impls/bnk/bnk.c

#include <petsctaolinesearch.h>
#include <../src/tao/bound/impls/bnk/bnk.h>
#include <petscksp.h>

static const char *BNK_INIT[64]   = {"constant", "direction", "interpolation"};
static const char *BNK_UPDATE[64] = {"step", "reduction", "interpolation"};
static const char *BNK_AS[64]     = {"none", "bertsekas"};

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

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

  PetscFunctionBegin;
  PetscCall(MatDestroy(&bnk->Hpre_inactive));
  PetscCall(MatDestroy(&bnk->H_inactive));
  if (bnk->active_idx) {
    PetscCall(MatCreateSubMatrix(tao->hessian, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->H_inactive));
    if (tao->hessian == tao->hessian_pre) {
      PetscCall(PetscObjectReference((PetscObject)bnk->H_inactive));
      bnk->Hpre_inactive = bnk->H_inactive;
    } else {
      PetscCall(MatCreateSubMatrix(tao->hessian_pre, bnk->inactive_idx, bnk->inactive_idx, MAT_INITIAL_MATRIX, &bnk->Hpre_inactive));
    }
    if (bnk->bfgs_pre) PetscCall(PCLMVMSetIS(bnk->bfgs_pre, bnk->inactive_idx));
  } else {
    PetscCall(PetscObjectReference((PetscObject)tao->hessian));
    bnk->H_inactive = tao->hessian;
    PetscCall(PetscObjectReference((PetscObject)tao->hessian_pre));
    bnk->Hpre_inactive = tao->hessian_pre;
    if (bnk->bfgs_pre) PetscCall(PCLMVMClearIS(bnk->bfgs_pre));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

PetscErrorCode TaoBNKInitialize(Tao tao, PetscInt initType, PetscBool *needH)
{
  TAO_BNK     *bnk = (TAO_BNK *)tao->data;
  PC           pc;
  PetscReal    f_min, ftrial, prered, actred, kappa, sigma, resnorm;
  PetscReal    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
  PetscBool    is_bfgs, is_jacobi, is_symmetric, sym_set;
  PetscInt     n, N, nDiff;
  PetscInt     i_max = 5;
  PetscInt     j_max = 1;
  PetscInt     i, j;
  PetscVoidFn *kspTR;

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

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

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

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

  /* Reset KSP stopping reason counters */
  bnk->ksp_atol = 0;
  bnk->ksp_rtol = 0;
  bnk->ksp_dtol = 0;
  bnk->ksp_ctol = 0;
  bnk->ksp_negc = 0;
  bnk->ksp_iter = 0;
  bnk->ksp_othr = 0;

  /* Reset accepted step type counters */
  bnk->tot_cg_its = 0;
  bnk->newt       = 0;
  bnk->bfgs       = 0;
  bnk->sgrad      = 0;
  bnk->grad       = 0;

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

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

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

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

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

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

        if (*needH) {
          /* Compute the Hessian at the new step, and extract the inactive subsystem */
          PetscCall((*bnk->computehessian)(tao));
          PetscCall(TaoBNKEstimateActiveSet(tao, BNK_AS_NONE));
          PetscCall(TaoBNKComputeSubHessian(tao));
          *needH = PETSC_FALSE;
        }

        for (i = 0; i < i_max; ++i) {
          /* Take a steepest descent step and snap it to bounds */
          PetscCall(VecCopy(tao->solution, bnk->Xold));
          PetscCall(VecAXPY(tao->solution, -tao->trust / bnk->gnorm, tao->gradient));
          PetscCall(TaoBoundSolution(tao->solution, tao->XL, tao->XU, 0.0, &nDiff, tao->solution));
          /* Compute the step we actually accepted */
          PetscCall(VecCopy(tao->solution, bnk->W));
          PetscCall(VecAXPY(bnk->W, -1.0, bnk->Xold));
          /* Compute the objective at the trial */
          PetscCall(TaoComputeObjective(tao, tao->solution, &ftrial));
          PetscCheck(!PetscIsInfOrNanReal(bnk->f), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
          PetscCall(VecCopy(bnk->Xold, tao->solution));
          if (PetscIsInfOrNanReal(ftrial)) {
            tau = bnk->gamma1_i;
          } else {
            if (ftrial < f_min) {
              f_min = ftrial;
              sigma = -tao->trust / bnk->gnorm;
            }

            /* Compute the predicted and actual reduction */
            if (bnk->active_idx) {
              PetscCall(VecGetSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive));
              PetscCall(VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
            } else {
              bnk->X_inactive    = bnk->W;
              bnk->inactive_work = bnk->Xwork;
            }
            PetscCall(MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work));
            PetscCall(VecDot(bnk->X_inactive, bnk->inactive_work, &prered));
            if (bnk->active_idx) {
              PetscCall(VecRestoreSubVector(bnk->W, bnk->inactive_idx, &bnk->X_inactive));
              PetscCall(VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
            }
            prered = tao->trust * (bnk->gnorm - 0.5 * tao->trust * prered / (bnk->gnorm * bnk->gnorm));
            actred = bnk->f - ftrial;
            if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
              kappa = 1.0;
            } else {
              kappa = actred / prered;
            }

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

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

              if (tau_max < 1.0) {
                tau = bnk->gamma3_i;
              } else if (tau_max > bnk->gamma4_i) {
                tau = bnk->gamma4_i;
              } else {
                tau = tau_max;
              }
            } else if (PetscAbsScalar(kappa - (PetscReal)1.0) <= bnk->mu2_i) {
              /*  Good agreement */
              max_radius = PetscMax(max_radius, tao->trust);

              if (tau_max < bnk->gamma2_i) {
                tau = bnk->gamma2_i;
              } else if (tau_max > bnk->gamma3_i) {
                tau = bnk->gamma3_i;
              } else {
                tau = tau_max;
              }
            } else {
              /*  Not good agreement */
              if (tau_min > 1.0) {
                tau = bnk->gamma2_i;
              } else if (tau_max < bnk->gamma1_i) {
                tau = bnk->gamma1_i;
              } else if ((tau_min < bnk->gamma1_i) && (tau_max >= 1.0)) {
                tau = bnk->gamma1_i;
              } else if ((tau_1 >= bnk->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
                tau = tau_1;
              } else if ((tau_2 >= bnk->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1_i) || (tau_2 >= 1.0))) {
                tau = tau_2;
              } else {
                tau = tau_max;
              }
            }
          }
          tao->trust = tau * tao->trust;
        }

        if (f_min < bnk->f) {
          /* We accidentally found a solution better than the initial, so accept it */
          bnk->f = f_min;
          PetscCall(VecCopy(tao->solution, bnk->Xold));
          PetscCall(VecAXPY(tao->solution, sigma, tao->gradient));
          PetscCall(TaoBoundSolution(tao->solution, tao->XL, tao->XU, 0.0, &nDiff, tao->solution));
          PetscCall(VecCopy(tao->solution, tao->stepdirection));
          PetscCall(VecAXPY(tao->stepdirection, -1.0, bnk->Xold));
          PetscCall(TaoComputeGradient(tao, tao->solution, bnk->unprojected_gradient));
          PetscCall(TaoBNKEstimateActiveSet(tao, bnk->as_type));
          PetscCall(VecCopy(bnk->unprojected_gradient, tao->gradient));
          if (bnk->active_idx) PetscCall(VecISSet(tao->gradient, bnk->active_idx, 0.0));
          /* Compute gradient at the new iterate and flip switch to compute the Hessian later */
          PetscCall(TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm));
          *needH = PETSC_TRUE;
          /* Test the new step for convergence */
          PetscCall(VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W));
          PetscCall(VecNorm(bnk->W, NORM_2, &resnorm));
          PetscCheck(!PetscIsInfOrNanReal(resnorm), PetscObjectComm((PetscObject)tao), PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
          PetscCall(TaoLogConvergenceHistory(tao, bnk->f, resnorm, 0.0, tao->ksp_its));
          PetscCall(TaoMonitor(tao, tao->niter, bnk->f, resnorm, 0.0, 1.0));
          PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
          if (tao->reason != TAO_CONTINUE_ITERATING) PetscFunctionReturn(PETSC_SUCCESS);
          /* active BNCG recycling early because we have a stepdirection computed */
          PetscCall(TaoSetRecycleHistory(bnk->bncg, PETSC_TRUE));
        }
      }
      tao->trust = PetscMax(tao->trust, max_radius);

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

    default:
      /* Norm of the first direction will initialize radius */
      tao->trust = 0.0;
      break;
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

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

PetscErrorCode TaoBNKComputeHessian(Tao tao)
{
  TAO_BNK *bnk = (TAO_BNK *)tao->data;

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

/*------------------------------------------------------------*/

/* Routine for estimating the active set */

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

  PetscFunctionBegin;
  hadactive = bnk->active_idx ? PETSC_TRUE : PETSC_FALSE;
  switch (asType) {
  case BNK_AS_NONE:
    PetscCall(ISDestroy(&bnk->inactive_idx));
    PetscCall(VecWhichInactive(tao->XL, tao->solution, bnk->unprojected_gradient, tao->XU, PETSC_TRUE, &bnk->inactive_idx));
    PetscCall(ISDestroy(&bnk->active_idx));
    PetscCall(ISComplementVec(bnk->inactive_idx, tao->solution, &bnk->active_idx));
    break;

  case BNK_AS_BERTSEKAS:
    /* Compute the trial step vector with which we will estimate the active set at the next iteration */
    if (bnk->M) {
      /* If the BFGS preconditioner matrix is available, we will construct a trial step with it */
      PetscCall(MatSolve(bnk->M, bnk->unprojected_gradient, bnk->W));
    } else {
      hessComputed = diagExists = PETSC_FALSE;
      if (tao->hessian) PetscCall(MatAssembled(tao->hessian, &hessComputed));
      if (hessComputed) PetscCall(MatHasOperation(tao->hessian, MATOP_GET_DIAGONAL, &diagExists));
      if (diagExists) {
        /* BFGS preconditioner doesn't exist so let's invert the absolute diagonal of the Hessian instead onto the gradient */
        PetscCall(MatGetDiagonal(tao->hessian, bnk->Xwork));
        PetscCall(VecAbs(bnk->Xwork));
        PetscCall(VecMedian(bnk->Diag_min, bnk->Xwork, bnk->Diag_max, bnk->Xwork));
        PetscCall(VecReciprocal(bnk->Xwork));
        PetscCall(VecPointwiseMult(bnk->W, bnk->Xwork, bnk->unprojected_gradient));
      } else {
        /* If the Hessian or its diagonal does not exist, we will simply use gradient step */
        PetscCall(VecCopy(bnk->unprojected_gradient, bnk->W));
      }
    }
    PetscCall(VecScale(bnk->W, -1.0));
    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));
    break;

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

/*------------------------------------------------------------*/

/* Routine for bounding the step direction */

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

  PetscFunctionBegin;
  switch (asType) {
  case BNK_AS_NONE:
    if (bnk->active_idx) PetscCall(VecISSet(step, bnk->active_idx, 0.0));
    break;
  case BNK_AS_BERTSEKAS:
    PetscCall(TaoBoundStep(tao->solution, tao->XL, tao->XU, bnk->active_lower, bnk->active_upper, bnk->active_fixed, 1.0, step));
    break;
  default:
    break;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

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

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

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

  PetscFunctionBegin;
  *terminate = PETSC_FALSE;
  if (bnk->max_cg_its > 0) {
    /* Copy the current function value (important vectors are already shared) */
    bnk->bncg_ctx->f = bnk->f;
    /* Take some small finite number of BNCG iterations */
    PetscCall(TaoSolve(bnk->bncg));
    /* Add the number of gradient and function evaluations to the total */
    tao->nfuncs += bnk->bncg->nfuncs;
    tao->nfuncgrads += bnk->bncg->nfuncgrads;
    tao->ngrads += bnk->bncg->ngrads;
    tao->nhess += bnk->bncg->nhess;
    bnk->tot_cg_its += bnk->bncg->niter;
    /* Extract the BNCG function value out and save it into BNK */
    bnk->f = bnk->bncg_ctx->f;
    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) {
      *terminate = PETSC_TRUE;
    } else {
      PetscCall(TaoBNKEstimateActiveSet(tao, bnk->as_type));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

/* Routine for computing the Newton step. */

PetscErrorCode TaoBNKComputeStep(Tao tao, PetscBool shift, KSPConvergedReason *ksp_reason, PetscInt *step_type)
{
  TAO_BNK     *bnk         = (TAO_BNK *)tao->data;
  PetscInt     bfgsUpdates = 0;
  PetscInt     kspits;
  PetscBool    is_lmvm;
  PetscVoidFn *kspTR;

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

  /* Shift the reduced Hessian matrix */
  if (shift && bnk->pert > 0) {
    PetscCall(PetscObjectTypeCompare((PetscObject)tao->hessian, MATLMVM, &is_lmvm));
    if (is_lmvm) {
      PetscCall(MatShift(tao->hessian, bnk->pert));
    } else {
      PetscCall(MatShift(bnk->H_inactive, bnk->pert));
      if (bnk->H_inactive != bnk->Hpre_inactive) PetscCall(MatShift(bnk->Hpre_inactive, bnk->pert));
    }
  }

  /* Solve the Newton system of equations */
  tao->ksp_its = 0;
  PetscCall(VecSet(tao->stepdirection, 0.0));
  if (bnk->resetksp) {
    PetscCall(KSPReset(tao->ksp));
    PetscCall(KSPResetFromOptions(tao->ksp));
    bnk->resetksp = PETSC_FALSE;
  }
  PetscCall(KSPSetOperators(tao->ksp, bnk->H_inactive, bnk->Hpre_inactive));
  PetscCall(VecCopy(bnk->unprojected_gradient, bnk->Gwork));
  if (bnk->active_idx) {
    PetscCall(VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
    PetscCall(VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
  } else {
    bnk->G_inactive = bnk->unprojected_gradient;
    bnk->X_inactive = tao->stepdirection;
  }
  PetscCall(KSPCGSetRadius(tao->ksp, tao->trust));
  PetscCall(KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive));
  PetscCall(KSPGetIterationNumber(tao->ksp, &kspits));
  tao->ksp_its += kspits;
  tao->ksp_tot_its += kspits;
  PetscCall(PetscObjectQueryFunction((PetscObject)tao->ksp, "KSPCGGetNormD_C", &kspTR));
  if (kspTR) {
    PetscCall(KSPCGGetNormD(tao->ksp, &bnk->dnorm));

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

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

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

        PetscCall(KSPCGSetRadius(tao->ksp, tao->trust));
        PetscCall(KSPSolve(tao->ksp, bnk->G_inactive, bnk->X_inactive));
        PetscCall(KSPGetIterationNumber(tao->ksp, &kspits));
        tao->ksp_its += kspits;
        tao->ksp_tot_its += kspits;
        PetscCall(KSPCGGetNormD(tao->ksp, &bnk->dnorm));

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

  /* Record convergence reasons */
  PetscCall(KSPGetConvergedReason(tao->ksp, ksp_reason));
  if (KSP_CONVERGED_ATOL == *ksp_reason) {
    ++bnk->ksp_atol;
  } else if (KSP_CONVERGED_RTOL == *ksp_reason) {
    ++bnk->ksp_rtol;
  } else if (KSP_CONVERGED_STEP_LENGTH == *ksp_reason) {
    ++bnk->ksp_ctol;
  } else if (KSP_CONVERGED_NEG_CURVE == *ksp_reason) {
    ++bnk->ksp_negc;
  } else if (KSP_DIVERGED_DTOL == *ksp_reason) {
    ++bnk->ksp_dtol;
  } else if (KSP_DIVERGED_ITS == *ksp_reason) {
    ++bnk->ksp_iter;
  } else {
    ++bnk->ksp_othr;
  }

  /* Make sure the BFGS preconditioner is healthy */
  if (bnk->M) {
    PetscCall(MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates));
    if ((KSP_DIVERGED_INDEFINITE_PC == *ksp_reason) && (bfgsUpdates > 0)) {
      /* Preconditioner is numerically indefinite; reset the approximation. */
      PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
      PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
    }
  }
  *step_type = BNK_NEWTON;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

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

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

  PetscFunctionBegin;
  /* Extract subvectors associated with the inactive set */
  if (bnk->active_idx) {
    PetscCall(VecGetSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
    PetscCall(VecGetSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
    PetscCall(VecGetSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
  } else {
    bnk->X_inactive    = tao->stepdirection;
    bnk->inactive_work = bnk->Xwork;
    bnk->G_inactive    = bnk->Gwork;
  }
  /* Recompute the predicted decrease based on the quadratic model */
  PetscCall(MatMult(bnk->H_inactive, bnk->X_inactive, bnk->inactive_work));
  PetscCall(VecAYPX(bnk->inactive_work, -0.5, bnk->G_inactive));
  PetscCall(VecDot(bnk->inactive_work, bnk->X_inactive, prered));
  /* Restore the sub vectors */
  if (bnk->active_idx) {
    PetscCall(VecRestoreSubVector(tao->stepdirection, bnk->inactive_idx, &bnk->X_inactive));
    PetscCall(VecRestoreSubVector(bnk->Xwork, bnk->inactive_idx, &bnk->inactive_work));
    PetscCall(VecRestoreSubVector(bnk->Gwork, bnk->inactive_idx, &bnk->G_inactive));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

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

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

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

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

        /* Initialize the perturbation */
        bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
        PetscCall(PetscObjectTypeCompare((PetscObject)(tao->ksp), KSPGLTR, &is_gltr));
        if (is_gltr) {
          PetscCall(KSPGLTRGetMinEig(tao->ksp, &e_min));
          bnk->pert = PetscMax(bnk->pert, -e_min);
        }
      } else {
        /* Increase the perturbation */
        bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
      }

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

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

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

          *stepType = BNK_SCALED_GRADIENT;
        } else {
          PetscCall(MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates));
          if (1 == bfgsUpdates) {
            /* The first BFGS direction is always the scaled gradient */
            *stepType = BNK_SCALED_GRADIENT;
          } else {
            *stepType = BNK_BFGS;
          }
        }
      }
      /* Make sure the safeguarded fall-back step is zero for actively bounded variables */
      PetscCall(VecScale(tao->stepdirection, -1.0));
      PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));
    } else {
      /* Computed Newton step is descent */
      switch (ksp_reason) {
      case KSP_DIVERGED_NANORINF:
      case KSP_DIVERGED_BREAKDOWN:
      case KSP_DIVERGED_INDEFINITE_MAT:
      case KSP_DIVERGED_INDEFINITE_PC:
      case KSP_CONVERGED_NEG_CURVE:
        /* Matrix or preconditioner is indefinite; increase perturbation */
        if (bnk->pert <= 0.0) {
          PetscBool is_gltr;

          /* Initialize the perturbation */
          bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
          PetscCall(PetscObjectTypeCompare((PetscObject)(tao->ksp), KSPGLTR, &is_gltr));
          if (is_gltr) {
            PetscCall(KSPGLTRGetMinEig(tao->ksp, &e_min));
            bnk->pert = PetscMax(bnk->pert, -e_min);
          }
        } else {
          /* Increase the perturbation */
          bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
        }
        break;

      default:
        /* Newton step computation is good; decrease perturbation */
        bnk->pert = PetscMin(bnk->psfac * bnk->pert, bnk->pmsfac * bnk->gnorm);
        if (bnk->pert < bnk->pmin) bnk->pert = 0.0;
        break;
      }
      *stepType = BNK_NEWTON;
    }
    break;

  case BNK_BFGS:
    /* Check for success (descent direction) */
    PetscCall(VecDot(tao->stepdirection, tao->gradient, &gdx));
    if (gdx >= 0 || PetscIsInfOrNanReal(gdx)) {
      /* Step is not descent or solve was not successful
         Use steepest descent direction (scaled) */
      PetscCall(MatLMVMReset(bnk->M, PETSC_FALSE));
      PetscCall(MatLMVMUpdate(bnk->M, tao->solution, bnk->unprojected_gradient));
      PetscCall(MatSolve(bnk->M, tao->gradient, tao->stepdirection));
      PetscCall(VecScale(tao->stepdirection, -1.0));
      PetscCall(TaoBNKBoundStep(tao, bnk->as_type, tao->stepdirection));
      *stepType = BNK_SCALED_GRADIENT;
    } else {
      *stepType = BNK_BFGS;
    }
    break;

  case BNK_SCALED_GRADIENT:
    break;

  default:
    break;
  }

  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

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

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

PetscErrorCode TaoBNKPerformLineSearch(Tao tao, PetscInt *stepType, PetscReal *steplen, TaoLineSearchConvergedReason *reason)
{
  TAO_BNK                     *bnk = (TAO_BNK *)tao->data;
  TaoLineSearchConvergedReason ls_reason;
  PetscReal                    e_min, gdx;
  PetscInt                     bfgsUpdates;

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

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

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

        /* Initialize the perturbation */
        bnk->pert = PetscMin(bnk->imax, PetscMax(bnk->imin, bnk->imfac * bnk->gnorm));
        PetscCall(PetscObjectTypeCompare((PetscObject)(tao->ksp), KSPGLTR, &is_gltr));
        if (is_gltr) {
          PetscCall(KSPGLTRGetMinEig(tao->ksp, &e_min));
          bnk->pert = PetscMax(bnk->pert, -e_min);
        }
      } else {
        /* Increase the perturbation */
        bnk->pert = PetscMin(bnk->pmax, PetscMax(bnk->pgfac * bnk->pert, bnk->pmgfac * bnk->gnorm));
      }

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

          bfgsUpdates = 1;
          *stepType   = BNK_SCALED_GRADIENT;
        } else {
          PetscCall(MatLMVMGetUpdateCount(bnk->M, &bfgsUpdates));
          if (1 == bfgsUpdates) {
            /* The first BFGS direction is always the scaled gradient */
            *stepType = BNK_SCALED_GRADIENT;
          } else {
            *stepType = BNK_BFGS;
          }
        }
      }
      break;

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

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

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

/*------------------------------------------------------------*/

/* Routine for updating the trust radius.

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

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

  PetscReal step, kappa;
  PetscReal gdx, tau_1, tau_2, tau_min, tau_max;

  PetscFunctionBegin;
  /* Update trust region radius */
  *accept = PETSC_FALSE;
  switch (updateType) {
  case BNK_UPDATE_STEP:
    *accept = PETSC_TRUE; /* always accept here because line search succeeded */
    if (stepType == BNK_NEWTON) {
      PetscCall(TaoLineSearchGetStepLength(tao->linesearch, &step));
      if (step < bnk->nu1) {
        /* Very bad step taken; reduce radius */
        tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
      } else if (step < bnk->nu2) {
        /* Reasonably bad step taken; reduce radius */
        tao->trust = bnk->omega2 * PetscMin(bnk->dnorm, tao->trust);
      } else if (step < bnk->nu3) {
        /*  Reasonable step was taken; leave radius alone */
        if (bnk->omega3 < 1.0) {
          tao->trust = bnk->omega3 * PetscMin(bnk->dnorm, tao->trust);
        } else if (bnk->omega3 > 1.0) {
          tao->trust = PetscMax(bnk->omega3 * bnk->dnorm, tao->trust);
        }
      } else if (step < bnk->nu4) {
        /*  Full step taken; increase the radius */
        tao->trust = PetscMax(bnk->omega4 * bnk->dnorm, tao->trust);
      } else {
        /*  More than full step taken; increase the radius */
        tao->trust = PetscMax(bnk->omega5 * bnk->dnorm, tao->trust);
      }
    } else {
      /*  Newton step was not good; reduce the radius */
      tao->trust = bnk->omega1 * PetscMin(bnk->dnorm, tao->trust);
    }
    break;

  case BNK_UPDATE_REDUCTION:
    if (stepType == BNK_NEWTON) {
      if ((prered < 0.0) || PetscIsInfOrNanReal(prered)) {
        /* The predicted reduction has the wrong sign.  This cannot
           happen in infinite precision arithmetic.  Step should
           be rejected! */
        tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
      } else {
        if (PetscIsInfOrNanReal(actred)) {
          tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
        } else {
          if ((PetscAbsScalar(actred) <= PetscMax(1.0, PetscAbsScalar(bnk->f)) * bnk->epsilon) && (PetscAbsScalar(prered) <= PetscMax(1.0, PetscAbsScalar(bnk->f)) * bnk->epsilon)) {
            kappa = 1.0;
          } else {
            kappa = actred / prered;
          }
          /* Accept or reject the step and update radius */
          if (kappa < bnk->eta1) {
            /* Reject the step */
            tao->trust = bnk->alpha1 * PetscMin(tao->trust, bnk->dnorm);
          } else {
            /* Accept the step */
            *accept = PETSC_TRUE;
            /* Update the trust region radius only if the computed step is at the trust radius boundary */
            if (bnk->dnorm == tao->trust) {
              if (kappa < bnk->eta2) {
                /* Marginal bad step */
                tao->trust = bnk->alpha2 * tao->trust;
              } else if (kappa < bnk->eta3) {
                /* Reasonable step */
                tao->trust = bnk->alpha3 * tao->trust;
              } else if (kappa < bnk->eta4) {
                /* Good step */
                tao->trust = bnk->alpha4 * tao->trust;
              } else {
                /* Very good step */
                tao->trust = bnk->alpha5 * tao->trust;
              }
            }
          }
        }
      }
    } else {
      /*  Newton step was not good; reduce the radius */
      tao->trust = bnk->alpha1 * PetscMin(bnk->dnorm, tao->trust);
    }
    break;

  default:
    if (stepType == BNK_NEWTON) {
      if (prered < 0.0) {
        /*  The predicted reduction has the wrong sign.  This cannot */
        /*  happen in infinite precision arithmetic.  Step should */
        /*  be rejected! */
        tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
      } else {
        if (PetscIsInfOrNanReal(actred)) {
          tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
        } else {
          if ((PetscAbsScalar(actred) <= bnk->epsilon) && (PetscAbsScalar(prered) <= bnk->epsilon)) {
            kappa = 1.0;
          } else {
            kappa = actred / prered;
          }

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

          if (kappa >= 1.0 - bnk->mu1) {
            /*  Great agreement */
            *accept = PETSC_TRUE;
            if (tau_max < 1.0) {
              tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
            } else if (tau_max > bnk->gamma4) {
              tao->trust = PetscMax(tao->trust, bnk->gamma4 * bnk->dnorm);
            } else {
              tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
            }
          } else if (kappa >= 1.0 - bnk->mu2) {
            /*  Good agreement */
            *accept = PETSC_TRUE;
            if (tau_max < bnk->gamma2) {
              tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
            } else if (tau_max > bnk->gamma3) {
              tao->trust = PetscMax(tao->trust, bnk->gamma3 * bnk->dnorm);
            } else if (tau_max < 1.0) {
              tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
            } else {
              tao->trust = PetscMax(tao->trust, tau_max * bnk->dnorm);
            }
          } else {
            /*  Not good agreement */
            if (tau_min > 1.0) {
              tao->trust = bnk->gamma2 * PetscMin(tao->trust, bnk->dnorm);
            } else if (tau_max < bnk->gamma1) {
              tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
            } else if ((tau_min < bnk->gamma1) && (tau_max >= 1.0)) {
              tao->trust = bnk->gamma1 * PetscMin(tao->trust, bnk->dnorm);
            } else if ((tau_1 >= bnk->gamma1) && (tau_1 < 1.0) && ((tau_2 < bnk->gamma1) || (tau_2 >= 1.0))) {
              tao->trust = tau_1 * PetscMin(tao->trust, bnk->dnorm);
            } else if ((tau_2 >= bnk->gamma1) && (tau_2 < 1.0) && ((tau_1 < bnk->gamma1) || (tau_2 >= 1.0))) {
              tao->trust = tau_2 * PetscMin(tao->trust, bnk->dnorm);
            } else {
              tao->trust = tau_max * PetscMin(tao->trust, bnk->dnorm);
            }
          }
        }
      }
    } else {
      /*  Newton step was not good; reduce the radius */
      tao->trust = bnk->gamma1 * PetscMin(bnk->dnorm, tao->trust);
    }
    break;
  }
  /* Make sure the radius does not violate min and max settings */
  tao->trust = PetscMin(tao->trust, bnk->max_radius);
  tao->trust = PetscMax(tao->trust, bnk->min_radius);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ---------------------------------------------------------- */

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

  PetscFunctionBegin;
  switch (stepType) {
  case BNK_NEWTON:
    ++bnk->newt;
    break;
  case BNK_BFGS:
    ++bnk->bfgs;
    break;
  case BNK_SCALED_GRADIENT:
    ++bnk->sgrad;
    break;
  case BNK_GRADIENT:
    ++bnk->grad;
    break;
  default:
    break;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ---------------------------------------------------------- */

PetscErrorCode TaoSetUp_BNK(Tao tao)
{
  TAO_BNK *bnk = (TAO_BNK *)tao->data;
  PetscInt i;

  PetscFunctionBegin;
  if (!tao->gradient) PetscCall(VecDuplicate(tao->solution, &tao->gradient));
  if (!tao->stepdirection) PetscCall(VecDuplicate(tao->solution, &tao->stepdirection));
  if (!bnk->W) PetscCall(VecDuplicate(tao->solution, &bnk->W));
  if (!bnk->Xold) PetscCall(VecDuplicate(tao->solution, &bnk->Xold));
  if (!bnk->Gold) PetscCall(VecDuplicate(tao->solution, &bnk->Gold));
  if (!bnk->Xwork) PetscCall(VecDuplicate(tao->solution, &bnk->Xwork));
  if (!bnk->Gwork) PetscCall(VecDuplicate(tao->solution, &bnk->Gwork));
  if (!bnk->unprojected_gradient) PetscCall(VecDuplicate(tao->solution, &bnk->unprojected_gradient));
  if (!bnk->unprojected_gradient_old) PetscCall(VecDuplicate(tao->solution, &bnk->unprojected_gradient_old));
  if (!bnk->Diag_min) PetscCall(VecDuplicate(tao->solution, &bnk->Diag_min));
  if (!bnk->Diag_max) PetscCall(VecDuplicate(tao->solution, &bnk->Diag_max));
  if (bnk->max_cg_its > 0) {
    /* Ensure that the important common vectors are shared between BNK and embedded BNCG */
    bnk->bncg_ctx = (TAO_BNCG *)bnk->bncg->data;
    PetscCall(PetscObjectReference((PetscObject)(bnk->unprojected_gradient_old)));
    PetscCall(VecDestroy(&bnk->bncg_ctx->unprojected_gradient_old));
    bnk->bncg_ctx->unprojected_gradient_old = bnk->unprojected_gradient_old;
    PetscCall(PetscObjectReference((PetscObject)(bnk->unprojected_gradient)));
    PetscCall(VecDestroy(&bnk->bncg_ctx->unprojected_gradient));
    bnk->bncg_ctx->unprojected_gradient = bnk->unprojected_gradient;
    PetscCall(PetscObjectReference((PetscObject)(bnk->Gold)));
    PetscCall(VecDestroy(&bnk->bncg_ctx->G_old));
    bnk->bncg_ctx->G_old = bnk->Gold;
    PetscCall(PetscObjectReference((PetscObject)(tao->gradient)));
    PetscCall(VecDestroy(&bnk->bncg->gradient));
    bnk->bncg->gradient = tao->gradient;
    PetscCall(PetscObjectReference((PetscObject)(tao->stepdirection)));
    PetscCall(VecDestroy(&bnk->bncg->stepdirection));
    bnk->bncg->stepdirection = tao->stepdirection;
    PetscCall(TaoSetSolution(bnk->bncg, tao->solution));
    /* Copy over some settings from BNK into BNCG */
    PetscCall(TaoSetMaximumIterations(bnk->bncg, bnk->max_cg_its));
    PetscCall(TaoSetTolerances(bnk->bncg, tao->gatol, tao->grtol, tao->gttol));
    PetscCall(TaoSetFunctionLowerBound(bnk->bncg, tao->fmin));
    PetscCall(TaoSetConvergenceTest(bnk->bncg, tao->ops->convergencetest, tao->cnvP));
    PetscCall(TaoSetObjective(bnk->bncg, tao->ops->computeobjective, tao->user_objP));
    PetscCall(TaoSetGradient(bnk->bncg, NULL, tao->ops->computegradient, tao->user_gradP));
    PetscCall(TaoSetObjectiveAndGradient(bnk->bncg, NULL, tao->ops->computeobjectiveandgradient, tao->user_objgradP));
    PetscCall(PetscObjectCopyFortranFunctionPointers((PetscObject)tao, (PetscObject)(bnk->bncg)));
    for (i = 0; i < tao->numbermonitors; ++i) {
      PetscCall(TaoMonitorSet(bnk->bncg, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]));
      PetscCall(PetscObjectReference((PetscObject)(tao->monitorcontext[i])));
    }
  }
  bnk->X_inactive    = NULL;
  bnk->G_inactive    = NULL;
  bnk->inactive_work = NULL;
  bnk->active_work   = NULL;
  bnk->inactive_idx  = NULL;
  bnk->active_idx    = NULL;
  bnk->active_lower  = NULL;
  bnk->active_upper  = NULL;
  bnk->active_fixed  = NULL;
  bnk->M             = NULL;
  bnk->H_inactive    = NULL;
  bnk->Hpre_inactive = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

PetscErrorCode TaoDestroy_BNK(Tao tao)
{
  TAO_BNK *bnk = (TAO_BNK *)tao->data;

  PetscFunctionBegin;
  PetscCall(VecDestroy(&bnk->W));
  PetscCall(VecDestroy(&bnk->Xold));
  PetscCall(VecDestroy(&bnk->Gold));
  PetscCall(VecDestroy(&bnk->Xwork));
  PetscCall(VecDestroy(&bnk->Gwork));
  PetscCall(VecDestroy(&bnk->unprojected_gradient));
  PetscCall(VecDestroy(&bnk->unprojected_gradient_old));
  PetscCall(VecDestroy(&bnk->Diag_min));
  PetscCall(VecDestroy(&bnk->Diag_max));
  PetscCall(ISDestroy(&bnk->active_lower));
  PetscCall(ISDestroy(&bnk->active_upper));
  PetscCall(ISDestroy(&bnk->active_fixed));
  PetscCall(ISDestroy(&bnk->active_idx));
  PetscCall(ISDestroy(&bnk->inactive_idx));
  PetscCall(MatDestroy(&bnk->Hpre_inactive));
  PetscCall(MatDestroy(&bnk->H_inactive));
  PetscCall(TaoDestroy(&bnk->bncg));
  PetscCall(KSPDestroy(&tao->ksp));
  PetscCall(PetscFree(tao->data));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

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

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "Newton-Krylov method for bound constrained optimization");
  PetscCall(PetscOptionsEList("-tao_bnk_init_type", "radius initialization type", "", BNK_INIT, BNK_INIT_TYPES, BNK_INIT[bnk->init_type], &bnk->init_type, NULL));
  PetscCall(PetscOptionsEList("-tao_bnk_update_type", "radius update type", "", BNK_UPDATE, BNK_UPDATE_TYPES, BNK_UPDATE[bnk->update_type], &bnk->update_type, NULL));
  PetscCall(PetscOptionsEList("-tao_bnk_as_type", "active set estimation method", "", BNK_AS, BNK_AS_TYPES, BNK_AS[bnk->as_type], &bnk->as_type, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_sval", "(developer) Hessian perturbation starting value", "", bnk->sval, &bnk->sval, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_imin", "(developer) minimum initial Hessian perturbation", "", bnk->imin, &bnk->imin, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_imax", "(developer) maximum initial Hessian perturbation", "", bnk->imax, &bnk->imax, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_imfac", "(developer) initial merit factor for Hessian perturbation", "", bnk->imfac, &bnk->imfac, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_pmin", "(developer) minimum Hessian perturbation", "", bnk->pmin, &bnk->pmin, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_pmax", "(developer) maximum Hessian perturbation", "", bnk->pmax, &bnk->pmax, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_pgfac", "(developer) Hessian perturbation growth factor", "", bnk->pgfac, &bnk->pgfac, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_psfac", "(developer) Hessian perturbation shrink factor", "", bnk->psfac, &bnk->psfac, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_pmgfac", "(developer) merit growth factor for Hessian perturbation", "", bnk->pmgfac, &bnk->pmgfac, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_pmsfac", "(developer) merit shrink factor for Hessian perturbation", "", bnk->pmsfac, &bnk->pmsfac, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_eta1", "(developer) threshold for rejecting step (-tao_bnk_update_type reduction)", "", bnk->eta1, &bnk->eta1, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_eta2", "(developer) threshold for accepting marginal step (-tao_bnk_update_type reduction)", "", bnk->eta2, &bnk->eta2, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_eta3", "(developer) threshold for accepting reasonable step (-tao_bnk_update_type reduction)", "", bnk->eta3, &bnk->eta3, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_eta4", "(developer) threshold for accepting good step (-tao_bnk_update_type reduction)", "", bnk->eta4, &bnk->eta4, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_alpha1", "(developer) radius reduction factor for rejected step (-tao_bnk_update_type reduction)", "", bnk->alpha1, &bnk->alpha1, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_alpha2", "(developer) radius reduction factor for marginally accepted bad step (-tao_bnk_update_type reduction)", "", bnk->alpha2, &bnk->alpha2, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_alpha3", "(developer) radius increase factor for reasonable accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha3, &bnk->alpha3, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_alpha4", "(developer) radius increase factor for good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha4, &bnk->alpha4, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_alpha5", "(developer) radius increase factor for very good accepted step (-tao_bnk_update_type reduction)", "", bnk->alpha5, &bnk->alpha5, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_nu1", "(developer) threshold for small line-search step length (-tao_bnk_update_type step)", "", bnk->nu1, &bnk->nu1, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_nu2", "(developer) threshold for reasonable line-search step length (-tao_bnk_update_type step)", "", bnk->nu2, &bnk->nu2, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_nu3", "(developer) threshold for large line-search step length (-tao_bnk_update_type step)", "", bnk->nu3, &bnk->nu3, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_nu4", "(developer) threshold for very large line-search step length (-tao_bnk_update_type step)", "", bnk->nu4, &bnk->nu4, NULL));
  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));
  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));
  PetscCall(PetscOptionsReal("-tao_bnk_omega3", "(developer) radius factor for decent line-search step length (-tao_bnk_update_type step)", "", bnk->omega3, &bnk->omega3, NULL));
  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));
  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));
  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));
  PetscCall(PetscOptionsReal("-tao_bnk_mu2_i", "(developer) threshold for accepting good step (-tao_bnk_init_type interpolation)", "", bnk->mu2_i, &bnk->mu2_i, NULL));
  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));
  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));
  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));
  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));
  PetscCall(PetscOptionsReal("-tao_bnk_theta_i", "(developer) trust region interpolation factor (-tao_bnk_init_type interpolation)", "", bnk->theta_i, &bnk->theta_i, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_mu1", "(developer) threshold for accepting very good step (-tao_bnk_update_type interpolation)", "", bnk->mu1, &bnk->mu1, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_mu2", "(developer) threshold for accepting good step (-tao_bnk_update_type interpolation)", "", bnk->mu2, &bnk->mu2, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_gamma1", "(developer) radius reduction factor for rejected very bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma1, &bnk->gamma1, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_gamma2", "(developer) radius reduction factor for rejected bad step (-tao_bnk_update_type interpolation)", "", bnk->gamma2, &bnk->gamma2, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_gamma3", "(developer) radius increase factor for accepted good step (-tao_bnk_update_type interpolation)", "", bnk->gamma3, &bnk->gamma3, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_gamma4", "(developer) radius increase factor for accepted very good step (-tao_bnk_update_type interpolation)", "", bnk->gamma4, &bnk->gamma4, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_theta", "(developer) trust region interpolation factor (-tao_bnk_update_type interpolation)", "", bnk->theta, &bnk->theta, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_min_radius", "(developer) lower bound on initial radius", "", bnk->min_radius, &bnk->min_radius, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_max_radius", "(developer) upper bound on radius", "", bnk->max_radius, &bnk->max_radius, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_epsilon", "(developer) tolerance used when computing actual and predicted reduction", "", bnk->epsilon, &bnk->epsilon, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_as_tol", "(developer) initial tolerance used when estimating actively bounded variables", "", bnk->as_tol, &bnk->as_tol, NULL));
  PetscCall(PetscOptionsReal("-tao_bnk_as_step", "(developer) step length used when estimating actively bounded variables", "", bnk->as_step, &bnk->as_step, NULL));
  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));
  PetscOptionsHeadEnd();

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

  PetscCall(KSPSetOptionsPrefix(tao->ksp, ((PetscObject)(tao))->prefix));
  PetscCall(KSPAppendOptionsPrefix(tao->ksp, "tao_bnk_"));
  PetscCall(KSPSetFromOptions(tao->ksp));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

PetscErrorCode TaoView_BNK(Tao tao, PetscViewer viewer)
{
  TAO_BNK  *bnk = (TAO_BNK *)tao->data;
  PetscInt  nrejects;
  PetscBool isascii;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
  if (isascii) {
    PetscCall(PetscViewerASCIIPushTab(viewer));
    PetscCall(TaoView(bnk->bncg, viewer));
    if (bnk->M) {
      PetscCall(MatLMVMGetRejectCount(bnk->M, &nrejects));
      PetscCall(PetscViewerASCIIPrintf(viewer, "Rejected BFGS updates: %" PetscInt_FMT "\n", nrejects));
    }
    PetscCall(PetscViewerASCIIPrintf(viewer, "CG steps: %" PetscInt_FMT "\n", bnk->tot_cg_its));
    PetscCall(PetscViewerASCIIPrintf(viewer, "Newton steps: %" PetscInt_FMT "\n", bnk->newt));
    if (bnk->M) PetscCall(PetscViewerASCIIPrintf(viewer, "BFGS steps: %" PetscInt_FMT "\n", bnk->bfgs));
    PetscCall(PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %" PetscInt_FMT "\n", bnk->sgrad));
    PetscCall(PetscViewerASCIIPrintf(viewer, "Gradient steps: %" PetscInt_FMT "\n", bnk->grad));
    PetscCall(PetscViewerASCIIPrintf(viewer, "KSP termination reasons:\n"));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  atol: %" PetscInt_FMT "\n", bnk->ksp_atol));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  rtol: %" PetscInt_FMT "\n", bnk->ksp_rtol));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  ctol: %" PetscInt_FMT "\n", bnk->ksp_ctol));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  negc: %" PetscInt_FMT "\n", bnk->ksp_negc));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  dtol: %" PetscInt_FMT "\n", bnk->ksp_dtol));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  iter: %" PetscInt_FMT "\n", bnk->ksp_iter));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  othr: %" PetscInt_FMT "\n", bnk->ksp_othr));
    PetscCall(PetscViewerASCIIPopTab(viewer));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ---------------------------------------------------------- */

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

    Options Database Keys:
+ -tao_bnk_max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
. -tao_bnk_init_type - trust radius initialization method ("constant", "direction", "interpolation")
. -tao_bnk_update_type - trust radius update method ("step", "direction", "interpolation")
. -tao_bnk_as_type - active-set estimation method ("none", "bertsekas")
. -tao_bnk_as_tol - (developer) initial tolerance used in estimating bounded active variables (-as_type bertsekas)
. -tao_bnk_as_step - (developer) trial step length used in estimating bounded active variables (-as_type bertsekas)
. -tao_bnk_sval - (developer) Hessian perturbation starting value
. -tao_bnk_imin - (developer) minimum initial Hessian perturbation
. -tao_bnk_imax - (developer) maximum initial Hessian perturbation
. -tao_bnk_pmin - (developer) minimum Hessian perturbation
. -tao_bnk_pmax - (developer) aximum Hessian perturbation
. -tao_bnk_pgfac - (developer) Hessian perturbation growth factor
. -tao_bnk_psfac - (developer) Hessian perturbation shrink factor
. -tao_bnk_imfac - (developer) initial merit factor for Hessian perturbation
. -tao_bnk_pmgfac - (developer) merit growth factor for Hessian perturbation
. -tao_bnk_pmsfac - (developer) merit shrink factor for Hessian perturbation
. -tao_bnk_eta1 - (developer) threshold for rejecting step (-update_type reduction)
. -tao_bnk_eta2 - (developer) threshold for accepting marginal step (-update_type reduction)
. -tao_bnk_eta3 - (developer) threshold for accepting reasonable step (-update_type reduction)
. -tao_bnk_eta4 - (developer) threshold for accepting good step (-update_type reduction)
. -tao_bnk_alpha1 - (developer) radius reduction factor for rejected step (-update_type reduction)
. -tao_bnk_alpha2 - (developer) radius reduction factor for marginally accepted bad step (-update_type reduction)
. -tao_bnk_alpha3 - (developer) radius increase factor for reasonable accepted step (-update_type reduction)
. -tao_bnk_alpha4 - (developer) radius increase factor for good accepted step (-update_type reduction)
. -tao_bnk_alpha5 - (developer) radius increase factor for very good accepted step (-update_type reduction)
. -tao_bnk_epsilon - (developer) tolerance for small pred/actual ratios that trigger automatic step acceptance (-update_type reduction)
. -tao_bnk_mu1 - (developer) threshold for accepting very good step (-update_type interpolation)
. -tao_bnk_mu2 - (developer) threshold for accepting good step (-update_type interpolation)
. -tao_bnk_gamma1 - (developer) radius reduction factor for rejected very bad step (-update_type interpolation)
. -tao_bnk_gamma2 - (developer) radius reduction factor for rejected bad step (-update_type interpolation)
. -tao_bnk_gamma3 - (developer) radius increase factor for accepted good step (-update_type interpolation)
. -tao_bnk_gamma4 - (developer) radius increase factor for accepted very good step (-update_type interpolation)
. -tao_bnk_theta - (developer) trust region interpolation factor (-update_type interpolation)
. -tao_bnk_nu1 - (developer) threshold for small line-search step length (-update_type step)
. -tao_bnk_nu2 - (developer) threshold for reasonable line-search step length (-update_type step)
. -tao_bnk_nu3 - (developer) threshold for large line-search step length (-update_type step)
. -tao_bnk_nu4 - (developer) threshold for very large line-search step length (-update_type step)
. -tao_bnk_omega1 - (developer) radius reduction factor for very small line-search step length (-update_type step)
. -tao_bnk_omega2 - (developer) radius reduction factor for small line-search step length (-update_type step)
. -tao_bnk_omega3 - (developer) radius factor for decent line-search step length (-update_type step)
. -tao_bnk_omega4 - (developer) radius increase factor for large line-search step length (-update_type step)
. -tao_bnk_omega5 - (developer) radius increase factor for very large line-search step length (-update_type step)
. -tao_bnk_mu1_i -  (developer) threshold for accepting very good step (-init_type interpolation)
. -tao_bnk_mu2_i -  (developer) threshold for accepting good step (-init_type interpolation)
. -tao_bnk_gamma1_i - (developer) radius reduction factor for rejected very bad step (-init_type interpolation)
. -tao_bnk_gamma2_i - (developer) radius reduction factor for rejected bad step (-init_type interpolation)
. -tao_bnk_gamma3_i - (developer) radius increase factor for accepted good step (-init_type interpolation)
. -tao_bnk_gamma4_i - (developer) radius increase factor for accepted very good step (-init_type interpolation)
- -tao_bnk_theta_i - (developer) trust region interpolation factor (-init_type interpolation)

  Level: beginner
M*/

PetscErrorCode TaoCreate_BNK(Tao tao)
{
  TAO_BNK *bnk;
  PC       pc;

  PetscFunctionBegin;
  PetscCall(PetscNew(&bnk));

  tao->ops->setup          = TaoSetUp_BNK;
  tao->ops->view           = TaoView_BNK;
  tao->ops->setfromoptions = TaoSetFromOptions_BNK;
  tao->ops->destroy        = TaoDestroy_BNK;

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

  tao->data = (void *)bnk;

  /*  Hessian shifting parameters */
  bnk->computehessian = TaoBNKComputeHessian;
  bnk->computestep    = TaoBNKComputeStep;

  bnk->sval  = 0.0;
  bnk->imin  = 1.0e-4;
  bnk->imax  = 1.0e+2;
  bnk->imfac = 1.0e-1;

  bnk->pmin   = 1.0e-12;
  bnk->pmax   = 1.0e+2;
  bnk->pgfac  = 1.0e+1;
  bnk->psfac  = 4.0e-1;
  bnk->pmgfac = 1.0e-1;
  bnk->pmsfac = 1.0e-1;

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

  bnk->omega1 = 0.25;
  bnk->omega2 = 0.50;
  bnk->omega3 = 1.00;
  bnk->omega4 = 2.00;
  bnk->omega5 = 4.00;

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

  bnk->alpha1 = 0.25;
  bnk->alpha2 = 0.50;
  bnk->alpha3 = 1.00;
  bnk->alpha4 = 2.00;
  bnk->alpha5 = 4.00;

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

  bnk->gamma1 = 0.25;
  bnk->gamma2 = 0.50;
  bnk->gamma3 = 2.00;
  bnk->gamma4 = 4.00;

  bnk->theta = 0.05;

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

  bnk->gamma1_i = 0.0625;
  bnk->gamma2_i = 0.5;
  bnk->gamma3_i = 2.0;
  bnk->gamma4_i = 5.0;

  bnk->theta_i = 0.25;

  /*  Remaining parameters */
  bnk->max_cg_its = 0;
  bnk->min_radius = 1.0e-10;
  bnk->max_radius = 1.0e10;
  bnk->epsilon    = PetscPowReal(PETSC_MACHINE_EPSILON, 2.0 / 3.0);
  bnk->as_tol     = 1.0e-3;
  bnk->as_step    = 1.0e-3;
  bnk->dmin       = 1.0e-6;
  bnk->dmax       = 1.0e6;

  bnk->M           = NULL;
  bnk->bfgs_pre    = NULL;
  bnk->init_type   = BNK_INIT_INTERPOLATION;
  bnk->update_type = BNK_UPDATE_REDUCTION;
  bnk->as_type     = BNK_AS_BERTSEKAS;

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

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

  /*  Set linear solver to default for symmetric matrices */
  PetscCall(KSPCreate(((PetscObject)tao)->comm, &tao->ksp));
  PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1));
  PetscCall(KSPSetType(tao->ksp, KSPSTCG));
  PetscCall(KSPGetPC(tao->ksp, &pc));
  PetscCall(PCSetType(pc, PCLMVM));
  PetscFunctionReturn(PETSC_SUCCESS);
}