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: /* Extracts from the full Hessian the part associated with the current bnk->inactive_idx and set the PCLMVM preconditioner */

     11: static PetscErrorCode TaoBNKComputeSubHessian(Tao tao)
     12: {
     13:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

     15:   PetscFunctionBegin;
     16:   PetscCall(MatDestroy(&bnk->Hpre_inactive));
     17:   PetscCall(MatDestroy(&bnk->H_inactive));
     18:   if (bnk->active_idx) {
     19:     PetscCall(MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive));
     20:     if (tao->hessian == tao->hessian_pre) {
     21:       PetscCall(PetscObjectReference((PetscObject)bnk->H_inactive));
     22:       bnk->Hpre_inactive = bnk->H_inactive;
     23:     } else {
     24:       PetscCall(MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive));
     25:     }
     26:     if (bnk->bfgs_pre) PetscCall(PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx));
     27:   } else {
     28:     PetscCall(PetscObjectReference((PetscObject)tao->hessian));
     29:     bnk->H_inactive = tao->hessian;
     30:     PetscCall(PetscObjectReference((PetscObject)tao->hessian_pre));
     31:     bnk->Hpre_inactive = tao->hessian_pre;
     32:     if (bnk->bfgs_pre) PetscCall(PCLMVMClearIS(bnk->bfgs_pre));
     33:   }
     34:   PetscFunctionReturn(PETSC_SUCCESS);
     35: }

     37: /* Initializes the KSP solver, the BFGS preconditioner, and the initial trust radius estimation */

     39: PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType, PetscBool *needH)
     40: {
     41:   TAO_BNK  *bnk = (TAO_BNK *)tao->data;
     42:   PC        pc;
     43:   PetscReal f_min, ftrial, prered, actred, kappa, sigma, resnorm;
     44:   PetscReal tau, tau_1, tau_2, tau_max, tau_min, max_radius;
     45:   PetscBool is_bfgs, is_jacobi, is_symmetric, sym_set;
     46:   PetscInt  n, N, nDiff;
     47:   PetscInt  i_max = 5;
     48:   PetscInt  j_max = 1;
     49:   PetscInt  i, j;
     50:   PetscBool kspTR;

     52:   PetscFunctionBegin;
     53:   /* Project the current point onto the feasible set */
     54:   PetscCall(TaoComputeVariableBounds(tao));
     55:   PetscCall(TaoSetVariableBounds(bnk->bncg, tao->XL, tao->XU));
     56:   if (tao->bounded) PetscCall(TaoLineSearchSetVariableBounds(tao->linesearch, tao->XL, tao->XU));

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

     61:   /* Check convergence criteria */
     62:   PetscCall(TaoComputeObjectiveAndGradient(tao, tao->solution, &bnk->f, bnk->unprojected_gradient));
     63:   PetscCall(TaoBNKEstimateActiveSet(tao, bnk->as_type));
     64:   PetscCall(VecCopy(bnk->unprojected_gradient, tao->gradient));
     65:   if (bnk->active_idx) PetscCall(VecISSet(tao->gradient, bnk->active_idx, 0.0));
     66:   PetscCall(TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm));

     68:   /* Test the initial point for convergence */
     69:   PetscCall(VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W));
     70:   PetscCall(VecNorm(bnk->W, NORM_2, &resnorm));
     71:   PetscCheck(!PetscIsInfOrNanReal(bnk->f) && !PetscIsInfOrNanReal(resnorm), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated infinity or NaN");
     72:   PetscCall(TaoLogConvergenceHistory(tao, bnk->f, resnorm, 0.0, tao->ksp_its));
     73:   PetscCall(TaoMonitor(tao, tao->niter, bnk->f, resnorm, 0.0, 1.0));
     74:   PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
     75:   if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(PETSC_SUCCESS);

     77:   /* Reset KSP stopping reason counters */
     78:   bnk->ksp_atol = 0;
     79:   bnk->ksp_rtol = 0;
     80:   bnk->ksp_dtol = 0;
     81:   bnk->ksp_ctol = 0;
     82:   bnk->ksp_negc = 0;
     83:   bnk->ksp_iter = 0;
     84:   bnk->ksp_othr = 0;

     86:   /* Reset accepted step type counters */
     87:   bnk->tot_cg_its = 0;
     88:   bnk->newt       = 0;
     89:   bnk->bfgs       = 0;
     90:   bnk->sgrad      = 0;
     91:   bnk->grad       = 0;

     93:   /* Initialize the Hessian perturbation */
     94:   bnk->pert = bnk->sval;

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

     99:   /* Allocate the vectors needed for the BFGS approximation */
    100:   PetscCall(KSPGetPC(tao->ksp, &pc));
    101:   PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCLMVM, &is_bfgs));
    102:   PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCJACOBI, &is_jacobi));
    103:   if (is_bfgs) {
    104:     bnk->bfgs_pre = pc;
    105:     PetscCall(PCLMVMGetMatLMVM(bnk->bfgs_pre, &bnk->M));
    106:     PetscCall(VecGetLocalSize(tao->solution, &n));
    107:     PetscCall(VecGetSize(tao->solution, &N));
    108:     PetscCall(MatSetSizes(bnk->M, n, n, N, N));
    109:     PetscCall(MatLMVMAllocate(bnk->M, tao->solution, bnk->unprojected_gradient));
    110:     PetscCall(MatIsSymmetricKnown(bnk->M, &sym_set, &is_symmetric));
    111:     PetscCheck(sym_set && is_symmetric, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_INCOMP, "LMVM matrix in the LMVM preconditioner must be symmetric.");
    112:   } else if (is_jacobi) PetscCall(PCJacobiSetUseAbs(pc, PETSC_TRUE));

    114:   /* Prepare the min/max vectors for safeguarding diagonal scales */
    115:   PetscCall(VecSet(bnk->Diag_min, bnk->dmin));
    116:   PetscCall(VecSet(bnk->Diag_max, bnk->dmax));

    118:   /* Initialize trust-region radius.  The initialization is only performed
    119:      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
    120:   *needH = PETSC_TRUE;
    121:   PetscCall(PetscObjectHasFunction((PetscObject)tao->ksp, "KSPCGSetRadius_C", &kspTR));
    122:   if (kspTR) {
    123:     switch (initType) {
    124:     case BNK_INIT_CONSTANT:
    125:       /* Use the initial radius specified */
    126:       tao->trust = tao->trust0;
    127:       break;

    129:     case BNK_INIT_INTERPOLATION:
    130:       /* Use interpolation based on the initial Hessian */
    131:       max_radius = 0.0;
    132:       tao->trust = tao->trust0;
    133:       for (j = 0; j < j_max; ++j) {
    134:         f_min = bnk->f;
    135:         sigma = 0.0;

    137:         if (*needH) {
    138:           /* Compute the Hessian at the new step, and extract the inactive subsystem */
    139:           PetscCall((*bnk->computehessian)(tao));
    140:           PetscCall(TaoBNKEstimateActiveSet(tao, BNK_AS_NONE));
    141:           PetscCall(TaoBNKComputeSubHessian(tao));
    142:           *needH = PETSC_FALSE;
    143:         }

    145:         for (i = 0; i < i_max; ++i) {
    146:           /* Take a steepest descent step and snap it to bounds */
    147:           PetscCall(VecCopy(tao->solution, bnk->Xold));
    148:           PetscCall(VecAXPY(tao->solution, -tao->trust / bnk->gnorm, tao->gradient));
    149:           PetscCall(TaoBoundSolution(tao->solution, tao->XL, tao->XU, 0.0, &nDiff, tao->solution));
    150:           /* Compute the step we actually accepted */
    151:           PetscCall(VecCopy(tao->solution, bnk->W));
    152:           PetscCall(VecAXPY(bnk->W, -1.0, bnk->Xold));
    153:           /* Compute the objective at the trial */
    154:           PetscCall(TaoComputeObjective(tao, tao->solution, &ftrial));
    155:           PetscCheck(!PetscIsInfOrNanReal(bnk->f), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated infinity or NaN");
    156:           PetscCall(VecCopy(bnk->Xold, tao->solution));
    157:           if (PetscIsInfOrNanReal(ftrial)) {
    158:             tau = bnk->gamma1_i;
    159:           } else {
    160:             if (ftrial < f_min) {
    161:               f_min = ftrial;
    162:               sigma = -tao->trust / bnk->gnorm;
    163:             }

    165:             /* Compute the predicted and actual reduction */
    166:             if (bnk->active_idx) {
    167:               PetscCall(VecGetSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive));
    168:               PetscCall(VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
    169:             } else {
    170:               bnk->X_inactive    = bnk->W;
    171:               bnk->inactive_work = bnk->Xwork;
    172:             }
    173:             PetscCall(MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work));
    174:             PetscCall(VecDot(bnk->X_inactive, bnk->inactive_work, &prered));
    175:             if (bnk->active_idx) {
    176:               PetscCall(VecRestoreSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive));
    177:               PetscCall(VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
    178:             }
    179:             prered = tao->trust * (bnk->gnorm - 0.5 * tao->trust * prered / (bnk->gnorm * bnk->gnorm));
    180:             actred = bnk->f - ftrial;
    181:             if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
    182:               kappa = 1.0;
    183:             } else {
    184:               kappa = actred / prered;
    185:             }

    187:             tau_1   = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust + (1.0 - bnk->theta_i) * prered - actred);
    188:             tau_2   = bnk->theta_i * bnk->gnorm * tao->trust / (bnk->theta_i * bnk->gnorm * tao->trust - (1.0 + bnk->theta_i) * prered + actred);
    189:             tau_min = PetscMin(tau_1, tau_2);
    190:             tau_max = PetscMax(tau_1, tau_2);

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

    196:               if (tau_max < 1.0) {
    197:                 tau = bnk->gamma3_i;
    198:               } else if (tau_max > bnk->gamma4_i) {
    199:                 tau = bnk->gamma4_i;
    200:               } else {
    201:                 tau = tau_max;
    202:               }
    203:             } else if (PetscAbsScalar(kappa - (PetscReal)1.0) <= bnk->mu2_i) {
    204:               /*  Good agreement */
    205:               max_radius = PetscMax(max_radius, tao->trust);

    207:               if (tau_max < bnk->gamma2_i) {
    208:                 tau = bnk->gamma2_i;
    209:               } else if (tau_max > bnk->gamma3_i) {
    210:                 tau = bnk->gamma3_i;
    211:               } else {
    212:                 tau = tau_max;
    213:               }
    214:             } else {
    215:               /*  Not good agreement */
    216:               if (tau_min > 1.0) {
    217:                 tau = bnk->gamma2_i;
    218:               } else if (tau_max < bnk->gamma1_i) {
    219:                 tau = bnk->gamma1_i;
    220:               } else if ((tau_min < bnk->gamma1_i) && (tau_max >= 1.0)) {
    221:                 tau = bnk->gamma1_i;
    222:               } else if ((tau_1 >= bnk->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
    223:                 tau = tau_1;
    224:               } else if ((tau_2 >= bnk->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
    225:                 tau = tau_2;
    226:               } else {
    227:                 tau = tau_max;
    228:               }
    229:             }
    230:           }
    231:           tao->trust = tau * tao->trust;
    232:         }

    234:         if (f_min < bnk->f) {
    235:           /* We accidentally found a solution better than the initial, so accept it */
    236:           bnk->f = f_min;
    237:           PetscCall(VecCopy(tao->solution, bnk->Xold));
    238:           PetscCall(VecAXPY(tao->solution, sigma, tao->gradient));
    239:           PetscCall(TaoBoundSolution(tao->solution, tao->XL, tao->XU, 0.0, &nDiff, tao->solution));
    240:           PetscCall(VecCopy(tao->solution, tao->stepdirection));
    241:           PetscCall(VecAXPY(tao->stepdirection, -1.0, bnk->Xold));
    242:           PetscCall(TaoComputeGradient(tao, tao->solution, bnk->unprojected_gradient));
    243:           PetscCall(TaoBNKEstimateActiveSet(tao, bnk->as_type));
    244:           PetscCall(VecCopy(bnk->unprojected_gradient, tao->gradient));
    245:           if (bnk->active_idx) PetscCall(VecISSet(tao->gradient, bnk->active_idx, 0.0));
    246:           /* Compute gradient at the new iterate and flip switch to compute the Hessian later */
    247:           PetscCall(TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm));
    248:           *needH = PETSC_TRUE;
    249:           /* Test the new step for convergence */
    250:           PetscCall(VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W));
    251:           PetscCall(VecNorm(bnk->W, NORM_2, &resnorm));
    252:           PetscCheck(!PetscIsInfOrNanReal(resnorm), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated infinity or NaN");
    253:           PetscCall(TaoLogConvergenceHistory(tao, bnk->f, resnorm, 0.0, tao->ksp_its));
    254:           PetscCall(TaoMonitor(tao, tao->niter, bnk->f, resnorm, 0.0, 1.0));
    255:           PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
    256:           if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(PETSC_SUCCESS);
    257:           /* active BNCG recycling early because we have a stepdirection computed */
    258:           PetscCall(TaoSetRecycleHistory(bnk->bncg, PETSC_TRUE));
    259:         }
    260:       }
    261:       tao->trust = PetscMax(tao->trust, max_radius);

    263:       /* Ensure that the trust radius is within the limits */
    264:       tao->trust = PetscMax(tao->trust, bnk->min_radius);
    265:       tao->trust = PetscMin(tao->trust, bnk->max_radius);
    266:       break;

    268:     default:
    269:       /* Norm of the first direction will initialize radius */
    270:       tao->trust = 0.0;
    271:       break;
    272:     }
    273:   }
    274:   PetscFunctionReturn(PETSC_SUCCESS);
    275: }

    277: /* Computes the exact Hessian and extracts its subHessian */

    279: PetscErrorCode TaoBNKComputeHessian(Tao tao)
    280: {
    281:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    283:   PetscFunctionBegin;
    284:   /* Compute the Hessian */
    285:   PetscCall(TaoComputeHessian(tao, tao->solution, tao->hessian, tao->hessian_pre));
    286:   /* Add a correction to the BFGS preconditioner */
    287:   if (bnk->M) PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    288:   /* Prepare the reduced sub-matrices for the inactive set */
    289:   PetscCall(TaoBNKComputeSubHessian(tao));
    290:   PetscFunctionReturn(PETSC_SUCCESS);
    291: }

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

    295: PetscErrorCode TaoBNKEstimateActiveSet(Tao tao, PetscInt asType)
    296: {
    297:   TAO_BNK  *bnk = (TAO_BNK *)tao->data;
    298:   PetscBool hessComputed, diagExists, hadactive;

    300:   PetscFunctionBegin;
    301:   hadactive = bnk->active_idx ? PETSC_TRUE : PETSC_FALSE;
    302:   switch (asType) {
    303:   case BNK_AS_NONE:
    304:     PetscCall(ISDestroy(&bnk->inactive_idx));
    305:     PetscCall(VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx));
    306:     PetscCall(ISDestroy(&bnk->active_idx));
    307:     PetscCall(ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx));
    308:     break;

    310:   case BNK_AS_BERTSEKAS:
    311:     /* Compute the trial step vector with which we will estimate the active set at the next iteration */
    312:     if (bnk->M) {
    313:       /* If the BFGS matrix is available, we will construct a trial step with it */
    314:       PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W));
    315:     } else {
    316:       hessComputed = diagExists = PETSC_FALSE;
    317:       if (tao->hessian) PetscCall(MatAssembled(tao->hessian, &hessComputed));
    318:       if (hessComputed) PetscCall(MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists));
    319:       if (diagExists) {
    320:         /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
    321:         PetscCall(MatGetDiagonal(tao->hessian, bnk->Xwork));
    322:         PetscCall(VecAbs(bnk->Xwork));
    323:         PetscCall(VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork));
    324:         PetscCall(VecReciprocal(bnk->Xwork));
    325:         PetscCall(VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient));
    326:       } else {
    327:         /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
    328:         PetscCall(VecCopy(bnk->unprojected_gradient, bnk->W));
    329:       }
    330:     }
    331:     PetscCall(VecScale(bnk->W, -1.0));
    332:     PetscCall(TaoEstimateActiveBounds(tao->solution, tao->XL, tao->XU, bnk->unprojected_gradient, bnk->W, bnk->Xwork, bnk->as_step, &bnk->as_tol, &bnk->active_lower, &bnk->active_upper, &bnk->active_fixed, &bnk->active_idx, &bnk->inactive_idx));
    333:     break;

    335:   default:
    336:     break;
    337:   }
    338:   bnk->resetksp = (PetscBool)(bnk->active_idx || hadactive); /* inactive Hessian size may have changed, need to reset operators */
    339:   PetscFunctionReturn(PETSC_SUCCESS);
    340: }

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

    344: PetscErrorCode TaoBNKBoundStep(Tao tao, PetscInt asType, Vec step)
    345: {
    346:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    348:   PetscFunctionBegin;
    349:   switch (asType) {
    350:   case BNK_AS_NONE:
    351:     if (bnk->active_idx) PetscCall(VecISSet(step, bnk->active_idx, 0.0));
    352:     break;
    353:   case BNK_AS_BERTSEKAS:
    354:     PetscCall(TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step));
    355:     break;
    356:   default:
    357:     break;
    358:   }
    359:   PetscFunctionReturn(PETSC_SUCCESS);
    360: }

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

    365:    In practice, this approach simply trades off Hessian evaluations
    366:    for more gradient evaluations.
    367: */

    369: PetscErrorCode TaoBNKTakeCGSteps(Tao tao, PetscBool *terminate)
    370: {
    371:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    373:   PetscFunctionBegin;
    374:   *terminate = PETSC_FALSE;
    375:   if (bnk->max_cg_its > 0) {
    376:     /* Copy the current function value (important vectors are already shared) */
    377:     bnk->bncg_ctx->f = bnk->f;
    378:     /* Take some small finite number of BNCG iterations */
    379:     PetscCall(TaoSolve(bnk->bncg));
    380:     /* Add the number of gradient and function evaluations to the total     *
    381:      * Note: nfuncs are not copied as tao and subsolvers share same TaoTerm */
    382:     bnk->tot_cg_its += bnk->bncg->niter;
    383:     /* Extract the BNCG function value out and save it into BNK */
    384:     bnk->f = bnk->bncg_ctx->f;
    385:     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) {
    386:       *terminate = PETSC_TRUE;
    387:     } else {
    388:       PetscCall(TaoBNKEstimateActiveSet(tao, bnk->as_type));
    389:     }
    390:   }
    391:   PetscFunctionReturn(PETSC_SUCCESS);
    392: }

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

    396: PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason, PetscInt *step_type)
    397: {
    398:   TAO_BNK  *bnk         = (TAO_BNK *)tao->data;
    399:   PetscInt  bfgsUpdates = 0;
    400:   PetscInt  kspits;
    401:   PetscBool is_lmvm;
    402:   PetscBool kspTR;

    404:   PetscFunctionBegin;
    405:   /* If there are no inactive variables left, save some computation and return an adjusted zero step
    406:      that has (l-x) and (u-x) for lower and upper bounded variables. */
    407:   if (!bnk->inactive_idx) {
    408:     PetscCall(VecSet(tao->stepdirection, 0.0));
    409:     PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));
    410:     PetscFunctionReturn(PETSC_SUCCESS);
    411:   }

    413:   /* Shift the reduced Hessian matrix */
    414:   if (shift && bnk->pert > 0) {
    415:     PetscCall(PetscObjectTypeCompare((PetscObject)tao->hessian, MATLMVM, &is_lmvm));
    416:     if (is_lmvm) {
    417:       PetscCall(MatShift(tao->hessian, bnk->pert));
    418:     } else {
    419:       PetscCall(MatShift(bnk->H_inactive, bnk->pert));
    420:       if (bnk->H_inactive != bnk->Hpre_inactive) PetscCall(MatShift(bnk->Hpre_inactive, bnk->pert));
    421:     }
    422:   }

    424:   /* Solve the Newton system of equations */
    425:   tao->ksp_its = 0;
    426:   PetscCall(VecSet(tao->stepdirection, 0.0));
    427:   if (bnk->resetksp) {
    428:     PetscCall(KSPReset(tao->ksp));
    429:     PetscCall(KSPResetFromOptions(tao->ksp));
    430:     bnk->resetksp = PETSC_FALSE;
    431:   }
    432:   PetscCall(KSPSetOperators(tao->ksp, bnk->H_inactive, bnk->Hpre_inactive));
    433:   PetscCall(VecCopy(bnk->unprojected_gradient, bnk->Gwork));
    434:   if (bnk->active_idx) {
    435:     PetscCall(VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
    436:     PetscCall(VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
    437:   } else {
    438:     bnk->G_inactive = bnk->unprojected_gradient;
    439:     bnk->X_inactive = tao->stepdirection;
    440:   }
    441:   PetscCall(KSPCGSetRadius(tao->ksp, tao->trust));
    442:   PetscCall(KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive));
    443:   PetscCall(KSPGetIterationNumber(tao->ksp, &kspits));
    444:   tao->ksp_its += kspits;
    445:   tao->ksp_tot_its += kspits;
    446:   PetscCall(PetscObjectHasFunction((PetscObject)tao->ksp, "KSPCGGetNormD_C", &kspTR));
    447:   if (kspTR) {
    448:     PetscCall(KSPCGGetNormD(tao->ksp, &bnk->dnorm));

    450:     if (0.0 == tao->trust) {
    451:       /* Radius was uninitialized; use the norm of the direction */
    452:       if (bnk->dnorm > 0.0) {
    453:         tao->trust = bnk->dnorm;

    455:         /* Modify the radius if it is too large or small */
    456:         tao->trust = PetscMax(tao->trust, bnk->min_radius);
    457:         tao->trust = PetscMin(tao->trust, bnk->max_radius);
    458:       } else {
    459:         /* The direction was bad; set radius to default value and re-solve
    460:            the trust-region subproblem to get a direction */
    461:         tao->trust = tao->trust0;

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

    467:         PetscCall(KSPCGSetRadius(tao->ksp, tao->trust));
    468:         PetscCall(KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive));
    469:         PetscCall(KSPGetIterationNumber(tao->ksp, &kspits));
    470:         tao->ksp_its += kspits;
    471:         tao->ksp_tot_its += kspits;
    472:         PetscCall(KSPCGGetNormD(tao->ksp, &bnk->dnorm));

    474:         PetscCheck(bnk->dnorm != 0.0, PetscObjectComm((PetscObject)tao), PETSC_ERR_PLIB, "Initial direction zero");
    475:       }
    476:     }
    477:   }
    478:   /* Restore sub vectors back */
    479:   if (bnk->active_idx) {
    480:     PetscCall(VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
    481:     PetscCall(VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
    482:   }
    483:   /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
    484:   PetscCall(VecScale(tao->stepdirection, -1.0));
    485:   PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));

    487:   /* Record convergence reasons */
    488:   PetscCall(KSPGetConvergedReason(tao->ksp, ksp_reason));
    489:   if (KSP_CONVERGED_ATOL == *ksp_reason) {
    490:     ++bnk->ksp_atol;
    491:   } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
    492:     ++bnk->ksp_rtol;
    493:   } else if (KSP_CONVERGED_STEP_LENGTH == *ksp_reason) {
    494:     ++bnk->ksp_ctol;
    495:   } else if (KSP_CONVERGED_NEG_CURVE == *ksp_reason) {
    496:     ++bnk->ksp_negc;
    497:   } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
    498:     ++bnk->ksp_dtol;
    499:   } else if (KSP_DIVERGED_ITS == *ksp_reason) {
    500:     ++bnk->ksp_iter;
    501:   } else {
    502:     ++bnk->ksp_othr;
    503:   }

    505:   /* Make sure the BFGS preconditioner is healthy */
    506:   if (bnk->M) {
    507:     PetscCall(MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates));
    508:     if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
    509:       /* Preconditioner is numerically indefinite; reset the approximation. */
    510:       PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
    511:       PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    512:     }
    513:   }
    514:   *step_type = BNK_NEWTON;
    515:   PetscFunctionReturn(PETSC_SUCCESS);
    516: }

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

    520: PetscErrorCode TaoBNKRecomputePred(Tao tao, Vec S, PetscReal *prered)
    521: {
    522:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    524:   PetscFunctionBegin;
    525:   /* Extract subvectors associated with the inactive set */
    526:   if (bnk->active_idx) {
    527:     PetscCall(VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
    528:     PetscCall(VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
    529:     PetscCall(VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
    530:   } else {
    531:     bnk->X_inactive    = tao->stepdirection;
    532:     bnk->inactive_work = bnk->Xwork;
    533:     bnk->G_inactive    = bnk->Gwork;
    534:   }
    535:   /* Recompute the predicted decrease based on the quadratic model */
    536:   PetscCall(MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work));
    537:   PetscCall(VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive));
    538:   PetscCall(VecDot(bnk->inactive_work, bnk->X_inactive, prered));
    539:   /* Restore the sub vectors */
    540:   if (bnk->active_idx) {
    541:     PetscCall(VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
    542:     PetscCall(VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
    543:     PetscCall(VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
    544:   }
    545:   PetscFunctionReturn(PETSC_SUCCESS);
    546: }

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

    550:    The step direction falls back onto BFGS, scaled gradient and gradient steps
    551:    in the event that the Newton step fails the test.
    552: */

    554: PetscErrorCode TaoBNKSafeguardStep(Tao tao, KSPConvergedReason ksp_reason, PetscInt *stepType)
    555: {
    556:   TAO_BNK  *bnk = (TAO_BNK *)tao->data;
    557:   PetscReal gdx, e_min;
    558:   PetscInt  bfgsUpdates;

    560:   PetscFunctionBegin;
    561:   switch (*stepType) {
    562:   case BNK_NEWTON:
    563:     PetscCall(VecDot(tao->stepdirection, tao->gradient, &gdx));
    564:     if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
    565:       /* Newton step is not descent or direction produced infinity or NaN
    566:         Update the perturbation for next time */
    567:       if (bnk->pert <= 0.0) {
    568:         PetscBool is_gltr;

    570:         /* Initialize the perturbation */
    571:         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
    572:         PetscCall(PetscObjectTypeCompare((PetscObject)tao->ksp, KSPGLTR, &is_gltr));
    573:         if (is_gltr) {
    574:           PetscCall(KSPGLTRGetMinEig(tao->ksp, &e_min));
    575:           bnk->pert = PetscMax(bnk->pert, -e_min);
    576:         }
    577:       } else {
    578:         /* Increase the perturbation */
    579:         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
    580:       }

    582:       if (!bnk->M) {
    583:         /* We don't have the bfgs matrix around and updated
    584:           Must use gradient direction in this case */
    585:         PetscCall(VecCopy(tao->gradient, tao->stepdirection));
    586:         *stepType = BNK_GRADIENT;
    587:       } else {
    588:         /* Attempt to use the BFGS direction */
    589:         PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection));

    591:         /* Check for success (descent direction)
    592:           NOTE: Negative gdx here means not a descent direction because
    593:           the fall-back step is missing a negative sign. */
    594:         PetscCall(VecDot(tao->gradient, tao->stepdirection, &gdx));
    595:         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
    596:           /* BFGS direction is not descent or direction produced not a number
    597:             We can assert bfgsUpdates > 1 in this case because
    598:             the first solve produces the scaled gradient direction,
    599:             which is guaranteed to be descent */

    601:           /* Use steepest descent direction (scaled) */
    602:           PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
    603:           PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    604:           PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection));

    606:           *stepType = BNK_SCALED_GRADIENT;
    607:         } else {
    608:           PetscCall(MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates));
    609:           if (1 == bfgsUpdates) {
    610:             /* The first BFGS direction is always the scaled gradient */
    611:             *stepType = BNK_SCALED_GRADIENT;
    612:           } else {
    613:             *stepType = BNK_BFGS;
    614:           }
    615:         }
    616:       }
    617:       /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
    618:       PetscCall(VecScale(tao->stepdirection, -1.0));
    619:       PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));
    620:     } else {
    621:       /* Computed Newton step is descent */
    622:       switch (ksp_reason) {
    623:       case KSP_DIVERGED_NANORINF:
    624:       case KSP_DIVERGED_BREAKDOWN:
    625:       case KSP_DIVERGED_INDEFINITE_MAT:
    626:       case KSP_DIVERGED_INDEFINITE_PC:
    627:       case KSP_CONVERGED_NEG_CURVE:
    628:         /* Matrix or preconditioner is indefinite; increase perturbation */
    629:         if (bnk->pert <= 0.0) {
    630:           PetscBool is_gltr;

    632:           /* Initialize the perturbation */
    633:           bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
    634:           PetscCall(PetscObjectTypeCompare((PetscObject)tao->ksp, KSPGLTR, &is_gltr));
    635:           if (is_gltr) {
    636:             PetscCall(KSPGLTRGetMinEig(tao->ksp, &e_min));
    637:             bnk->pert = PetscMax(bnk->pert, -e_min);
    638:           }
    639:         } else {
    640:           /* Increase the perturbation */
    641:           bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
    642:         }
    643:         break;

    645:       default:
    646:         /* Newton step computation is good; decrease perturbation */
    647:         bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
    648:         if (bnk->pert < bnk->pmin) bnk->pert = 0.0;
    649:         break;
    650:       }
    651:       *stepType = BNK_NEWTON;
    652:     }
    653:     break;

    655:   case BNK_BFGS:
    656:     /* Check for success (descent direction) */
    657:     PetscCall(VecDot(tao->stepdirection, tao->gradient, &gdx));
    658:     if (gdx >= 0 || PetscIsInfOrNanReal(gdx)) {
    659:       /* Step is not descent or solve was not successful
    660:          Use steepest descent direction (scaled) */
    661:       PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
    662:       PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    663:       PetscCall(MatSolve(bnk->M, tao->gradient, tao->stepdirection));
    664:       PetscCall(VecScale(tao->stepdirection, -1.0));
    665:       PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));
    666:       *stepType = BNK_SCALED_GRADIENT;
    667:     } else {
    668:       *stepType = BNK_BFGS;
    669:     }
    670:     break;

    672:   case BNK_SCALED_GRADIENT:
    673:     break;

    675:   default:
    676:     break;
    677:   }
    678:   PetscFunctionReturn(PETSC_SUCCESS);
    679: }

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

    683:   Includes fallbacks to BFGS, scaled gradient, and unscaled gradient steps if the
    684:   Newton step does not produce a valid step length.
    685: */

    687: PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
    688: {
    689:   TAO_BNK                     *bnk = (TAO_BNK *)tao->data;
    690:   TaoLineSearchConvergedReason ls_reason;
    691:   PetscReal                    e_min, gdx;
    692:   PetscInt                     bfgsUpdates;

    694:   PetscFunctionBegin;
    695:   /* Perform the linesearch */
    696:   PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason));
    697:   PetscCall(TaoAddLineSearchCounts(tao));

    699:   while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && *stepType != BNK_SCALED_GRADIENT && *stepType != BNK_GRADIENT) {
    700:     /* Linesearch failed, revert solution */
    701:     bnk->f = bnk->fold;
    702:     PetscCall(VecCopy(bnk->Xold, tao->solution));
    703:     PetscCall(VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient));

    705:     switch (*stepType) {
    706:     case BNK_NEWTON:
    707:       /* Failed to obtain acceptable iterate with Newton step
    708:          Update the perturbation for next time */
    709:       if (bnk->pert <= 0.0) {
    710:         PetscBool is_gltr;

    712:         /* Initialize the perturbation */
    713:         bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
    714:         PetscCall(PetscObjectTypeCompare((PetscObject)tao->ksp, KSPGLTR, &is_gltr));
    715:         if (is_gltr) {
    716:           PetscCall(KSPGLTRGetMinEig(tao->ksp, &e_min));
    717:           bnk->pert = PetscMax(bnk->pert, -e_min);
    718:         }
    719:       } else {
    720:         /* Increase the perturbation */
    721:         bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
    722:       }

    724:       if (!bnk->M) {
    725:         /* We don't have the bfgs matrix around and being updated
    726:            Must use gradient direction in this case */
    727:         PetscCall(VecCopy(bnk->unprojected_gradient, tao->stepdirection));
    728:         *stepType = BNK_GRADIENT;
    729:       } else {
    730:         /* Attempt to use the BFGS direction */
    731:         PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection));
    732:         /* Check for success (descent direction)
    733:            NOTE: Negative gdx means not a descent direction because the step here is missing a negative sign. */
    734:         PetscCall(VecDot(tao->gradient, tao->stepdirection, &gdx));
    735:         if ((gdx <= 0.0) || PetscIsInfOrNanReal(gdx)) {
    736:           /* BFGS direction is not descent or direction produced not a number
    737:              We can assert bfgsUpdates > 1 in this case
    738:              Use steepest descent direction (scaled) */
    739:           PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
    740:           PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    741:           PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection));

    743:           bfgsUpdates = 1;
    744:           *stepType   = BNK_SCALED_GRADIENT;
    745:         } else {
    746:           PetscCall(MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates));
    747:           if (1 == bfgsUpdates) {
    748:             /* The first BFGS direction is always the scaled gradient */
    749:             *stepType = BNK_SCALED_GRADIENT;
    750:           } else {
    751:             *stepType = BNK_BFGS;
    752:           }
    753:         }
    754:       }
    755:       break;

    757:     case BNK_BFGS:
    758:       /* Can only enter if pc_type == BNK_PC_BFGS
    759:          Failed to obtain acceptable iterate with BFGS step
    760:          Attempt to use the scaled gradient direction */
    761:       PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
    762:       PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    763:       PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, tao->stepdirection));

    765:       bfgsUpdates = 1;
    766:       *stepType   = BNK_SCALED_GRADIENT;
    767:       break;
    768:     }
    769:     /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
    770:     PetscCall(VecScale(tao->stepdirection, -1.0));
    771:     PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));

    773:     /* Perform one last line search with the fall-back step */
    774:     PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &bnk->f, bnk->unprojected_gradient, tao->stepdirection, steplen, &ls_reason));
    775:     PetscCall(TaoAddLineSearchCounts(tao));
    776:   }
    777:   *reason = ls_reason;
    778:   PetscFunctionReturn(PETSC_SUCCESS);
    779: }

    781: /* Routine for updating the trust radius.

    783:   Function features three different update methods:
    784:   1) Line-search step length based
    785:   2) Predicted decrease on the CG quadratic model
    786:   3) Interpolation
    787: */

    789: PetscErrorCode TaoBNKUpdateTrustRadius(Tao tao, PetscReal prered, PetscReal actred, PetscInt updateType, PetscInt stepType, PetscBool *accept)
    790: {
    791:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    793:   PetscReal step, kappa;
    794:   PetscReal gdx, tau_1, tau_2, tau_min, tau_max;

    796:   PetscFunctionBegin;
    797:   /* Update trust region radius */
    798:   *accept = PETSC_FALSE;
    799:   switch (updateType) {
    800:   case BNK_UPDATE_STEP:
    801:     *accept = PETSC_TRUE; /* always accept here because line search succeeded */
    802:     if (stepType == BNK_NEWTON) {
    803:       PetscCall(TaoLineSearchGetStepLength(tao->linesearch, &step));
    804:       if (step < bnk->nu1) {
    805:         /* Very bad step taken; reduce radius */
    806:         tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
    807:       } else if (step < bnk->nu2) {
    808:         /* Reasonably bad step taken; reduce radius */
    809:         tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
    810:       } else if (step < bnk->nu3) {
    811:         /*  Reasonable step was taken; leave radius alone */
    812:         if (bnk->omega3 < 1.0) {
    813:           tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
    814:         } else if (bnk->omega3 > 1.0) {
    815:           tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
    816:         }
    817:       } else if (step < bnk->nu4) {
    818:         /*  Full step taken; increase the radius */
    819:         tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
    820:       } else {
    821:         /*  More than full step taken; increase the radius */
    822:         tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
    823:       }
    824:     } else {
    825:       /*  Newton step was not good; reduce the radius */
    826:       tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
    827:     }
    828:     break;

    830:   case BNK_UPDATE_REDUCTION:
    831:     if (stepType == BNK_NEWTON) {
    832:       if ((prered < 0.0) || PetscIsInfOrNanReal(prered)) {
    833:         /* The predicted reduction has the wrong sign.  This cannot
    834:            happen in infinite precision arithmetic.  Step should
    835:            be rejected! */
    836:         tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
    837:       } else {
    838:         if (PetscIsInfOrNanReal(actred)) {
    839:           tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
    840:         } else {
    841:           if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f)) * bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f)) * bnk->epsilon)) {
    842:             kappa = 1.0;
    843:           } else {
    844:             kappa = actred / prered;
    845:           }
    846:           /* Accept or reject the step and update radius */
    847:           if (kappa < bnk->eta1) {
    848:             /* Reject the step */
    849:             tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
    850:           } else {
    851:             /* Accept the step */
    852:             *accept = PETSC_TRUE;
    853:             /* Update the trust region radius only if the computed step is at the trust radius boundary */
    854:             if (bnk->dnorm == tao->trust) {
    855:               if (kappa < bnk->eta2) {
    856:                 /* Marginal bad step */
    857:                 tao->trust = bnk->alpha2 * tao->trust;
    858:               } else if (kappa < bnk->eta3) {
    859:                 /* Reasonable step */
    860:                 tao->trust = bnk->alpha3 * tao->trust;
    861:               } else if (kappa < bnk->eta4) {
    862:                 /* Good step */
    863:                 tao->trust = bnk->alpha4 * tao->trust;
    864:               } else {
    865:                 /* Very good step */
    866:                 tao->trust = bnk->alpha5 * tao->trust;
    867:               }
    868:             }
    869:           }
    870:         }
    871:       }
    872:     } else {
    873:       /*  Newton step was not good; reduce the radius */
    874:       tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
    875:     }
    876:     break;

    878:   default:
    879:     if (stepType == BNK_NEWTON) {
    880:       if (prered < 0.0) {
    881:         /*  The predicted reduction has the wrong sign.  This cannot */
    882:         /*  happen in infinite precision arithmetic.  Step should */
    883:         /*  be rejected! */
    884:         tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    885:       } else {
    886:         if (PetscIsInfOrNanReal(actred)) {
    887:           tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    888:         } else {
    889:           if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
    890:             kappa = 1.0;
    891:           } else {
    892:             kappa = actred / prered;
    893:           }

    895:           PetscCall(VecDot(tao->gradient, tao->stepdirection, &gdx));
    896:           tau_1   = bnk->theta * gdx / (bnk->theta * gdx - (1.0 - bnk->theta) * prered + actred);
    897:           tau_2   = bnk->theta * gdx / (bnk->theta * gdx + (1.0 + bnk->theta) * prered - actred);
    898:           tau_min = PetscMin(tau_1, tau_2);
    899:           tau_max = PetscMax(tau_1, tau_2);

    901:           if (kappa >= 1.0 - bnk->mu1) {
    902:             /*  Great agreement */
    903:             *accept = PETSC_TRUE;
    904:             if (tau_max < 1.0) {
    905:               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
    906:             } else if (tau_max > bnk->gamma4) {
    907:               tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
    908:             } else {
    909:               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
    910:             }
    911:           } else if (kappa >= 1.0 - bnk->mu2) {
    912:             /*  Good agreement */
    913:             *accept = PETSC_TRUE;
    914:             if (tau_max < bnk->gamma2) {
    915:               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
    916:             } else if (tau_max > bnk->gamma3) {
    917:               tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
    918:             } else if (tau_max < 1.0) {
    919:               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
    920:             } else {
    921:               tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
    922:             }
    923:           } else {
    924:             /*  Not good agreement */
    925:             if (tau_min > 1.0) {
    926:               tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
    927:             } else if (tau_max < bnk->gamma1) {
    928:               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    929:             } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
    930:               tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
    931:             } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
    932:               tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
    933:             } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
    934:               tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
    935:             } else {
    936:               tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
    937:             }
    938:           }
    939:         }
    940:       }
    941:     } else {
    942:       /*  Newton step was not good; reduce the radius */
    943:       tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
    944:     }
    945:     break;
    946:   }
    947:   /* Make sure the radius does not violate min and max settings */
    948:   tao->trust = PetscMin(tao->trust, bnk->max_radius);
    949:   tao->trust = PetscMax(tao->trust, bnk->min_radius);
    950:   PetscFunctionReturn(PETSC_SUCCESS);
    951: }

    953: PetscErrorCode TaoBNKAddStepCounts(Tao tao, PetscInt stepType)
    954: {
    955:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    957:   PetscFunctionBegin;
    958:   switch (stepType) {
    959:   case BNK_NEWTON:
    960:     ++bnk->newt;
    961:     break;
    962:   case BNK_BFGS:
    963:     ++bnk->bfgs;
    964:     break;
    965:   case BNK_SCALED_GRADIENT:
    966:     ++bnk->sgrad;
    967:     break;
    968:   case BNK_GRADIENT:
    969:     ++bnk->grad;
    970:     break;
    971:   default:
    972:     break;
    973:   }
    974:   PetscFunctionReturn(PETSC_SUCCESS);
    975: }

    977: PetscErrorCode TaoSetUp_BNK(Tao tao)
    978: {
    979:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    981:   PetscFunctionBegin;
    982:   if (!tao->gradient) PetscCall(VecDuplicate(tao->solution, &tao->gradient));
    983:   if (!tao->stepdirection) PetscCall(VecDuplicate(tao->solution, &tao->stepdirection));
    984:   if (!bnk->W) PetscCall(VecDuplicate(tao->solution, &bnk->W));
    985:   if (!bnk->Xold) PetscCall(VecDuplicate(tao->solution, &bnk->Xold));
    986:   if (!bnk->Gold) PetscCall(VecDuplicate(tao->solution, &bnk->Gold));
    987:   if (!bnk->Xwork) PetscCall(VecDuplicate(tao->solution, &bnk->Xwork));
    988:   if (!bnk->Gwork) PetscCall(VecDuplicate(tao->solution, &bnk->Gwork));
    989:   if (!bnk->unprojected_gradient) PetscCall(VecDuplicate(tao->solution, &bnk->unprojected_gradient));
    990:   if (!bnk->unprojected_gradient_old) PetscCall(VecDuplicate(tao->solution, &bnk->unprojected_gradient_old));
    991:   if (!bnk->Diag_min) PetscCall(VecDuplicate(tao->solution, &bnk->Diag_min));
    992:   if (!bnk->Diag_max) PetscCall(VecDuplicate(tao->solution, &bnk->Diag_max));
    993:   PetscCall(TaoSetSolution(bnk->bncg, tao->solution));
    994:   if (bnk->max_cg_its > 0) {
    995:     /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
    996:     bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
    997:     PetscCall(PetscObjectReference((PetscObject)bnk->unprojected_gradient_old));
    998:     PetscCall(VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old));
    999:     bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
    1000:     PetscCall(PetscObjectReference((PetscObject)bnk->unprojected_gradient));
    1001:     PetscCall(VecDestroy(&bnk->bncg_ctx->unprojected_gradient));
    1002:     bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
    1003:     PetscCall(PetscObjectReference((PetscObject)bnk->Gold));
    1004:     PetscCall(VecDestroy(&bnk->bncg_ctx->G_old));
    1005:     bnk->bncg_ctx->G_old = bnk->Gold;
    1006:     PetscCall(PetscObjectReference((PetscObject)tao->gradient));
    1007:     PetscCall(VecDestroy(&bnk->bncg->gradient));
    1008:     bnk->bncg->gradient = tao->gradient;
    1009:     PetscCall(PetscObjectReference((PetscObject)tao->stepdirection));
    1010:     PetscCall(VecDestroy(&bnk->bncg->stepdirection));
    1011:     bnk->bncg->stepdirection = tao->stepdirection;
    1012:     /* Copy over some settings from BNK into BNCG */
    1013:     PetscCall(TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its));
    1014:     PetscCall(TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol));
    1015:     PetscCall(TaoSetFunctionLowerBound(bnk->bncg, tao->fmin));
    1016:     PetscCall(TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP));
    1017:     {
    1018:       TaoTerm   term;
    1019:       Vec       params;
    1020:       PetscReal scale;
    1021:       Mat       map;

    1023:       // Note: tao->objective_term.term and bnk->bncg->objective_term.term will point to same address
    1024:       PetscCall(TaoGetTerm(tao, &scale, &term, &params, &map));
    1025:       PetscCall(TaoAddTerm(bnk->bncg, NULL, scale, term, params, map));
    1026:     }
    1027:     PetscCall(PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)bnk->bncg));
    1028:   }
    1029:   bnk->X_inactive    = NULL;
    1030:   bnk->G_inactive    = NULL;
    1031:   bnk->inactive_work = NULL;
    1032:   bnk->active_work   = NULL;
    1033:   bnk->inactive_idx  = NULL;
    1034:   bnk->active_idx    = NULL;
    1035:   bnk->active_lower  = NULL;
    1036:   bnk->active_upper  = NULL;
    1037:   bnk->active_fixed  = NULL;
    1038:   bnk->M             = NULL;
    1039:   bnk->H_inactive    = NULL;
    1040:   bnk->Hpre_inactive = NULL;
    1041:   PetscFunctionReturn(PETSC_SUCCESS);
    1042: }

    1044: PetscErrorCode TaoDestroy_BNK(Tao tao)
    1045: {
    1046:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    1048:   PetscFunctionBegin;
    1049:   PetscCall(VecDestroy(&bnk->W));
    1050:   PetscCall(VecDestroy(&bnk->Xold));
    1051:   PetscCall(VecDestroy(&bnk->Gold));
    1052:   PetscCall(VecDestroy(&bnk->Xwork));
    1053:   PetscCall(VecDestroy(&bnk->Gwork));
    1054:   PetscCall(VecDestroy(&bnk->unprojected_gradient));
    1055:   PetscCall(VecDestroy(&bnk->unprojected_gradient_old));
    1056:   PetscCall(VecDestroy(&bnk->Diag_min));
    1057:   PetscCall(VecDestroy(&bnk->Diag_max));
    1058:   PetscCall(ISDestroy(&bnk->active_lower));
    1059:   PetscCall(ISDestroy(&bnk->active_upper));
    1060:   PetscCall(ISDestroy(&bnk->active_fixed));
    1061:   PetscCall(ISDestroy(&bnk->active_idx));
    1062:   PetscCall(ISDestroy(&bnk->inactive_idx));
    1063:   PetscCall(MatDestroy(&bnk->Hpre_inactive));
    1064:   PetscCall(MatDestroy(&bnk->H_inactive));
    1065:   PetscCall(TaoDestroy(&bnk->bncg));
    1066:   PetscCall(KSPDestroy(&tao->ksp));
    1067:   PetscCall(PetscFree(tao->data));
    1068:   PetscFunctionReturn(PETSC_SUCCESS);
    1069: }

    1071: PetscErrorCode TaoSetFromOptions_BNK(Tao tao, PetscOptionItems PetscOptionsObject)
    1072: {
    1073:   TAO_BNK *bnk = (TAO_BNK *)tao->data;

    1075:   PetscFunctionBegin;
    1076:   PetscOptionsHeadBegin(PetscOptionsObject, "Newton-Krylov method for bound constrained optimization");
    1077:   PetscCall(PetscOptionsEList("-tao_bnk_init_type", "radius initialization type", "", BNK_INIT, BNK_INIT_TYPES, BNK_INIT[bnk->init_type], &bnk->init_type, NULL));
    1078:   PetscCall(PetscOptionsEList("-tao_bnk_update_type", "radius update type", "", BNK_UPDATE, BNK_UPDATE_TYPES, BNK_UPDATE[bnk->update_type], &bnk->update_type, NULL));
    1079:   PetscCall(PetscOptionsEList("-tao_bnk_as_type", "active set estimation method", "", BNK_AS, BNK_AS_TYPES, BNK_AS[bnk->as_type], &bnk->as_type, NULL));
    1080:   PetscCall(PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval, NULL));
    1081:   PetscCall(PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin, NULL));
    1082:   PetscCall(PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax, NULL));
    1083:   PetscCall(PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac, NULL));
    1084:   PetscCall(PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin, NULL));
    1085:   PetscCall(PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax, NULL));
    1086:   PetscCall(PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac, NULL));
    1087:   PetscCall(PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac, NULL));
    1088:   PetscCall(PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac, NULL));
    1089:   PetscCall(PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac, NULL));
    1090:   PetscCall(PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1, NULL));
    1091:   PetscCall(PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2, NULL));
    1092:   PetscCall(PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3, NULL));
    1093:   PetscCall(PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4, NULL));
    1094:   PetscCall(PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1, NULL));
    1095:   PetscCall(PetscOptionsReal("-tao_bnk_alpha2", "(developer) radius reduction factor for marginally accepted bad step (-tao_bnk_update_type reduction)", "", bnk->alpha2, &bnk->alpha2, NULL));
    1096:   PetscCall(PetscOptionsReal("-tao_bnk_alpha3", "(developer) radius increase factor for reasonable accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha3, &bnk->alpha3, NULL));
    1097:   PetscCall(PetscOptionsReal("-tao_bnk_alpha4", "(developer) radius increase factor for good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha4, &bnk->alpha4, NULL));
    1098:   PetscCall(PetscOptionsReal("-tao_bnk_alpha5", "(developer) radius increase factor for very good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha5, &bnk->alpha5, NULL));
    1099:   PetscCall(PetscOptionsReal("-tao_bnk_nu1", "(developer) threshold for small line-search step length (-tao_bnk_update_type step)", "", bnk->nu1, &bnk->nu1, NULL));
    1100:   PetscCall(PetscOptionsReal("-tao_bnk_nu2", "(developer) threshold for reasonable line-search step length (-tao_bnk_update_type step)", "", bnk->nu2, &bnk->nu2, NULL));
    1101:   PetscCall(PetscOptionsReal("-tao_bnk_nu3", "(developer) threshold for large line-search step length (-tao_bnk_update_type step)", "", bnk->nu3, &bnk->nu3, NULL));
    1102:   PetscCall(PetscOptionsReal("-tao_bnk_nu4", "(developer) threshold for very large line-search step length (-tao_bnk_update_type step)", "", bnk->nu4, &bnk->nu4, NULL));
    1103:   PetscCall(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));
    1104:   PetscCall(PetscOptionsReal("-tao_bnk_omega2", "(developer) radius reduction factor for small line-search step length (-tao_bnk_update_type step)", "", bnk->omega2, &bnk->omega2, NULL));
    1105:   PetscCall(PetscOptionsReal("-tao_bnk_omega3", "(developer) radius factor for decent line-search step length (-tao_bnk_update_type step)", "", bnk->omega3, &bnk->omega3, NULL));
    1106:   PetscCall(PetscOptionsReal("-tao_bnk_omega4", "(developer) radius increase factor for large line-search step length (-tao_bnk_update_type step)", "", bnk->omega4, &bnk->omega4, NULL));
    1107:   PetscCall(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));
    1108:   PetscCall(PetscOptionsReal("-tao_bnk_mu1_i", "(developer) threshold for accepting very good step (-tao_bnk_init_type interpolation)", "", bnk->mu1_i, &bnk->mu1_i, NULL));
    1109:   PetscCall(PetscOptionsReal("-tao_bnk_mu2_i", "(developer) threshold for accepting good step (-tao_bnk_init_type interpolation)", "", bnk->mu2_i, &bnk->mu2_i, NULL));
    1110:   PetscCall(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));
    1111:   PetscCall(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));
    1112:   PetscCall(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));
    1113:   PetscCall(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));
    1114:   PetscCall(PetscOptionsReal("-tao_bnk_theta_i", "(developer) trust region interpolation factor (-tao_bnk_init_type interpolation)", "", bnk->theta_i, &bnk->theta_i, NULL));
    1115:   PetscCall(PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1, NULL));
    1116:   PetscCall(PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2, NULL));
    1117:   PetscCall(PetscOptionsReal("-tao_bnk_gamma1", "(developer) radius reduction factor for rejected very bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma1, &bnk->gamma1, NULL));
    1118:   PetscCall(PetscOptionsReal("-tao_bnk_gamma2", "(developer) radius reduction factor for rejected bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma2, &bnk->gamma2, NULL));
    1119:   PetscCall(PetscOptionsReal("-tao_bnk_gamma3", "(developer) radius increase factor for accepted good step (-tao_bnk_update_type interpolation)", "", bnk->gamma3, &bnk->gamma3, NULL));
    1120:   PetscCall(PetscOptionsReal("-tao_bnk_gamma4", "(developer) radius increase factor for accepted very good step (-tao_bnk_update_type interpolation)", "", bnk->gamma4, &bnk->gamma4, NULL));
    1121:   PetscCall(PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta, NULL));
    1122:   PetscCall(PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius, NULL));
    1123:   PetscCall(PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius, NULL));
    1124:   PetscCall(PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon, NULL));
    1125:   PetscCall(PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol, NULL));
    1126:   PetscCall(PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step, NULL));
    1127:   PetscCall(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));
    1128:   PetscOptionsHeadEnd();

    1130:   PetscCall(TaoSetOptionsPrefix(bnk->bncg, ((PetscObject)tao)->prefix));
    1131:   PetscCall(TaoAppendOptionsPrefix(bnk->bncg, "tao_bnk_cg_"));
    1132:   PetscCall(TaoSetFromOptions(bnk->bncg));

    1134:   PetscCall(KSPSetOptionsPrefix(tao->ksp, ((PetscObject)tao)->prefix));
    1135:   PetscCall(KSPAppendOptionsPrefix(tao->ksp, "tao_bnk_"));
    1136:   PetscCall(KSPSetFromOptions(tao->ksp));
    1137:   PetscFunctionReturn(PETSC_SUCCESS);
    1138: }

    1140: PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
    1141: {
    1142:   TAO_BNK  *bnk = (TAO_BNK *)tao->data;
    1143:   PetscInt  nrejects;
    1144:   PetscBool isascii;

    1146:   PetscFunctionBegin;
    1147:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
    1148:   if (isascii) {
    1149:     PetscCall(PetscViewerASCIIPushTab(viewer));
    1150:     PetscCall(TaoView(bnk->bncg, viewer));
    1151:     if (bnk->M) {
    1152:       PetscCall(MatLMVMGetRejectCount(bnk->M, &nrejects));
    1153:       PetscCall(PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %" PetscInt_FMT "\n", nrejects));
    1154:     }
    1155:     PetscCall(PetscViewerASCIIPrintf(viewer, "CG steps: %" PetscInt_FMT "\n", bnk->tot_cg_its));
    1156:     PetscCall(PetscViewerASCIIPrintf(viewer, "Newton steps: %" PetscInt_FMT "\n", bnk->newt));
    1157:     if (bnk->M) PetscCall(PetscViewerASCIIPrintf(viewer, "BFGS steps: %" PetscInt_FMT "\n", bnk->bfgs));
    1158:     PetscCall(PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %" PetscInt_FMT "\n", bnk->sgrad));
    1159:     PetscCall(PetscViewerASCIIPrintf(viewer, "Gradient steps: %" PetscInt_FMT "\n", bnk->grad));
    1160:     PetscCall(PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n"));
    1161:     PetscCall(PetscViewerASCIIPrintf(viewer, "  atol: %" PetscInt_FMT "\n", bnk->ksp_atol));
    1162:     PetscCall(PetscViewerASCIIPrintf(viewer, "  rtol: %" PetscInt_FMT "\n", bnk->ksp_rtol));
    1163:     PetscCall(PetscViewerASCIIPrintf(viewer, "  ctol: %" PetscInt_FMT "\n", bnk->ksp_ctol));
    1164:     PetscCall(PetscViewerASCIIPrintf(viewer, "  negc: %" PetscInt_FMT "\n", bnk->ksp_negc));
    1165:     PetscCall(PetscViewerASCIIPrintf(viewer, "  dtol: %" PetscInt_FMT "\n", bnk->ksp_dtol));
    1166:     PetscCall(PetscViewerASCIIPrintf(viewer, "  iter: %" PetscInt_FMT "\n", bnk->ksp_iter));
    1167:     PetscCall(PetscViewerASCIIPrintf(viewer, "  othr: %" PetscInt_FMT "\n", bnk->ksp_othr));
    1168:     PetscCall(PetscViewerASCIIPopTab(viewer));
    1169:   }
    1170:   PetscFunctionReturn(PETSC_SUCCESS);
    1171: }

    1173: /*MC
    1174:   TAOBNK - Shared base-type for Bounded Newton-Krylov type algorithms.
    1175:   At each iteration, the BNK methods solve the symmetric
    1176:   system of equations to obtain the step direction $d_k$:
    1177:              $ H_k d_k = -g_k $
    1178:   for free variables only. The step can be globalized either through
    1179:   trust-region methods, or a line search, or a heuristic mixture of both.

    1181:     Options Database Keys:
    1182: + -tao_bnk_max_cg_its  - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
    1183: . -tao_bnk_init_type   - trust radius initialization method ("constant", "direction", "interpolation")
    1184: . -tao_bnk_update_type - trust radius update method ("step", "direction", "interpolation")
    1185: . -tao_bnk_as_type     - active-set estimation method ("none", "bertsekas")
    1186: . -tao_bnk_as_tol      - (developer) initial tolerance used in estimating bounded active variables (-as_type bertsekas)
    1187: . -tao_bnk_as_step     - (developer) trial step length used in estimating bounded active variables (-as_type bertsekas)
    1188: . -tao_bnk_sval        - (developer) Hessian perturbation starting value
    1189: . -tao_bnk_imin        - (developer) minimum initial Hessian perturbation
    1190: . -tao_bnk_imax        - (developer) maximum initial Hessian perturbation
    1191: . -tao_bnk_pmin        - (developer) minimum Hessian perturbation
    1192: . -tao_bnk_pmax        - (developer) aximum Hessian perturbation
    1193: . -tao_bnk_pgfac       - (developer) Hessian perturbation growth factor
    1194: . -tao_bnk_psfac       - (developer) Hessian perturbation shrink factor
    1195: . -tao_bnk_imfac       - (developer) initial merit factor for Hessian perturbation
    1196: . -tao_bnk_pmgfac      - (developer) merit growth factor for Hessian perturbation
    1197: . -tao_bnk_pmsfac      - (developer) merit shrink factor for Hessian perturbation
    1198: . -tao_bnk_eta1        - (developer) threshold for rejecting step (-update_type reduction)
    1199: . -tao_bnk_eta2        - (developer) threshold for accepting marginal step (-update_type reduction)
    1200: . -tao_bnk_eta3        - (developer) threshold for accepting reasonable step (-update_type reduction)
    1201: . -tao_bnk_eta4        - (developer) threshold for accepting good step (-update_type reduction)
    1202: . -tao_bnk_alpha1      - (developer) radius reduction factor for rejected step (-update_type reduction)
    1203: . -tao_bnk_alpha2      - (developer) radius reduction factor for marginally accepted bad step (-update_type reduction)
    1204: . -tao_bnk_alpha3      - (developer) radius increase factor for reasonable accepted step (-update_type reduction)
    1205: . -tao_bnk_alpha4      - (developer) radius increase factor for good accepted step (-update_type reduction)
    1206: . -tao_bnk_alpha5      - (developer) radius increase factor for very good accepted step (-update_type reduction)
    1207: . -tao_bnk_epsilon     - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-update_type reduction)
    1208: . -tao_bnk_mu1         - (developer) threshold for accepting very good step (-update_type interpolation)
    1209: . -tao_bnk_mu2         - (developer) threshold for accepting good step (-update_type interpolation)
    1210: . -tao_bnk_gamma1      - (developer) radius reduction factor for rejected very bad step (-update_type interpolation)
    1211: . -tao_bnk_gamma2      - (developer) radius reduction factor for rejected bad step (-update_type interpolation)
    1212: . -tao_bnk_gamma3      - (developer) radius increase factor for accepted good step (-update_type interpolation)
    1213: . -tao_bnk_gamma4      - (developer) radius increase factor for accepted very good step (-update_type interpolation)
    1214: . -tao_bnk_theta       - (developer) trust region interpolation factor (-update_type interpolation)
    1215: . -tao_bnk_nu1         - (developer) threshold for small line-search step length (-update_type step)
    1216: . -tao_bnk_nu2         - (developer) threshold for reasonable line-search step length (-update_type step)
    1217: . -tao_bnk_nu3         - (developer) threshold for large line-search step length (-update_type step)
    1218: . -tao_bnk_nu4         - (developer) threshold for very large line-search step length (-update_type step)
    1219: . -tao_bnk_omega1      - (developer) radius reduction factor for very small line-search step length (-update_type step)
    1220: . -tao_bnk_omega2      - (developer) radius reduction factor for small line-search step length (-update_type step)
    1221: . -tao_bnk_omega3      - (developer) radius factor for decent line-search step length (-update_type step)
    1222: . -tao_bnk_omega4      - (developer) radius increase factor for large line-search step length (-update_type step)
    1223: . -tao_bnk_omega5      - (developer) radius increase factor for very large line-search step length (-update_type step)
    1224: . -tao_bnk_mu1_i       - (developer) threshold for accepting very good step (-init_type interpolation)
    1225: . -tao_bnk_mu2_i       - (developer) threshold for accepting good step (-init_type interpolation)
    1226: . -tao_bnk_gamma1_i    - (developer) radius reduction factor for rejected very bad step (-init_type interpolation)
    1227: . -tao_bnk_gamma2_i    - (developer) radius reduction factor for rejected bad step (-init_type interpolation)
    1228: . -tao_bnk_gamma3_i    - (developer) radius increase factor for accepted good step (-init_type interpolation)
    1229: . -tao_bnk_gamma4_i    - (developer) radius increase factor for accepted very good step (-init_type interpolation)
    1230: - -tao_bnk_theta_i     - (developer) trust region interpolation factor (-init_type interpolation)

    1232:   Level: beginner

    1234:   The various algorithmic factors can only be supplied via the options database

    1236: .seealso: `Tao`, `TAONLS`, `TAONTL`, `TAONM`, `TaoType`, `TaoCreate()`
    1237: M*/

    1239: PetscErrorCode TaoCreate_BNK(Tao tao)
    1240: {
    1241:   TAO_BNK *bnk;
    1242:   PC       pc;

    1244:   PetscFunctionBegin;
    1245:   PetscCall(PetscNew(&bnk));

    1247:   tao->ops->setup            = TaoSetUp_BNK;
    1248:   tao->ops->view             = TaoView_BNK;
    1249:   tao->ops->setfromoptions   = TaoSetFromOptions_BNK;
    1250:   tao->ops->destroy          = TaoDestroy_BNK;
    1251:   tao->uses_gradient         = PETSC_TRUE;
    1252:   tao->uses_hessian_matrices = PETSC_TRUE;

    1254:   /*  Override default settings (unless already changed) */
    1255:   PetscCall(TaoParametersInitialize(tao));
    1256:   PetscObjectParameterSetDefault(tao, max_it, 50);
    1257:   PetscObjectParameterSetDefault(tao, trust0, 100.0);

    1259:   tao->data = (void *)bnk;

    1261:   /*  Hessian shifting parameters */
    1262:   bnk->computehessian = TaoBNKComputeHessian;
    1263:   bnk->computestep    = TaoBNKComputeStep;

    1265:   bnk->sval  = 0.0;
    1266:   bnk->imin  = 1.0e-4;
    1267:   bnk->imax  = 1.0e+2;
    1268:   bnk->imfac = 1.0e-1;

    1270:   bnk->pmin   = 1.0e-12;
    1271:   bnk->pmax   = 1.0e+2;
    1272:   bnk->pgfac  = 1.0e+1;
    1273:   bnk->psfac  = 4.0e-1;
    1274:   bnk->pmgfac = 1.0e-1;
    1275:   bnk->pmsfac = 1.0e-1;

    1277:   /*  Default values for trust-region radius update based on steplength */
    1278:   bnk->nu1 = 0.25;
    1279:   bnk->nu2 = 0.50;
    1280:   bnk->nu3 = 1.00;
    1281:   bnk->nu4 = 1.25;

    1283:   bnk->omega1 = 0.25;
    1284:   bnk->omega2 = 0.50;
    1285:   bnk->omega3 = 1.00;
    1286:   bnk->omega4 = 2.00;
    1287:   bnk->omega5 = 4.00;

    1289:   /*  Default values for trust-region radius update based on reduction */
    1290:   bnk->eta1 = 1.0e-4;
    1291:   bnk->eta2 = 0.25;
    1292:   bnk->eta3 = 0.50;
    1293:   bnk->eta4 = 0.90;

    1295:   bnk->alpha1 = 0.25;
    1296:   bnk->alpha2 = 0.50;
    1297:   bnk->alpha3 = 1.00;
    1298:   bnk->alpha4 = 2.00;
    1299:   bnk->alpha5 = 4.00;

    1301:   /*  Default values for trust-region radius update based on interpolation */
    1302:   bnk->mu1 = 0.10;
    1303:   bnk->mu2 = 0.50;

    1305:   bnk->gamma1 = 0.25;
    1306:   bnk->gamma2 = 0.50;
    1307:   bnk->gamma3 = 2.00;
    1308:   bnk->gamma4 = 4.00;

    1310:   bnk->theta = 0.05;

    1312:   /*  Default values for trust region initialization based on interpolation */
    1313:   bnk->mu1_i = 0.35;
    1314:   bnk->mu2_i = 0.50;

    1316:   bnk->gamma1_i = 0.0625;
    1317:   bnk->gamma2_i = 0.5;
    1318:   bnk->gamma3_i = 2.0;
    1319:   bnk->gamma4_i = 5.0;

    1321:   bnk->theta_i = 0.25;

    1323:   /*  Remaining parameters */
    1324:   bnk->max_cg_its = 0;
    1325:   bnk->min_radius = 1.0e-10;
    1326:   bnk->max_radius = 1.0e10;
    1327:   bnk->epsilon    = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0 / 3.0);
    1328:   bnk->as_tol     = 1.0e-3;
    1329:   bnk->as_step    = 1.0e-3;
    1330:   bnk->dmin       = 1.0e-6;
    1331:   bnk->dmax       = 1.0e6;

    1333:   bnk->M           = NULL;
    1334:   bnk->bfgs_pre    = NULL;
    1335:   bnk->init_type   = BNK_INIT_INTERPOLATION;
    1336:   bnk->update_type = BNK_UPDATE_REDUCTION;
    1337:   bnk->as_type     = BNK_AS_BERTSEKAS;

    1339:   /* Create the embedded BNCG solver */
    1340:   PetscCall(TaoCreate(PetscObjectComm((PetscObject)tao), &bnk->bncg));
    1341:   PetscCall(PetscObjectIncrementTabLevel((PetscObject)bnk->bncg, (PetscObject)tao, 1));
    1342:   PetscCall(TaoSetType(bnk->bncg, TAOBNCG));

    1344:   /* Create the line search */
    1345:   PetscCall(TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch));
    1346:   PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1));
    1347:   PetscCall(TaoLineSearchSetType(tao->linesearch, TAOLINESEARCHMT));
    1348:   PetscCall(TaoLineSearchUseTaoRoutines(tao->linesearch, tao));

    1350:   /*  Set linear solver to default for symmetric matrices */
    1351:   PetscCall(KSPCreate(((PetscObject)tao)->comm, &tao->ksp));
    1352:   PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1));
    1353:   PetscCall(KSPSetType(tao->ksp, KSPSTCG));
    1354:   PetscCall(KSPGetPC(tao->ksp, &pc));
    1355:   PetscCall(PCSetType(pc, PCLMVM));
    1356:   PetscFunctionReturn(PETSC_SUCCESS);
    1357: }