Actual source code: bnk.c

      1: #include <petsctaolinesearch.h>
      2: #include <../src/tao/bound/impls/bnk/bnk.h>
      3: #include <petscksp.h>

      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"};

      9: /*------------------------------------------------------------*/

     11: /* Routine for initializing the KSP solver, the BFGS preconditioner, and the initial trust radius estimation */

     13: PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType, PetscBool *needH)
     14: {
     15:   TAO_BNK           *bnk = (TAO_BNK *)tao->data;
     16:   PC                pc;
     17:   PetscReal         f_min, ftrial, prered, actred, kappa, sigma, resnorm;
     18:   PetscReal         tau, tau_1, tau_2, tau_max, tau_min, max_radius;
     19:   PetscBool         is_bfgs, is_jacobi, is_symmetric, sym_set;
     20:   PetscInt          n, N, nDiff;
     21:   PetscInt          i_max = 5;
     22:   PetscInt          j_max = 1;
     23:   PetscInt          i, j;
     24:   PetscVoidFunction kspTR;

     26:   /* Project the current point onto the feasible set */
     27:   TaoComputeVariableBounds(tao);
     28:   TaoSetVariableBounds(bnk->bncg, tao->XL, tao->XU);
     29:   if (tao->bounded) {
     30:     TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);
     31:   }

     33:   /* Project the initial point onto the feasible region */
     34:   TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);

     36:   /* Check convergence criteria */
     37:   TaoComputeObjectiveAndGradient(tao, tao->solution, &bnk->f, bnk->unprojected_gradient);
     38:   TaoBNKEstimateActiveSet(tao, bnk->as_type);
     39:   VecCopy(bnk->unprojected_gradient, tao->gradient);
     40:   VecISSet(tao->gradient, bnk->active_idx, 0.0);
     41:   TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm);

     43:   /* Test the initial point for convergence */
     44:   VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);
     45:   VecNorm(bnk->W, NORM_2, &resnorm);
     47:   TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);
     48:   TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);
     49:   (*tao->ops->convergencetest)(tao,tao->cnvP);
     50:   if (tao->reason != TAO_CONTINUE_ITERATING) return 0;

     52:   /* Reset KSP stopping reason counters */
     53:   bnk->ksp_atol = 0;
     54:   bnk->ksp_rtol = 0;
     55:   bnk->ksp_dtol = 0;
     56:   bnk->ksp_ctol = 0;
     57:   bnk->ksp_negc = 0;
     58:   bnk->ksp_iter = 0;
     59:   bnk->ksp_othr = 0;

     61:   /* Reset accepted step type counters */
     62:   bnk->tot_cg_its = 0;
     63:   bnk->newt = 0;
     64:   bnk->bfgs = 0;
     65:   bnk->sgrad = 0;
     66:   bnk->grad = 0;

     68:   /* Initialize the Hessian perturbation */
     69:   bnk->pert = bnk->sval;

     71:   /* Reset initial steplength to zero (this helps BNCG reset its direction internally) */
     72:   VecSet(tao->stepdirection, 0.0);

     74:   /* Allocate the vectors needed for the BFGS approximation */
     75:   KSPGetPC(tao->ksp, &pc);
     76:   PetscObjectTypeCompare((PetscObject)pc, PCLMVM, &is_bfgs);
     77:   PetscObjectTypeCompare((PetscObject)pc, PCJACOBI, &is_jacobi);
     78:   if (is_bfgs) {
     79:     bnk->bfgs_pre = pc;
     80:     PCLMVMGetMatLMVM(bnk->bfgs_pre, &bnk->M);
     81:     VecGetLocalSize(tao->solution, &n);
     82:     VecGetSize(tao->solution, &N);
     83:     MatSetSizes(bnk->M, n, n, N, N);
     84:     MatLMVMAllocate(bnk->M, tao->solution, bnk->unprojected_gradient);
     85:     MatIsSymmetricKnown(bnk->M, &sym_set, &is_symmetric);
     87:   } else if (is_jacobi) {
     88:     PCJacobiSetUseAbs(pc,PETSC_TRUE);
     89:   }

     91:   /* Prepare the min/max vectors for safeguarding diagonal scales */
     92:   VecSet(bnk->Diag_min, bnk->dmin);
     93:   VecSet(bnk->Diag_max, bnk->dmax);

     95:   /* Initialize trust-region radius.  The initialization is only performed
     96:      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
     97:   *needH = PETSC_TRUE;
     98:   PetscObjectQueryFunction((PetscObject)tao->ksp,"KSPCGSetRadius_C",&kspTR);
     99:   if (kspTR) {
    100:     switch (initType) {
    101:     case BNK_INIT_CONSTANT:
    102:       /* Use the initial radius specified */
    103:       tao->trust = tao->trust0;
    104:       break;

    106:     case BNK_INIT_INTERPOLATION:
    107:       /* Use interpolation based on the initial Hessian */
    108:       max_radius = 0.0;
    109:       tao->trust = tao->trust0;
    110:       for (j = 0; j < j_max; ++j) {
    111:         f_min = bnk->f;
    112:         sigma = 0.0;

    114:         if (*needH) {
    115:           /* Compute the Hessian at the new step, and extract the inactive subsystem */
    116:           (*bnk->computehessian)(tao);
    117:           TaoBNKEstimateActiveSet(tao, BNK_AS_NONE);
    118:           MatDestroy(&bnk->H_inactive);
    119:           if (bnk->active_idx) {
    120:             MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);
    121:           } else {
    122:             PetscObjectReference((PetscObject)tao->hessian);
    123:             bnk->H_inactive = tao->hessian;
    124:           }
    125:           *needH = PETSC_FALSE;
    126:         }

    128:         for (i = 0; i < i_max; ++i) {
    129:           /* Take a steepest descent step and snap it to bounds */
    130:           VecCopy(tao->solution, bnk->Xold);
    131:           VecAXPY(tao->solution, -tao->trust/bnk->gnorm, tao->gradient);
    132:           TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);
    133:           /* Compute the step we actually accepted */
    134:           VecCopy(tao->solution, bnk->W);
    135:           VecAXPY(bnk->W, -1.0, bnk->Xold);
    136:           /* Compute the objective at the trial */
    137:           TaoComputeObjective(tao, tao->solution, &ftrial);
    139:           VecCopy(bnk->Xold, tao->solution);
    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:             }

    148:             /* Compute the predicted and actual reduction */
    149:             if (bnk->active_idx) {
    150:               VecGetSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);
    151:               VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);
    152:             } else {
    153:               bnk->X_inactive = bnk->W;
    154:               bnk->inactive_work = bnk->Xwork;
    155:             }
    156:             MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);
    157:             VecDot(bnk->X_inactive, bnk->inactive_work, &prered);
    158:             if (bnk->active_idx) {
    159:               VecRestoreSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive);
    160:               VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);
    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:             }

    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);

    175:             if (PetscAbsScalar(kappa - (PetscReal)1.0) <= bnk->mu1_i) {
    176:               /*  Great agreement */
    177:               max_radius = PetscMax(max_radius, tao->trust);

    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);

    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:         }

    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:           VecCopy(tao->solution, bnk->Xold);
    221:           VecAXPY(tao->solution,sigma,tao->gradient);
    222:           TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);
    223:           VecCopy(tao->solution, tao->stepdirection);
    224:           VecAXPY(tao->stepdirection, -1.0, bnk->Xold);
    225:           TaoComputeGradient(tao,tao->solution,bnk->unprojected_gradient);
    226:           TaoBNKEstimateActiveSet(tao, bnk->as_type);
    227:           VecCopy(bnk->unprojected_gradient, tao->gradient);
    228:           VecISSet(tao->gradient, bnk->active_idx, 0.0);
    229:           /* Compute gradient at the new iterate and flip switch to compute the Hessian later */
    230:           TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm);
    231:           *needH = PETSC_TRUE;
    232:           /* Test the new step for convergence */
    233:           VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);
    234:           VecNorm(bnk->W, NORM_2, &resnorm);
    236:           TaoLogConvergenceHistory(tao,bnk->f,resnorm,0.0,tao->ksp_its);
    237:           TaoMonitor(tao,tao->niter,bnk->f,resnorm,0.0,1.0);
    238:           (*tao->ops->convergencetest)(tao,tao->cnvP);
    239:           if (tao->reason != TAO_CONTINUE_ITERATING) return 0;
    240:           /* active BNCG recycling early because we have a stepdirection computed */
    241:           TaoSetRecycleHistory(bnk->bncg, PETSC_TRUE);
    242:         }
    243:       }
    244:       tao->trust = PetscMax(tao->trust, max_radius);

    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;

    251:     default:
    252:       /* Norm of the first direction will initialize radius */
    253:       tao->trust = 0.0;
    254:       break;
    255:     }
    256:   }
    257:   return 0;
    258: }

    260: /*------------------------------------------------------------*/

    262: /* Routine for computing the exact Hessian and preparing the preconditioner at the new iterate */

    264: PetscErrorCode TaoBNKComputeHessian(Tao tao)
    265: {
    266:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    268:   /* Compute the Hessian */
    269:   TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);
    270:   /* Add a correction to the BFGS preconditioner */
    271:   if (bnk->M) {
    272:     MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);
    273:   }
    274:   /* Prepare the reduced sub-matrices for the inactive set */
    275:   MatDestroy(&bnk->Hpre_inactive);
    276:   MatDestroy(&bnk->H_inactive);
    277:   if (bnk->active_idx) {
    278:     MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive);
    279:     if (tao->hessian == tao->hessian_pre) {
    280:       PetscObjectReference((PetscObject)bnk->H_inactive);
    281:       bnk->Hpre_inactive = bnk->H_inactive;
    282:     } else {
    283:       MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive);
    284:     }
    285:     if (bnk->bfgs_pre) {
    286:       PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx);
    287:     }
    288:   } else {
    289:     PetscObjectReference((PetscObject)tao->hessian);
    290:     bnk->H_inactive = tao->hessian;
    291:     if (tao->hessian == tao->hessian_pre) {
    292:       PetscObjectReference((PetscObject)bnk->H_inactive);
    293:       bnk->Hpre_inactive = bnk->H_inactive;
    294:     } else {
    295:       PetscObjectReference((PetscObject)tao->hessian_pre);
    296:       bnk->Hpre_inactive = tao->hessian_pre;
    297:     }
    298:     if (bnk->bfgs_pre) {
    299:       PCLMVMClearIS(bnk->bfgs_pre);
    300:     }
    301:   }
    302:   return 0;
    303: }

    305: /*------------------------------------------------------------*/

    307: /* Routine for estimating the active set */

    309: PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
    310: {
    312:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
    313:   PetscBool      hessComputed, diagExists;

    315:   switch (asType) {
    316:   case BNK_AS_NONE:
    317:     ISDestroy(&bnk->inactive_idx);
    318:     VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx);
    319:     ISDestroy(&bnk->active_idx);
    320:     ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx);
    321:     break;

    323:   case BNK_AS_BERTSEKAS:
    324:     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
    325:     if (bnk->M) {
    326:       /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
    327:       MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W);
    328:     } else {
    329:       hessComputed = diagExists = PETSC_FALSE;
    330:       if (tao->hessian) {
    331:         MatAssembled(tao->hessian, &hessComputed);
    332:       }
    333:       if (hessComputed) {
    334:         MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists);
    335:       }
    336:       if (diagExists) {
    337:         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
    338:         MatGetDiagonal(tao->hessian, bnk->Xwork);
    339:         VecAbs(bnk->Xwork);
    340:         VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork);
    341:         VecReciprocal(bnk->Xwork);
    342:         VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient);
    343:       } else {
    344:         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
    345:         VecCopy(bnk->unprojected_gradient, bnk->W);
    346:       }
    347:     }
    348:     VecScale(bnk->W, -1.0);
    349:     TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol,
    350:                                    &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx);
    351:     break;

    353:   default:
    354:     break;
    355:   }
    356:   return 0;
    357: }

    359: /*------------------------------------------------------------*/

    361: /* Routine for bounding the step direction */

    363: PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
    364: {
    365:   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;

    367:   switch (asType) {
    368:   case BNK_AS_NONE:
    369:     VecISSet(step, bnk->active_idx, 0.0);
    370:     break;

    372:   case BNK_AS_BERTSEKAS:
    373:     TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step);
    374:     break;

    376:   default:
    377:     break;
    378:   }
    379:   return 0;
    380: }

    382: /*------------------------------------------------------------*/

    384: /* Routine for taking a finite number of BNCG iterations to
    385:    accelerate Newton convergence.

    387:    In practice, this approach simply trades off Hessian evaluations
    388:    for more gradient evaluations.
    389: */

    391: PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
    392: {
    393:   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;

    395:   *terminate = PETSC_FALSE;
    396:   if (bnk->max_cg_its > 0) {
    397:     /* Copy the current function value (important vectors are already shared) */
    398:     bnk->bncg_ctx->f = bnk->f;
    399:     /* Take some small finite number of BNCG iterations */
    400:     TaoSolve(bnk->bncg);
    401:     /* Add the number of gradient and function evaluations to the total */
    402:     tao->nfuncs += bnk->bncg->nfuncs;
    403:     tao->nfuncgrads += bnk->bncg->nfuncgrads;
    404:     tao->ngrads += bnk->bncg->ngrads;
    405:     tao->nhess += bnk->bncg->nhess;
    406:     bnk->tot_cg_its += bnk->bncg->niter;
    407:     /* Extract the BNCG function value out and save it into BNK */
    408:     bnk->f = bnk->bncg_ctx->f;
    409:     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) {
    410:       *terminate = PETSC_TRUE;
    411:     } else {
    412:       TaoBNKEstimateActiveSet(tao, bnk->as_type);
    413:     }
    414:   }
    415:   return 0;
    416: }

    418: /*------------------------------------------------------------*/

    420: /* Routine for computing the Newton step. */

    422: PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason, PetscInt *step_type)
    423: {
    424:   TAO_BNK           *bnk = (TAO_BNK *)tao->data;
    425:   PetscInt          bfgsUpdates = 0;
    426:   PetscInt          kspits;
    427:   PetscBool         is_lmvm;
    428:   PetscVoidFunction kspTR;

    430:   /* If there are no inactive variables left, save some computation and return an adjusted zero step
    431:      that has (l-x) and (u-x) for lower and upper bounded variables. */
    432:   if (!bnk->inactive_idx) {
    433:     VecSet(tao->stepdirection, 0.0);
    434:     TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);
    435:     return 0;
    436:   }

    438:   /* Shift the reduced Hessian matrix */
    439:   if (shift && bnk->pert > 0) {
    440:     PetscObjectTypeCompare((PetscObject)tao->hessian, MATLMVM, &is_lmvm);
    441:     if (is_lmvm) {
    442:       MatShift(tao->hessian, bnk->pert);
    443:     } else {
    444:       MatShift(bnk->H_inactive, bnk->pert);
    445:       if (bnk->H_inactive != bnk->Hpre_inactive) {
    446:         MatShift(bnk->Hpre_inactive, bnk->pert);
    447:       }
    448:     }
    449:   }

    451:   /* Solve the Newton system of equations */
    452:   tao->ksp_its = 0;
    453:   VecSet(tao->stepdirection, 0.0);
    454:   KSPReset(tao->ksp);
    455:   KSPResetFromOptions(tao->ksp);
    456:   KSPSetOperators(tao->ksp,bnk->H_inactive,bnk->Hpre_inactive);
    457:   VecCopy(bnk->unprojected_gradient, bnk->Gwork);
    458:   if (bnk->active_idx) {
    459:     VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);
    460:     VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);
    461:   } else {
    462:     bnk->G_inactive = bnk->unprojected_gradient;
    463:     bnk->X_inactive = tao->stepdirection;
    464:   }
    465:   PetscObjectQueryFunction((PetscObject)tao->ksp,"KSPCGSetRadius_C",&kspTR);
    466:   if (kspTR) {
    467:     KSPCGSetRadius(tao->ksp,tao->trust);
    468:     KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);
    469:     KSPGetIterationNumber(tao->ksp,&kspits);
    470:     tao->ksp_its += kspits;
    471:     tao->ksp_tot_its += kspits;
    472:     KSPCGGetNormD(tao->ksp,&bnk->dnorm);

    474:     if (0.0 == tao->trust) {
    475:       /* Radius was uninitialized; use the norm of the direction */
    476:       if (bnk->dnorm > 0.0) {
    477:         tao->trust = bnk->dnorm;

    479:         /* Modify the radius if it is too large or small */
    480:         tao->trust = PetscMax(tao->trust, bnk->min_radius);
    481:         tao->trust = PetscMin(tao->trust, bnk->max_radius);
    482:       } else {
    483:         /* The direction was bad; set radius to default value and re-solve
    484:            the trust-region subproblem to get a direction */
    485:         tao->trust = tao->trust0;

    487:         /* Modify the radius if it is too large or small */
    488:         tao->trust = PetscMax(tao->trust, bnk->min_radius);
    489:         tao->trust = PetscMin(tao->trust, bnk->max_radius);

    491:         KSPCGSetRadius(tao->ksp,tao->trust);
    492:         KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);
    493:         KSPGetIterationNumber(tao->ksp,&kspits);
    494:         tao->ksp_its += kspits;
    495:         tao->ksp_tot_its += kspits;
    496:         KSPCGGetNormD(tao->ksp,&bnk->dnorm);

    499:       }
    500:     }
    501:   } else {
    502:     KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive);
    503:     KSPGetIterationNumber(tao->ksp, &kspits);
    504:     tao->ksp_its += kspits;
    505:     tao->ksp_tot_its += kspits;
    506:   }
    507:   /* Restore sub vectors back */
    508:   if (bnk->active_idx) {
    509:     VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);
    510:     VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);
    511:   }
    512:   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
    513:   VecScale(tao->stepdirection, -1.0);
    514:   TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);

    516:   /* Record convergence reasons */
    517:   KSPGetConvergedReason(tao->ksp, ksp_reason);
    518:   if (KSP_CONVERGED_ATOL == *ksp_reason) {
    519:     ++bnk->ksp_atol;
    520:   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
    521:     ++bnk->ksp_rtol;
    522:   } else if (KSP_CONVERGED_CG_CONSTRAINED == *ksp_reason) {
    523:     ++bnk->ksp_ctol;
    524:   } else if (KSP_CONVERGED_CG_NEG_CURVE == *ksp_reason) {
    525:     ++bnk->ksp_negc;
    526:   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
    527:     ++bnk->ksp_dtol;
    528:   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
    529:     ++bnk->ksp_iter;
    530:   } else {
    531:     ++bnk->ksp_othr;
    532:   }

    534:   /* Make sure the BFGS preconditioner is healthy */
    535:   if (bnk->M) {
    536:     MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);
    537:     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
    538:       /* Preconditioner is numerically indefinite; reset the approximation. */
    539:       MatLMVMReset(bnk->M, PETSC_FALSE);
    540:       MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);
    541:     }
    542:   }
    543:   *step_type = BNK_NEWTON;
    544:   return 0;
    545: }

    547: /*------------------------------------------------------------*/

    549: /* Routine for recomputing the predicted reduction for a given step vector */

    551: PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
    552: {
    553:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    555:   /* Extract subvectors associated with the inactive set */
    556:   if (bnk->active_idx) {
    557:     VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);
    558:     VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);
    559:     VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);
    560:   } else {
    561:     bnk->X_inactive = tao->stepdirection;
    562:     bnk->inactive_work = bnk->Xwork;
    563:     bnk->G_inactive = bnk->Gwork;
    564:   }
    565:   /* Recompute the predicted decrease based on the quadratic model */
    566:   MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work);
    567:   VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive);
    568:   VecDot(bnk->inactive_work, bnk->X_inactive, prered);
    569:   /* Restore the sub vectors */
    570:   if (bnk->active_idx) {
    571:     VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive);
    572:     VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work);
    573:     VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive);
    574:   }
    575:   return 0;
    576: }

    578: /*------------------------------------------------------------*/

    580: /* Routine for ensuring that the Newton step is a descent direction.

    582:    The step direction falls back onto BFGS, scaled gradient and gradient steps
    583:    in the event that the Newton step fails the test.
    584: */

    586: PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
    587: {
    588:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
    589:   PetscReal      gdx, e_min;
    590:   PetscInt       bfgsUpdates;

    592:   switch (*stepType) {
    593:   case BNK_NEWTON:
    594:     VecDot(tao->stepdirection, tao->gradient, &gdx);
    595:     if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
    596:       /* Newton step is not descent or direction produced Inf or NaN
    597:         Update the perturbation for next time */
    598:       if (bnk->pert <= 0.0) {
    599:         PetscBool is_gltr;

    601:         /* Initialize the perturbation */
    602:         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
    603:         PetscObjectTypeCompare((PetscObject)(tao->ksp),KSPGLTR,&is_gltr);
    604:         if (is_gltr) {
    605:           KSPGLTRGetMinEig(tao->ksp,&e_min);
    606:           bnk->pert = PetscMax(bnk->pert, -e_min);
    607:         }
    608:       } else {
    609:         /* Increase the perturbation */
    610:         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
    611:       }

    613:       if (!bnk->M) {
    614:         /* We don't have the bfgs matrix around and updated
    615:           Must use gradient direction in this case */
    616:         VecCopy(tao->gradient, tao->stepdirection);
    617:         *stepType = BNK_GRADIENT;
    618:       } else {
    619:         /* Attempt to use the BFGS direction */
    620:         MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);

    622:         /* Check for success (descent direction)
    623:           NOTE: Negative gdx here means not a descent direction because
    624:           the fall-back step is missing a negative sign. */
    625:         VecDot(tao->gradient, tao->stepdirection, &gdx);
    626:         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
    627:           /* BFGS direction is not descent or direction produced not a number
    628:             We can assert bfgsUpdates > 1 in this case because
    629:             the first solve produces the scaled gradient direction,
    630:             which is guaranteed to be descent */

    632:           /* Use steepest descent direction (scaled) */
    633:           MatLMVMReset(bnk->M, PETSC_FALSE);
    634:           MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);
    635:           MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);

    637:           *stepType = BNK_SCALED_GRADIENT;
    638:         } else {
    639:           MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);
    640:           if (1 == bfgsUpdates) {
    641:             /* The first BFGS direction is always the scaled gradient */
    642:             *stepType = BNK_SCALED_GRADIENT;
    643:           } else {
    644:             *stepType = BNK_BFGS;
    645:           }
    646:         }
    647:       }
    648:       /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
    649:       VecScale(tao->stepdirection, -1.0);
    650:       TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);
    651:     } else {
    652:       /* Computed Newton step is descent */
    653:       switch (ksp_reason) {
    654:       case KSP_DIVERGED_NANORINF:
    655:       case KSP_DIVERGED_BREAKDOWN:
    656:       case KSP_DIVERGED_INDEFINITE_MAT:
    657:       case KSP_DIVERGED_INDEFINITE_PC:
    658:       case KSP_CONVERGED_CG_NEG_CURVE:
    659:         /* Matrix or preconditioner is indefinite; increase perturbation */
    660:         if (bnk->pert <= 0.0) {
    661:           PetscBool is_gltr;

    663:           /* Initialize the perturbation */
    664:           bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
    665:           PetscObjectTypeCompare((PetscObject)(tao->ksp),KSPGLTR,&is_gltr);
    666:           if (is_gltr) {
    667:             KSPGLTRGetMinEig(tao->ksp, &e_min);
    668:             bnk->pert = PetscMax(bnk->pert, -e_min);
    669:           }
    670:         } else {
    671:           /* Increase the perturbation */
    672:           bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
    673:         }
    674:         break;

    676:       default:
    677:         /* Newton step computation is good; decrease perturbation */
    678:         bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
    679:         if (bnk->pert < bnk->pmin) {
    680:           bnk->pert = 0.0;
    681:         }
    682:         break;
    683:       }
    684:       *stepType = BNK_NEWTON;
    685:     }
    686:     break;

    688:   case BNK_BFGS:
    689:     /* Check for success (descent direction) */
    690:     VecDot(tao->stepdirection, tao->gradient, &gdx);
    691:     if (gdx >= 0 || PetscIsInfOrNanReal(gdx)) {
    692:       /* Step is not descent or solve was not successful
    693:          Use steepest descent direction (scaled) */
    694:       MatLMVMReset(bnk->M, PETSC_FALSE);
    695:       MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);
    696:       MatSolve(bnk->M, tao->gradient, tao->stepdirection);
    697:       VecScale(tao->stepdirection,-1.0);
    698:       TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);
    699:       *stepType = BNK_SCALED_GRADIENT;
    700:     } else {
    701:       *stepType = BNK_BFGS;
    702:     }
    703:     break;

    705:   case BNK_SCALED_GRADIENT:
    706:     break;

    708:   default:
    709:     break;
    710:   }

    712:   return 0;
    713: }

    715: /*------------------------------------------------------------*/

    717: /* Routine for performing a bound-projected More-Thuente line search.

    719:   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
    720:   Newton step does not produce a valid step length.
    721: */

    723: PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
    724: {
    725:   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
    726:   TaoLineSearchConvergedReason ls_reason;
    727:   PetscReal                    e_min, gdx;
    728:   PetscInt                     bfgsUpdates;

    730:   /* Perform the linesearch */
    731:   TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);
    732:   TaoAddLineSearchCounts(tao);

    734:   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_SCALED_GRADIENT && *stepType != BNK_GRADIENT) {
    735:     /* Linesearch failed, revert solution */
    736:     bnk->f = bnk->fold;
    737:     VecCopy(bnk->Xold, tao->solution);
    738:     VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);

    740:     switch(*stepType) {
    741:     case BNK_NEWTON:
    742:       /* Failed to obtain acceptable iterate with Newton step
    743:          Update the perturbation for next time */
    744:       if (bnk->pert <= 0.0) {
    745:         PetscBool is_gltr;

    747:         /* Initialize the perturbation */
    748:         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
    749:         PetscObjectTypeCompare((PetscObject)(tao->ksp),KSPGLTR,&is_gltr);
    750:         if (is_gltr) {
    751:           KSPGLTRGetMinEig(tao->ksp,&e_min);
    752:           bnk->pert = PetscMax(bnk->pert, -e_min);
    753:         }
    754:       } else {
    755:         /* Increase the perturbation */
    756:         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
    757:       }

    759:       if (!bnk->M) {
    760:         /* We don't have the bfgs matrix around and being updated
    761:            Must use gradient direction in this case */
    762:         VecCopy(bnk->unprojected_gradient, tao->stepdirection);
    763:         *stepType = BNK_GRADIENT;
    764:       } else {
    765:         /* Attempt to use the BFGS direction */
    766:         MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);
    767:         /* Check for success (descent direction)
    768:            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
    769:         VecDot(tao->gradient, tao->stepdirection, &gdx);
    770:         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
    771:           /* BFGS direction is not descent or direction produced not a number
    772:              We can assert bfgsUpdates > 1 in this case
    773:              Use steepest descent direction (scaled) */
    774:           MatLMVMReset(bnk->M, PETSC_FALSE);
    775:           MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);
    776:           MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);

    778:           bfgsUpdates = 1;
    779:           *stepType = BNK_SCALED_GRADIENT;
    780:         } else {
    781:           MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates);
    782:           if (1 == bfgsUpdates) {
    783:             /* The first BFGS direction is always the scaled gradient */
    784:             *stepType = BNK_SCALED_GRADIENT;
    785:           } else {
    786:             *stepType = BNK_BFGS;
    787:           }
    788:         }
    789:       }
    790:       break;

    792:     case BNK_BFGS:
    793:       /* Can only enter if pc_type == BNK_PC_BFGS
    794:          Failed to obtain acceptable iterate with BFGS step
    795:          Attempt to use the scaled gradient direction */
    796:       MatLMVMReset(bnk->M, PETSC_FALSE);
    797:       MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient);
    798:       MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection);

    800:       bfgsUpdates = 1;
    801:       *stepType = BNK_SCALED_GRADIENT;
    802:       break;
    803:     }
    804:     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
    805:     VecScale(tao->stepdirection, -1.0);
    806:     TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection);

    808:     /* Perform one last line search with the fall-back step */
    809:     TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason);
    810:     TaoAddLineSearchCounts(tao);
    811:   }
    812:   *reason = ls_reason;
    813:   return 0;
    814: }

    816: /*------------------------------------------------------------*/

    818: /* Routine for updating the trust radius.

    820:   Function features three different update methods:
    821:   1) Line-search step length based
    822:   2) Predicted decrease on the CG quadratic model
    823:   3) Interpolation
    824: */

    826: PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
    827: {
    828:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    830:   PetscReal      step, kappa;
    831:   PetscReal      gdx, tau_1, tau_2, tau_min, tau_max;

    833:   /* Update trust region radius */
    834:   *accept = PETSC_FALSE;
    835:   switch(updateType) {
    836:   case BNK_UPDATE_STEP:
    837:     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
    838:     if (stepType == BNK_NEWTON) {
    839:       TaoLineSearchGetStepLength(tao->linesearch, &step);
    840:       if (step < bnk->nu1) {
    841:         /* Very bad step taken; reduce radius */
    842:         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
    843:       } else if (step < bnk->nu2) {
    844:         /* Reasonably bad step taken; reduce radius */
    845:         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
    846:       } else if (step < bnk->nu3) {
    847:         /*  Reasonable step was taken; leave radius alone */
    848:         if (bnk->omega3 < 1.0) {
    849:           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
    850:         } else if (bnk->omega3 > 1.0) {
    851:           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
    852:         }
    853:       } else if (step < bnk->nu4) {
    854:         /*  Full step taken; increase the radius */
    855:         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
    856:       } else {
    857:         /*  More than full step taken; increase the radius */
    858:         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
    859:       }
    860:     } else {
    861:       /*  Newton step was not good; reduce the radius */
    862:       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
    863:     }
    864:     break;

    866:   case BNK_UPDATE_REDUCTION:
    867:     if (stepType == BNK_NEWTON) {
    868:       if ((prered < 0.0) || PetscIsInfOrNanReal(prered)) {
    869:         /* The predicted reduction has the wrong sign.  This cannot
    870:            happen in infinite precision arithmetic.  Step should
    871:            be rejected! */
    872:         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
    873:       } else {
    874:         if (PetscIsInfOrNanReal(actred)) {
    875:           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
    876:         } else {
    877:           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f))*bnk->epsilon)) {
    878:             kappa = 1.0;
    879:           } else {
    880:             kappa = actred / prered;
    881:           }
    882:           /* Accept or reject the step and update radius */
    883:           if (kappa < bnk->eta1) {
    884:             /* Reject the step */
    885:             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
    886:           } else {
    887:             /* Accept the step */
    888:             *accept = PETSC_TRUE;
    889:             /* Update the trust region radius only if the computed step is at the trust radius boundary */
    890:             if (bnk->dnorm == tao->trust) {
    891:               if (kappa < bnk->eta2) {
    892:                 /* Marginal bad step */
    893:                 tao->trust = bnk->alpha2 * tao->trust;
    894:               } else if (kappa < bnk->eta3) {
    895:                 /* Reasonable step */
    896:                 tao->trust = bnk->alpha3 * tao->trust;
    897:               } else if (kappa < bnk->eta4) {
    898:                 /* Good step */
    899:                 tao->trust = bnk->alpha4 * tao->trust;
    900:               } else {
    901:                 /* Very good step */
    902:                 tao->trust = bnk->alpha5 * tao->trust;
    903:               }
    904:             }
    905:           }
    906:         }
    907:       }
    908:     } else {
    909:       /*  Newton step was not good; reduce the radius */
    910:       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
    911:     }
    912:     break;

    914:   default:
    915:     if (stepType == BNK_NEWTON) {
    916:       if (prered < 0.0) {
    917:         /*  The predicted reduction has the wrong sign.  This cannot */
    918:         /*  happen in infinite precision arithmetic.  Step should */
    919:         /*  be rejected! */
    920:         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    921:       } else {
    922:         if (PetscIsInfOrNanReal(actred)) {
    923:           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    924:         } else {
    925:           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
    926:             kappa = 1.0;
    927:           } else {
    928:             kappa = actred / prered;
    929:           }

    931:           VecDot(tao->gradient, tao->stepdirection, &gdx);
    932:           tau_1 = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
    933:           tau_2 = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
    934:           tau_min = PetscMin(tau_1, tau_2);
    935:           tau_max = PetscMax(tau_1, tau_2);

    937:           if (kappa >= 1.0 - bnk->mu1) {
    938:             /*  Great agreement */
    939:             *accept = PETSC_TRUE;
    940:             if (tau_max < 1.0) {
    941:               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
    942:             } else if (tau_max > bnk->gamma4) {
    943:               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
    944:             } else {
    945:               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
    946:             }
    947:           } else if (kappa >= 1.0 - bnk->mu2) {
    948:             /*  Good agreement */
    949:             *accept = PETSC_TRUE;
    950:             if (tau_max < bnk->gamma2) {
    951:               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
    952:             } else if (tau_max > bnk->gamma3) {
    953:               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
    954:             } else if (tau_max < 1.0) {
    955:               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
    956:             } else {
    957:               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
    958:             }
    959:           } else {
    960:             /*  Not good agreement */
    961:             if (tau_min > 1.0) {
    962:               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
    963:             } else if (tau_max < bnk->gamma1) {
    964:               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    965:             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
    966:               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    967:             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
    968:               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
    969:             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
    970:               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
    971:             } else {
    972:               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
    973:             }
    974:           }
    975:         }
    976:       }
    977:     } else {
    978:       /*  Newton step was not good; reduce the radius */
    979:       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
    980:     }
    981:     break;
    982:   }
    983:   /* Make sure the radius does not violate min and max settings */
    984:   tao->trust = PetscMin(tao->trust, bnk->max_radius);
    985:   tao->trust = PetscMax(tao->trust, bnk->min_radius);
    986:   return 0;
    987: }

    989: /* ---------------------------------------------------------- */

    991: PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
    992: {
    993:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    995:   switch (stepType) {
    996:   case BNK_NEWTON:
    997:     ++bnk->newt;
    998:     break;
    999:   case BNK_BFGS:
    1000:     ++bnk->bfgs;
    1001:     break;
    1002:   case BNK_SCALED_GRADIENT:
    1003:     ++bnk->sgrad;
    1004:     break;
    1005:   case BNK_GRADIENT:
    1006:     ++bnk->grad;
    1007:     break;
    1008:   default:
    1009:     break;
    1010:   }
    1011:   return 0;
    1012: }

    1014: /* ---------------------------------------------------------- */

    1016: PetscErrorCode TaoSetUp_BNK(Tao tao)
    1017: {
    1018:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
    1019:   PetscInt       i;

    1021:   if (!tao->gradient) {
    1022:     VecDuplicate(tao->solution,&tao->gradient);
    1023:   }
    1024:   if (!tao->stepdirection) {
    1025:     VecDuplicate(tao->solution,&tao->stepdirection);
    1026:   }
    1027:   if (!bnk->W) {
    1028:     VecDuplicate(tao->solution,&bnk->W);
    1029:   }
    1030:   if (!bnk->Xold) {
    1031:     VecDuplicate(tao->solution,&bnk->Xold);
    1032:   }
    1033:   if (!bnk->Gold) {
    1034:     VecDuplicate(tao->solution,&bnk->Gold);
    1035:   }
    1036:   if (!bnk->Xwork) {
    1037:     VecDuplicate(tao->solution,&bnk->Xwork);
    1038:   }
    1039:   if (!bnk->Gwork) {
    1040:     VecDuplicate(tao->solution,&bnk->Gwork);
    1041:   }
    1042:   if (!bnk->unprojected_gradient) {
    1043:     VecDuplicate(tao->solution,&bnk->unprojected_gradient);
    1044:   }
    1045:   if (!bnk->unprojected_gradient_old) {
    1046:     VecDuplicate(tao->solution,&bnk->unprojected_gradient_old);
    1047:   }
    1048:   if (!bnk->Diag_min) {
    1049:     VecDuplicate(tao->solution,&bnk->Diag_min);
    1050:   }
    1051:   if (!bnk->Diag_max) {
    1052:     VecDuplicate(tao->solution,&bnk->Diag_max);
    1053:   }
    1054:   if (bnk->max_cg_its > 0) {
    1055:     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
    1056:     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
    1057:     PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old));
    1058:     VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old);
    1059:     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
    1060:     PetscObjectReference((PetscObject)(bnk->unprojected_gradient));
    1061:     VecDestroy(&bnk->bncg_ctx->unprojected_gradient);
    1062:     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
    1063:     PetscObjectReference((PetscObject)(bnk->Gold));
    1064:     VecDestroy(&bnk->bncg_ctx->G_old);
    1065:     bnk->bncg_ctx->G_old = bnk->Gold;
    1066:     PetscObjectReference((PetscObject)(tao->gradient));
    1067:     VecDestroy(&bnk->bncg->gradient);
    1068:     bnk->bncg->gradient = tao->gradient;
    1069:     PetscObjectReference((PetscObject)(tao->stepdirection));
    1070:     VecDestroy(&bnk->bncg->stepdirection);
    1071:     bnk->bncg->stepdirection = tao->stepdirection;
    1072:     TaoSetSolution(bnk->bncg, tao->solution);
    1073:     /* Copy over some settings from BNK into BNCG */
    1074:     TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its);
    1075:     TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol);
    1076:     TaoSetFunctionLowerBound(bnk->bncg, tao->fmin);
    1077:     TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP);
    1078:     TaoSetObjective(bnk->bncg, tao->ops->computeobjective, tao->user_objP);
    1079:     TaoSetGradient(bnk->bncg, NULL, tao->ops->computegradient, tao->user_gradP);
    1080:     TaoSetObjectiveAndGradient(bnk->bncg, NULL, tao->ops->computeobjectiveandgradient, tao->user_objgradP);
    1081:     PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg));
    1082:     for (i=0; i<tao->numbermonitors; ++i) {
    1083:       TaoSetMonitor(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);
    1084:       PetscObjectReference((PetscObject)(tao->monitorcontext[i]));
    1085:     }
    1086:   }
    1087:   bnk->X_inactive = NULL;
    1088:   bnk->G_inactive = NULL;
    1089:   bnk->inactive_work = NULL;
    1090:   bnk->active_work = NULL;
    1091:   bnk->inactive_idx = NULL;
    1092:   bnk->active_idx = NULL;
    1093:   bnk->active_lower = NULL;
    1094:   bnk->active_upper = NULL;
    1095:   bnk->active_fixed = NULL;
    1096:   bnk->M = NULL;
    1097:   bnk->H_inactive = NULL;
    1098:   bnk->Hpre_inactive = NULL;
    1099:   return 0;
    1100: }

    1102: /*------------------------------------------------------------*/

    1104: PetscErrorCode TaoDestroy_BNK(Tao tao)
    1105: {
    1106:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    1108:   if (tao->setupcalled) {
    1109:     VecDestroy(&bnk->W);
    1110:     VecDestroy(&bnk->Xold);
    1111:     VecDestroy(&bnk->Gold);
    1112:     VecDestroy(&bnk->Xwork);
    1113:     VecDestroy(&bnk->Gwork);
    1114:     VecDestroy(&bnk->unprojected_gradient);
    1115:     VecDestroy(&bnk->unprojected_gradient_old);
    1116:     VecDestroy(&bnk->Diag_min);
    1117:     VecDestroy(&bnk->Diag_max);
    1118:   }
    1119:   ISDestroy(&bnk->active_lower);
    1120:   ISDestroy(&bnk->active_upper);
    1121:   ISDestroy(&bnk->active_fixed);
    1122:   ISDestroy(&bnk->active_idx);
    1123:   ISDestroy(&bnk->inactive_idx);
    1124:   MatDestroy(&bnk->Hpre_inactive);
    1125:   MatDestroy(&bnk->H_inactive);
    1126:   TaoDestroy(&bnk->bncg);
    1127:   PetscFree(tao->data);
    1128:   return 0;
    1129: }

    1131: /*------------------------------------------------------------*/

    1133: PetscErrorCode TaoSetFromOptions_BNK(PetscOptionItems *PetscOptionsObject,Tao tao)
    1134: {
    1135:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    1137:   PetscOptionsHead(PetscOptionsObject,"Newton-Krylov method for bound constrained optimization");
    1138:   PetscOptionsEList("-tao_bnk_init_type", "radius initialization type", "", BNK_INIT, BNK_INIT_TYPES, BNK_INIT[bnk->init_type], &bnk->init_type, NULL);
    1139:   PetscOptionsEList("-tao_bnk_update_type", "radius update type", "", BNK_UPDATE, BNK_UPDATE_TYPES, BNK_UPDATE[bnk->update_type], &bnk->update_type, NULL);
    1140:   PetscOptionsEList("-tao_bnk_as_type", "active set estimation method", "", BNK_AS, BNK_AS_TYPES, BNK_AS[bnk->as_type], &bnk->as_type, NULL);
    1141:   PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval,NULL);
    1142:   PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin,NULL);
    1143:   PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax,NULL);
    1144:   PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac,NULL);
    1145:   PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin,NULL);
    1146:   PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax,NULL);
    1147:   PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac,NULL);
    1148:   PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac,NULL);
    1149:   PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac,NULL);
    1150:   PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac,NULL);
    1151:   PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1,NULL);
    1152:   PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2,NULL);
    1153:   PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3,NULL);
    1154:   PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4,NULL);
    1155:   PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1,NULL);
    1156:   PetscOptionsReal("-tao_bnk_alpha2", "(developer) radius reduction factor for marginally accepted bad step (-tao_bnk_update_type reduction)", "", bnk->alpha2, &bnk->alpha2,NULL);
    1157:   PetscOptionsReal("-tao_bnk_alpha3", "(developer) radius increase factor for reasonable accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha3, &bnk->alpha3,NULL);
    1158:   PetscOptionsReal("-tao_bnk_alpha4", "(developer) radius increase factor for good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha4, &bnk->alpha4,NULL);
    1159:   PetscOptionsReal("-tao_bnk_alpha5", "(developer) radius increase factor for very good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha5, &bnk->alpha5,NULL);
    1160:   PetscOptionsReal("-tao_bnk_nu1", "(developer) threshold for small line-search step length (-tao_bnk_update_type step)", "", bnk->nu1, &bnk->nu1,NULL);
    1161:   PetscOptionsReal("-tao_bnk_nu2", "(developer) threshold for reasonable line-search step length (-tao_bnk_update_type step)", "", bnk->nu2, &bnk->nu2,NULL);
    1162:   PetscOptionsReal("-tao_bnk_nu3", "(developer) threshold for large line-search step length (-tao_bnk_update_type step)", "", bnk->nu3, &bnk->nu3,NULL);
    1163:   PetscOptionsReal("-tao_bnk_nu4", "(developer) threshold for very large line-search step length (-tao_bnk_update_type step)", "", bnk->nu4, &bnk->nu4,NULL);
    1164:   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);
    1165:   PetscOptionsReal("-tao_bnk_omega2", "(developer) radius reduction factor for small line-search step length (-tao_bnk_update_type step)", "", bnk->omega2, &bnk->omega2,NULL);
    1166:   PetscOptionsReal("-tao_bnk_omega3", "(developer) radius factor for decent line-search step length (-tao_bnk_update_type step)", "", bnk->omega3, &bnk->omega3,NULL);
    1167:   PetscOptionsReal("-tao_bnk_omega4", "(developer) radius increase factor for large line-search step length (-tao_bnk_update_type step)", "", bnk->omega4, &bnk->omega4,NULL);
    1168:   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);
    1169:   PetscOptionsReal("-tao_bnk_mu1_i", "(developer) threshold for accepting very good step (-tao_bnk_init_type interpolation)", "", bnk->mu1_i, &bnk->mu1_i,NULL);
    1170:   PetscOptionsReal("-tao_bnk_mu2_i", "(developer) threshold for accepting good step (-tao_bnk_init_type interpolation)", "", bnk->mu2_i, &bnk->mu2_i,NULL);
    1171:   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);
    1172:   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);
    1173:   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);
    1174:   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);
    1175:   PetscOptionsReal("-tao_bnk_theta_i", "(developer) trust region interpolation factor (-tao_bnk_init_type interpolation)", "", bnk->theta_i, &bnk->theta_i,NULL);
    1176:   PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1,NULL);
    1177:   PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2,NULL);
    1178:   PetscOptionsReal("-tao_bnk_gamma1", "(developer) radius reduction factor for rejected very bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma1, &bnk->gamma1,NULL);
    1179:   PetscOptionsReal("-tao_bnk_gamma2", "(developer) radius reduction factor for rejected bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma2, &bnk->gamma2,NULL);
    1180:   PetscOptionsReal("-tao_bnk_gamma3", "(developer) radius increase factor for accepted good step (-tao_bnk_update_type interpolation)", "", bnk->gamma3, &bnk->gamma3,NULL);
    1181:   PetscOptionsReal("-tao_bnk_gamma4", "(developer) radius increase factor for accepted very good step (-tao_bnk_update_type interpolation)", "", bnk->gamma4, &bnk->gamma4,NULL);
    1182:   PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta,NULL);
    1183:   PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius,NULL);
    1184:   PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius,NULL);
    1185:   PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon,NULL);
    1186:   PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol,NULL);
    1187:   PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step,NULL);
    1188:   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);
    1189:   PetscOptionsTail();

    1191:   TaoSetOptionsPrefix(bnk->bncg,((PetscObject)(tao))->prefix);
    1192:   TaoAppendOptionsPrefix(bnk->bncg,"tao_bnk_cg_");
    1193:   TaoSetFromOptions(bnk->bncg);

    1195:   KSPSetOptionsPrefix(tao->ksp,((PetscObject)(tao))->prefix);
    1196:   KSPAppendOptionsPrefix(tao->ksp,"tao_bnk_");
    1197:   KSPSetFromOptions(tao->ksp);
    1198:   return 0;
    1199: }

    1201: /*------------------------------------------------------------*/

    1203: PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
    1204: {
    1205:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;
    1206:   PetscInt       nrejects;
    1207:   PetscBool      isascii;

    1209:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    1210:   if (isascii) {
    1211:     PetscViewerASCIIPushTab(viewer);
    1212:     if (bnk->M) {
    1213:       MatLMVMGetRejectCount(bnk->M,&nrejects);
    1214:       PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %D\n",nrejects);
    1215:     }
    1216:     PetscViewerASCIIPrintf(viewer, "CG steps: %D\n", bnk->tot_cg_its);
    1217:     PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", bnk->newt);
    1218:     if (bnk->M) {
    1219:       PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", bnk->bfgs);
    1220:     }
    1221:     PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", bnk->sgrad);
    1222:     PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", bnk->grad);
    1223:     PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n");
    1224:     PetscViewerASCIIPrintf(viewer, "  atol: %D\n", bnk->ksp_atol);
    1225:     PetscViewerASCIIPrintf(viewer, "  rtol: %D\n", bnk->ksp_rtol);
    1226:     PetscViewerASCIIPrintf(viewer, "  ctol: %D\n", bnk->ksp_ctol);
    1227:     PetscViewerASCIIPrintf(viewer, "  negc: %D\n", bnk->ksp_negc);
    1228:     PetscViewerASCIIPrintf(viewer, "  dtol: %D\n", bnk->ksp_dtol);
    1229:     PetscViewerASCIIPrintf(viewer, "  iter: %D\n", bnk->ksp_iter);
    1230:     PetscViewerASCIIPrintf(viewer, "  othr: %D\n", bnk->ksp_othr);
    1231:     PetscViewerASCIIPopTab(viewer);
    1232:   }
    1233:   return 0;
    1234: }

    1236: /* ---------------------------------------------------------- */

    1238: /*MC
    1239:   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
    1240:   At each iteration, the BNK methods solve the symmetric
    1241:   system of equations to obtain the step diretion dk:
    1242:               Hk dk = -gk
    1243:   for free variables only. The step can be globalized either through
    1244:   trust-region methods, or a line search, or a heuristic mixture of both.

    1246:     Options Database Keys:
    1247: + -tao_bnk_max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
    1248: . -tao_bnk_init_type - trust radius initialization method ("constant", "direction", "interpolation")
    1249: . -tao_bnk_update_type - trust radius update method ("step", "direction", "interpolation")
    1250: . -tao_bnk_as_type - active-set estimation method ("none", "bertsekas")
    1251: . -tao_bnk_as_tol - (developer) initial tolerance used in estimating bounded active variables (-as_type bertsekas)
    1252: . -tao_bnk_as_step - (developer) trial step length used in estimating bounded active variables (-as_type bertsekas)
    1253: . -tao_bnk_sval - (developer) Hessian perturbation starting value
    1254: . -tao_bnk_imin - (developer) minimum initial Hessian perturbation
    1255: . -tao_bnk_imax - (developer) maximum initial Hessian perturbation
    1256: . -tao_bnk_pmin - (developer) minimum Hessian perturbation
    1257: . -tao_bnk_pmax - (developer) aximum Hessian perturbation
    1258: . -tao_bnk_pgfac - (developer) Hessian perturbation growth factor
    1259: . -tao_bnk_psfac - (developer) Hessian perturbation shrink factor
    1260: . -tao_bnk_imfac - (developer) initial merit factor for Hessian perturbation
    1261: . -tao_bnk_pmgfac - (developer) merit growth factor for Hessian perturbation
    1262: . -tao_bnk_pmsfac - (developer) merit shrink factor for Hessian perturbation
    1263: . -tao_bnk_eta1 - (developer) threshold for rejecting step (-update_type reduction)
    1264: . -tao_bnk_eta2 - (developer) threshold for accepting marginal step (-update_type reduction)
    1265: . -tao_bnk_eta3 - (developer) threshold for accepting reasonable step (-update_type reduction)
    1266: . -tao_bnk_eta4 - (developer) threshold for accepting good step (-update_type reduction)
    1267: . -tao_bnk_alpha1 - (developer) radius reduction factor for rejected step (-update_type reduction)
    1268: . -tao_bnk_alpha2 - (developer) radius reduction factor for marginally accepted bad step (-update_type reduction)
    1269: . -tao_bnk_alpha3 - (developer) radius increase factor for reasonable accepted step (-update_type reduction)
    1270: . -tao_bnk_alpha4 - (developer) radius increase factor for good accepted step (-update_type reduction)
    1271: . -tao_bnk_alpha5 - (developer) radius increase factor for very good accepted step (-update_type reduction)
    1272: . -tao_bnk_epsilon - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-update_type reduction)
    1273: . -tao_bnk_mu1 - (developer) threshold for accepting very good step (-update_type interpolation)
    1274: . -tao_bnk_mu2 - (developer) threshold for accepting good step (-update_type interpolation)
    1275: . -tao_bnk_gamma1 - (developer) radius reduction factor for rejected very bad step (-update_type interpolation)
    1276: . -tao_bnk_gamma2 - (developer) radius reduction factor for rejected bad step (-update_type interpolation)
    1277: . -tao_bnk_gamma3 - (developer) radius increase factor for accepted good step (-update_type interpolation)
    1278: . -tao_bnk_gamma4 - (developer) radius increase factor for accepted very good step (-update_type interpolation)
    1279: . -tao_bnk_theta - (developer) trust region interpolation factor (-update_type interpolation)
    1280: . -tao_bnk_nu1 - (developer) threshold for small line-search step length (-update_type step)
    1281: . -tao_bnk_nu2 - (developer) threshold for reasonable line-search step length (-update_type step)
    1282: . -tao_bnk_nu3 - (developer) threshold for large line-search step length (-update_type step)
    1283: . -tao_bnk_nu4 - (developer) threshold for very large line-search step length (-update_type step)
    1284: . -tao_bnk_omega1 - (developer) radius reduction factor for very small line-search step length (-update_type step)
    1285: . -tao_bnk_omega2 - (developer) radius reduction factor for small line-search step length (-update_type step)
    1286: . -tao_bnk_omega3 - (developer) radius factor for decent line-search step length (-update_type step)
    1287: . -tao_bnk_omega4 - (developer) radius increase factor for large line-search step length (-update_type step)
    1288: . -tao_bnk_omega5 - (developer) radius increase factor for very large line-search step length (-update_type step)
    1289: . -tao_bnk_mu1_i -  (developer) threshold for accepting very good step (-init_type interpolation)
    1290: . -tao_bnk_mu2_i -  (developer) threshold for accepting good step (-init_type interpolation)
    1291: . -tao_bnk_gamma1_i - (developer) radius reduction factor for rejected very bad step (-init_type interpolation)
    1292: . -tao_bnk_gamma2_i - (developer) radius reduction factor for rejected bad step (-init_type interpolation)
    1293: . -tao_bnk_gamma3_i - (developer) radius increase factor for accepted good step (-init_type interpolation)
    1294: . -tao_bnk_gamma4_i - (developer) radius increase factor for accepted very good step (-init_type interpolation)
    1295: - -tao_bnk_theta_i - (developer) trust region interpolation factor (-init_type interpolation)

    1297:   Level: beginner
    1298: M*/

    1300: PetscErrorCode TaoCreate_BNK(Tao tao)
    1301: {
    1302:   TAO_BNK        *bnk;
    1303:   const char     *morethuente_type = TAOLINESEARCHMT;
    1304:   PC             pc;

    1306:   PetscNewLog(tao,&bnk);

    1308:   tao->ops->setup = TaoSetUp_BNK;
    1309:   tao->ops->view = TaoView_BNK;
    1310:   tao->ops->setfromoptions = TaoSetFromOptions_BNK;
    1311:   tao->ops->destroy = TaoDestroy_BNK;

    1313:   /*  Override default settings (unless already changed) */
    1314:   if (!tao->max_it_changed) tao->max_it = 50;
    1315:   if (!tao->trust0_changed) tao->trust0 = 100.0;

    1317:   tao->data = (void*)bnk;

    1319:   /*  Hessian shifting parameters */
    1320:   bnk->computehessian = TaoBNKComputeHessian;
    1321:   bnk->computestep = TaoBNKComputeStep;

    1323:   bnk->sval   = 0.0;
    1324:   bnk->imin   = 1.0e-4;
    1325:   bnk->imax   = 1.0e+2;
    1326:   bnk->imfac  = 1.0e-1;

    1328:   bnk->pmin   = 1.0e-12;
    1329:   bnk->pmax   = 1.0e+2;
    1330:   bnk->pgfac  = 1.0e+1;
    1331:   bnk->psfac  = 4.0e-1;
    1332:   bnk->pmgfac = 1.0e-1;
    1333:   bnk->pmsfac = 1.0e-1;

    1335:   /*  Default values for trust-region radius update based on steplength */
    1336:   bnk->nu1 = 0.25;
    1337:   bnk->nu2 = 0.50;
    1338:   bnk->nu3 = 1.00;
    1339:   bnk->nu4 = 1.25;

    1341:   bnk->omega1 = 0.25;
    1342:   bnk->omega2 = 0.50;
    1343:   bnk->omega3 = 1.00;
    1344:   bnk->omega4 = 2.00;
    1345:   bnk->omega5 = 4.00;

    1347:   /*  Default values for trust-region radius update based on reduction */
    1348:   bnk->eta1 = 1.0e-4;
    1349:   bnk->eta2 = 0.25;
    1350:   bnk->eta3 = 0.50;
    1351:   bnk->eta4 = 0.90;

    1353:   bnk->alpha1 = 0.25;
    1354:   bnk->alpha2 = 0.50;
    1355:   bnk->alpha3 = 1.00;
    1356:   bnk->alpha4 = 2.00;
    1357:   bnk->alpha5 = 4.00;

    1359:   /*  Default values for trust-region radius update based on interpolation */
    1360:   bnk->mu1 = 0.10;
    1361:   bnk->mu2 = 0.50;

    1363:   bnk->gamma1 = 0.25;
    1364:   bnk->gamma2 = 0.50;
    1365:   bnk->gamma3 = 2.00;
    1366:   bnk->gamma4 = 4.00;

    1368:   bnk->theta = 0.05;

    1370:   /*  Default values for trust region initialization based on interpolation */
    1371:   bnk->mu1_i = 0.35;
    1372:   bnk->mu2_i = 0.50;

    1374:   bnk->gamma1_i = 0.0625;
    1375:   bnk->gamma2_i = 0.5;
    1376:   bnk->gamma3_i = 2.0;
    1377:   bnk->gamma4_i = 5.0;

    1379:   bnk->theta_i = 0.25;

    1381:   /*  Remaining parameters */
    1382:   bnk->max_cg_its = 0;
    1383:   bnk->min_radius = 1.0e-10;
    1384:   bnk->max_radius = 1.0e10;
    1385:   bnk->epsilon = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0/3.0);
    1386:   bnk->as_tol = 1.0e-3;
    1387:   bnk->as_step = 1.0e-3;
    1388:   bnk->dmin = 1.0e-6;
    1389:   bnk->dmax = 1.0e6;

    1391:   bnk->M           = NULL;
    1392:   bnk->bfgs_pre    = NULL;
    1393:   bnk->init_type   = BNK_INIT_INTERPOLATION;
    1394:   bnk->update_type = BNK_UPDATE_REDUCTION;
    1395:   bnk->as_type     = BNK_AS_BERTSEKAS;

    1397:   /* Create the embedded BNCG solver */
    1398:   TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg);
    1399:   PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1);
    1400:   TaoSetType(bnk->bncg, TAOBNCG);

    1402:   /* Create the line search */
    1403:   TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);
    1404:   PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);
    1405:   TaoLineSearchSetType(tao->linesearch,morethuente_type);
    1406:   TaoLineSearchUseTaoRoutines(tao->linesearch,tao);

    1408:   /*  Set linear solver to default for symmetric matrices */
    1409:   KSPCreate(((PetscObject)tao)->comm,&tao->ksp);
    1410:   PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);
    1411:   KSPSetType(tao->ksp,KSPSTCG);
    1412:   KSPGetPC(tao->ksp, &pc);
    1413:   PCSetType(pc, PCLMVM);
    1414:   return 0;
    1415: }