
1190   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);
1191   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);
1192   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);
1193   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);
1194   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);
1195   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);
1196   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);
1197   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);
1198   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);
1199   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);
1200   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);
1201   ierr = PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1,NULL);CHKERRQ(ierr);
1202   ierr = PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2,NULL);CHKERRQ(ierr);
1203   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);
1204   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);
1205   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);
1206   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);
1207   ierr = PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta,NULL);CHKERRQ(ierr);
1208   ierr = PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);CHKERRQ(ierr);
1209   ierr = PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);CHKERRQ(ierr);
1210   ierr = PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);CHKERRQ(ierr);
1211   ierr = PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);CHKERRQ(ierr);
1212   ierr = PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);CHKERRQ(ierr);
1213   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);
1214   ierr = PetscOptionsTail();CHKERRQ(ierr);
1215   ierr = TaoSetFromOptions(bnk->bncg);CHKERRQ(ierr);
1216   ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr);
1217   ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr);
1218   PetscFunctionReturn(0);
1219 }
1220 
1221 /*------------------------------------------------------------*/
1222 
1223 PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
1224 {
1225   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
1226   PetscInt       nrejects;
1227   PetscBool      isascii;
1228   PetscErrorCode ierr;
1229 
1230   PetscFunctionBegin;
1231   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1232   if (isascii) {
1233     ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
1234     if (bnk->M) {
1235       ierr = MatLMVMGetRejectCount(bnk->M,&nrejects);CHKERRQ(ierr);
1236       ierr = PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %D\n",nrejects);CHKERRQ(ierr);
1237     }
1238     ierr = PetscViewerASCIIPrintf(viewer, "CG steps: %D\n", bnk->tot_cg_its);CHKERRQ(ierr);
1239     ierr = PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);CHKERRQ(ierr);
1240     if (bnk->M) {
1241       ierr = PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);CHKERRQ(ierr);
1242     }
1243     ierr = PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);CHKERRQ(ierr);
1244     ierr = PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);CHKERRQ(ierr);
1245     ierr = PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");CHKERRQ(ierr);
1246     ierr = PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);CHKERRQ(ierr);
1247     ierr = PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);CHKERRQ(ierr);
1248     ierr = PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);CHKERRQ(ierr);
1249     ierr = PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);CHKERRQ(ierr);
1250     ierr = PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);CHKERRQ(ierr);
1251     ierr = PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);CHKERRQ(ierr);
1252     ierr = PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);CHKERRQ(ierr);
1253     ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
1254   }
1255   PetscFunctionReturn(0);
1256 }
1257 
1258 /* ---------------------------------------------------------- */
1259 
1260 /*MC
1261   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
1262   At each iteration, the BNK methods solve the symmetric
1263   system of equations to obtain the step diretion dk:
1264               Hk dk = -gk
1265   for free variables only. The step can be globalized either through
1266   trust-region methods, or a line search, or a heuristic mixture of both.
1267 
1268     Options Database Keys:
1269 + -max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
1270 . -init_type - trust radius initialization method ("constant", "direction", "interpolation")
1271 . -update_type - trust radius update method ("step", "direction", "interpolation")
1272 . -as_type - active-set estimation method ("none", "bertsekas")
1273 . -as_tol - (developer) initial tolerance used in estimating bounded active variables (-as_type bertsekas)
1274 . -as_step - (developer) trial step length used in estimating bounded active variables (-as_type bertsekas)
1275 . -sval - (developer) Hessian perturbation starting value
1276 . -imin - (developer) minimum initial Hessian perturbation
1277 . -imax - (developer) maximum initial Hessian perturbation
1278 . -pmin - (developer) minimum Hessian perturbation
1279 . -pmax - (developer) aximum Hessian perturbation
1280 . -pgfac - (developer) Hessian perturbation growth factor
1281 . -psfac - (developer) Hessian perturbation shrink factor
1282 . -imfac - (developer) initial merit factor for Hessian perturbation
1283 . -pmgfac - (developer) merit growth factor for Hessian perturbation
1284 . -pmsfac - (developer) merit shrink factor for Hessian perturbation
1285 . -eta1 - (developer) threshold for rejecting step (-update_type reduction)
1286 . -eta2 - (developer) threshold for accepting marginal step (-update_type reduction)
1287 . -eta3 - (developer) threshold for accepting reasonable step (-update_type reduction)
1288 . -eta4 - (developer) threshold for accepting good step (-update_type reduction)
1289 . -alpha1 - (developer) radius reduction factor for rejected step (-update_type reduction)
1290 . -alpha2 - (developer) radius reduction factor for marginally accepted bad step (-update_type reduction)
1291 . -alpha3 - (developer) radius increase factor for reasonable accepted step (-update_type reduction)
1292 . -alpha4 - (developer) radius increase factor for good accepted step (-update_type reduction)
1293 . -alpha5 - (developer) radius increase factor for very good accepted step (-update_type reduction)
1294 . -epsilon - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-update_type reduction)
1295 . -mu1 - (developer) threshold for accepting very good step (-update_type interpolation)
1296 . -mu2 - (developer) threshold for accepting good step (-update_type interpolation)
1297 . -gamma1 - (developer) radius reduction factor for rejected very bad step (-update_type interpolation)
1298 . -gamma2 - (developer) radius reduction factor for rejected bad step (-update_type interpolation)
1299 . -gamma3 - (developer) radius increase factor for accepted good step (-update_type interpolation)
1300 . -gamma4 - (developer) radius increase factor for accepted very good step (-update_type interpolation)
1301 . -theta - (developer) trust region interpolation factor (-update_type interpolation)
1302 . -nu1 - (developer) threshold for small line-search step length (-update_type step)
1303 . -nu2 - (developer) threshold for reasonable line-search step length (-update_type step)
1304 . -nu3 - (developer) threshold for large line-search step length (-update_type step)
1305 . -nu4 - (developer) threshold for very large line-search step length (-update_type step)
1306 . -omega1 - (developer) radius reduction factor for very small line-search step length (-update_type step)
1307 . -omega2 - (developer) radius reduction factor for small line-search step length (-update_type step)
1308 . -omega3 - (developer) radius factor for decent line-search step length (-update_type step)
1309 . -omega4 - (developer) radius increase factor for large line-search step length (-update_type step)
1310 . -omega5 - (developer) radius increase factor for very large line-search step length (-update_type step)
1311 . -mu1_i -  (developer) threshold for accepting very good step (-init_type interpolation)
1312 . -mu2_i -  (developer) threshold for accepting good step (-init_type interpolation)
1313 . -gamma1_i - (developer) radius reduction factor for rejected very bad step (-init_type interpolation)
1314 . -gamma2_i - (developer) radius reduction factor for rejected bad step (-init_type interpolation)
1315 . -gamma3_i - (developer) radius increase factor for accepted good step (-init_type interpolation)
1316 . -gamma4_i - (developer) radius increase factor for accepted very good step (-init_type interpolation)
1317 - -theta_i - (developer) trust region interpolation factor (-init_type interpolation)
1318 
1319   Level: beginner
1320 M*/
1321 
1322 PetscErrorCode TaoCreate_BNK(Tao tao)
1323 {
1324   TAO_BNK        *bnk;
1325   const char     *morethuente_type = TAOLINESEARCHMT;
1326   PetscErrorCode ierr;
1327   PC             pc;
1328 
1329   PetscFunctionBegin;
1330   ierr = PetscNewLog(tao,&bnk);CHKERRQ(ierr);
1331 
1332   tao->ops->setup = TaoSetUp_BNK;
1333   tao->ops->view = TaoView_BNK;
1334   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
1335   tao->ops->destroy = TaoDestroy_BNK;
1336 
1337   /*  Override default settings (unless already changed) */
1338   if (!tao->max_it_changed) tao->max_it = 50;
1339   if (!tao->trust0_changed) tao->trust0 = 100.0;
1340 
1341   tao->data = (void*)bnk;
1342 
1343   /*  Hessian shifting parameters */
1344   bnk->computehessian = TaoBNKComputeHessian;
1345   bnk->computestep = TaoBNKComputeStep;
1346 
1347   bnk->sval   = 0.0;
1348   bnk->imin   = 1.0e-4;
1349   bnk->imax   = 1.0e+2;
1350   bnk->imfac  = 1.0e-1;
1351 
1352   bnk->pmin   = 1.0e-12;
1353   bnk->pmax   = 1.0e+2;
1354   bnk->pgfac  = 1.0e+1;
1355   bnk->psfac  = 4.0e-1;
1356   bnk->pmgfac = 1.0e-1;
1357   bnk->pmsfac = 1.0e-1;
1358 
1359   /*  Default values for trust-region radius update based on steplength */
1360   bnk->nu1 = 0.25;
1361   bnk->nu2 = 0.50;
1362   bnk->nu3 = 1.00;
1363   bnk->nu4 = 1.25;
1364 
1365   bnk->omega1 = 0.25;
1366   bnk->omega2 = 0.50;
1367   bnk->omega3 = 1.00;
1368   bnk->omega4 = 2.00;
1369   bnk->omega5 = 4.00;
1370 
1371   /*  Default values for trust-region radius update based on reduction */
1372   bnk->eta1 = 1.0e-4;
1373   bnk->eta2 = 0.25;
1374   bnk->eta3 = 0.50;
1375   bnk->eta4 = 0.90;
1376 
1377   bnk->alpha1 = 0.25;
1378   bnk->alpha2 = 0.50;
1379   bnk->alpha3 = 1.00;
1380   bnk->alpha4 = 2.00;
1381   bnk->alpha5 = 4.00;
1382 
1383   /*  Default values for trust-region radius update based on interpolation */
1384   bnk->mu1 = 0.10;
1385   bnk->mu2 = 0.50;
1386 
1387   bnk->gamma1 = 0.25;
1388   bnk->gamma2 = 0.50;
1389   bnk->gamma3 = 2.00;
1390   bnk->gamma4 = 4.00;
1391 
1392   bnk->theta = 0.05;
1393 
1394   /*  Default values for trust region initialization based on interpolation */
1395   bnk->mu1_i = 0.35;
1396   bnk->mu2_i = 0.50;
1397 
1398   bnk->gamma1_i = 0.0625;
1399   bnk->gamma2_i = 0.5;
1400   bnk->gamma3_i = 2.0;
1401   bnk->gamma4_i = 5.0;
1402 
1403   bnk->theta_i = 0.25;
1404 
1405   /*  Remaining parameters */
1406   bnk->max_cg_its = 0;
1407   bnk->min_radius = 1.0e-10;
1408   bnk->max_radius = 1.0e10;
1409   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
1410   bnk->as_tol = 1.0e-3;
1411   bnk->as_step = 1.0e-3;
1412   bnk->dmin = 1.0e-6;
1413   bnk->dmax = 1.0e6;
1414 
1415   bnk->M               = NULL;
1416   bnk->bfgs_pre        = NULL;
1417   bnk->init_type       = BNK_INIT_INTERPOLATION;
1418   bnk->update_type     = BNK_UPDATE_REDUCTION;
1419   bnk->as_type         = BNK_AS_BERTSEKAS;
1420 
1421   bnk->is_stcg         = PETSC_FALSE;
1422   bnk->is_gltr         = PETSC_FALSE;
1423   bnk->is_nash         = PETSC_FALSE;
1424 
1425   /* Create the embedded BNCG solver */
1426   ierr = TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg);CHKERRQ(ierr);
1427   ierr = PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1);CHKERRQ(ierr);
1428   ierr = TaoSetOptionsPrefix(bnk->bncg, "tao_bnk_");CHKERRQ(ierr);
1429   ierr = TaoSetType(bnk->bncg, TAOBNCG);CHKERRQ(ierr);
1430 
1431   /* Create the line search */
1432   ierr = TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);CHKERRQ(ierr);
1433   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr);
1434   ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr);
1435   ierr = TaoLineSearchSetType(tao->linesearch,morethuente_type);CHKERRQ(ierr);
1436   ierr = TaoLineSearchUseTaoRoutines(tao->linesearch,tao);CHKERRQ(ierr);
1437 
1438   /*  Set linear solver to default for symmetric matrices */
1439   ierr = KSPCreate(((PetscObject)tao)->comm,&tao->ksp);CHKERRQ(ierr);
1440   ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr);
1441   ierr = KSPSetOptionsPrefix(tao->ksp,"tao_bnk_");CHKERRQ(ierr);
1442   ierr = KSPSetType(tao->ksp,KSPSTCG);CHKERRQ(ierr);
1443   ierr = KSPGetPC(tao->ksp, &pc);CHKERRQ(ierr);
1444   ierr = PCSetType(pc, PCLMVM);CHKERRQ(ierr);
1445   PetscFunctionReturn(0);
1446 }
1447