impls/ipm/pdipm.c

#include <../src/tao/constrained/impls/ipm/pdipm.h>

/*
   TaoPDIPMEvaluateFunctionsAndJacobians - Evaluate the objective function f, gradient fx, constraints, and all the Jacobians at current vector

   Collective

   Input Parameter:
+  tao - solver context
-  x - vector at which all objects to be evaluated

   Level: beginner

.seealso: `TAOPDIPM`, `TaoPDIPMUpdateConstraints()`, `TaoPDIPMSetUpBounds()`
*/
static PetscErrorCode TaoPDIPMEvaluateFunctionsAndJacobians(Tao tao, Vec x)
{
  TAO_PDIPM *pdipm = (TAO_PDIPM *)tao->data;

  PetscFunctionBegin;
  /* Compute user objective function and gradient */
  PetscCall(TaoComputeObjectiveAndGradient(tao, x, &pdipm->obj, tao->gradient));

  /* Equality constraints and Jacobian */
  if (pdipm->Ng) {
    PetscCall(TaoComputeEqualityConstraints(tao, x, tao->constraints_equality));
    PetscCall(TaoComputeJacobianEquality(tao, x, tao->jacobian_equality, tao->jacobian_equality_pre));
  }

  /* Inequality constraints and Jacobian */
  if (pdipm->Nh) {
    PetscCall(TaoComputeInequalityConstraints(tao, x, tao->constraints_inequality));
    PetscCall(TaoComputeJacobianInequality(tao, x, tao->jacobian_inequality, tao->jacobian_inequality_pre));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  TaoPDIPMUpdateConstraints - Update the vectors ce and ci at x

  Collective

  Input Parameter:
+ tao - Tao context
- x - vector at which constraints to be evaluated

   Level: beginner

.seealso: `TAOPDIPM`, `TaoPDIPMEvaluateFunctionsAndJacobians()`
*/
static PetscErrorCode TaoPDIPMUpdateConstraints(Tao tao, Vec x)
{
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  PetscInt           i, offset, offset1, k, xstart;
  PetscScalar       *carr;
  const PetscInt    *ubptr, *lbptr, *bxptr, *fxptr;
  const PetscScalar *xarr, *xuarr, *xlarr, *garr, *harr;

  PetscFunctionBegin;
  PetscCall(VecGetOwnershipRange(x, &xstart, NULL));
  PetscCall(VecGetArrayRead(x, &xarr));
  PetscCall(VecGetArrayRead(tao->XU, &xuarr));
  PetscCall(VecGetArrayRead(tao->XL, &xlarr));

  /* (1) Update ce vector */
  PetscCall(VecGetArrayWrite(pdipm->ce, &carr));

  if (pdipm->Ng) {
    /* (1.a) Inserting updated g(x) */
    PetscCall(VecGetArrayRead(tao->constraints_equality, &garr));
    PetscCall(PetscMemcpy(carr, garr, pdipm->ng * sizeof(PetscScalar)));
    PetscCall(VecRestoreArrayRead(tao->constraints_equality, &garr));
  }

  /* (1.b) Update xfixed */
  if (pdipm->Nxfixed) {
    offset = pdipm->ng;
    PetscCall(ISGetIndices(pdipm->isxfixed, &fxptr)); /* global indices in x */
    for (k = 0; k < pdipm->nxfixed; k++) {
      i                = fxptr[k] - xstart;
      carr[offset + k] = xarr[i] - xuarr[i];
    }
  }
  PetscCall(VecRestoreArrayWrite(pdipm->ce, &carr));

  /* (2) Update ci vector */
  PetscCall(VecGetArrayWrite(pdipm->ci, &carr));

  if (pdipm->Nh) {
    /* (2.a) Inserting updated h(x) */
    PetscCall(VecGetArrayRead(tao->constraints_inequality, &harr));
    PetscCall(PetscMemcpy(carr, harr, pdipm->nh * sizeof(PetscScalar)));
    PetscCall(VecRestoreArrayRead(tao->constraints_inequality, &harr));
  }

  /* (2.b) Update xub */
  offset = pdipm->nh;
  if (pdipm->Nxub) {
    PetscCall(ISGetIndices(pdipm->isxub, &ubptr));
    for (k = 0; k < pdipm->nxub; k++) {
      i                = ubptr[k] - xstart;
      carr[offset + k] = xuarr[i] - xarr[i];
    }
  }

  if (pdipm->Nxlb) {
    /* (2.c) Update xlb */
    offset += pdipm->nxub;
    PetscCall(ISGetIndices(pdipm->isxlb, &lbptr)); /* global indices in x */
    for (k = 0; k < pdipm->nxlb; k++) {
      i                = lbptr[k] - xstart;
      carr[offset + k] = xarr[i] - xlarr[i];
    }
  }

  if (pdipm->Nxbox) {
    /* (2.d) Update xbox */
    offset += pdipm->nxlb;
    offset1 = offset + pdipm->nxbox;
    PetscCall(ISGetIndices(pdipm->isxbox, &bxptr)); /* global indices in x */
    for (k = 0; k < pdipm->nxbox; k++) {
      i                 = bxptr[k] - xstart; /* local indices in x */
      carr[offset + k]  = xuarr[i] - xarr[i];
      carr[offset1 + k] = xarr[i] - xlarr[i];
    }
  }
  PetscCall(VecRestoreArrayWrite(pdipm->ci, &carr));

  /* Restoring Vectors */
  PetscCall(VecRestoreArrayRead(x, &xarr));
  PetscCall(VecRestoreArrayRead(tao->XU, &xuarr));
  PetscCall(VecRestoreArrayRead(tao->XL, &xlarr));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   TaoPDIPMSetUpBounds - Create upper and lower bound vectors of x

   Collective

   Input Parameter:
.  tao - holds pdipm and XL & XU

   Level: beginner

.seealso: `TAOPDIPM`, `TaoPDIPMUpdateConstraints`
*/
static PetscErrorCode TaoPDIPMSetUpBounds(Tao tao)
{
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  const PetscScalar *xl, *xu;
  PetscInt           n, *ixlb, *ixub, *ixfixed, *ixfree, *ixbox, i, low, high, idx;
  MPI_Comm           comm;
  PetscInt           sendbuf[5], recvbuf[5];

  PetscFunctionBegin;
  /* Creates upper and lower bounds vectors on x, if not created already */
  PetscCall(TaoComputeVariableBounds(tao));

  PetscCall(VecGetLocalSize(tao->XL, &n));
  PetscCall(PetscMalloc5(n, &ixlb, n, &ixub, n, &ixfree, n, &ixfixed, n, &ixbox));

  PetscCall(VecGetOwnershipRange(tao->XL, &low, &high));
  PetscCall(VecGetArrayRead(tao->XL, &xl));
  PetscCall(VecGetArrayRead(tao->XU, &xu));
  for (i = 0; i < n; i++) {
    idx = low + i;
    if ((PetscRealPart(xl[i]) > PETSC_NINFINITY) && (PetscRealPart(xu[i]) < PETSC_INFINITY)) {
      if (PetscRealPart(xl[i]) == PetscRealPart(xu[i])) {
        ixfixed[pdipm->nxfixed++] = idx;
      } else ixbox[pdipm->nxbox++] = idx;
    } else {
      if ((PetscRealPart(xl[i]) > PETSC_NINFINITY) && (PetscRealPart(xu[i]) >= PETSC_INFINITY)) {
        ixlb[pdipm->nxlb++] = idx;
      } else if ((PetscRealPart(xl[i]) <= PETSC_NINFINITY) && (PetscRealPart(xu[i]) < PETSC_INFINITY)) {
        ixub[pdipm->nxlb++] = idx;
      } else ixfree[pdipm->nxfree++] = idx;
    }
  }
  PetscCall(VecRestoreArrayRead(tao->XL, &xl));
  PetscCall(VecRestoreArrayRead(tao->XU, &xu));

  PetscCall(PetscObjectGetComm((PetscObject)tao, &comm));
  sendbuf[0] = pdipm->nxlb;
  sendbuf[1] = pdipm->nxub;
  sendbuf[2] = pdipm->nxfixed;
  sendbuf[3] = pdipm->nxbox;
  sendbuf[4] = pdipm->nxfree;

  PetscCall(MPIU_Allreduce(sendbuf, recvbuf, 5, MPIU_INT, MPI_SUM, comm));
  pdipm->Nxlb    = recvbuf[0];
  pdipm->Nxub    = recvbuf[1];
  pdipm->Nxfixed = recvbuf[2];
  pdipm->Nxbox   = recvbuf[3];
  pdipm->Nxfree  = recvbuf[4];

  if (pdipm->Nxlb) PetscCall(ISCreateGeneral(comm, pdipm->nxlb, ixlb, PETSC_COPY_VALUES, &pdipm->isxlb));
  if (pdipm->Nxub) PetscCall(ISCreateGeneral(comm, pdipm->nxub, ixub, PETSC_COPY_VALUES, &pdipm->isxub));
  if (pdipm->Nxfixed) PetscCall(ISCreateGeneral(comm, pdipm->nxfixed, ixfixed, PETSC_COPY_VALUES, &pdipm->isxfixed));
  if (pdipm->Nxbox) PetscCall(ISCreateGeneral(comm, pdipm->nxbox, ixbox, PETSC_COPY_VALUES, &pdipm->isxbox));
  if (pdipm->Nxfree) PetscCall(ISCreateGeneral(comm, pdipm->nxfree, ixfree, PETSC_COPY_VALUES, &pdipm->isxfree));
  PetscCall(PetscFree5(ixlb, ixub, ixfixed, ixbox, ixfree));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   TaoPDIPMInitializeSolution - Initialize `TAOPDIPM` solution X = [x; lambdae; lambdai; z].
   X consists of four subvectors in the order [x; lambdae; lambdai; z]. These
     four subvectors need to be initialized and its values copied over to X. Instead
     of copying, we use `VecPlaceArray()`/`VecResetArray()` functions to share the memory locations for
     X and the subvectors

   Collective

   Input Parameter:
.  tao - Tao context

   Level: beginner
*/
static PetscErrorCode TaoPDIPMInitializeSolution(Tao tao)
{
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  PetscScalar       *Xarr, *z, *lambdai;
  PetscInt           i;
  const PetscScalar *xarr, *h;

  PetscFunctionBegin;
  PetscCall(VecGetArrayWrite(pdipm->X, &Xarr));

  /* Set Initialize X.x = tao->solution */
  PetscCall(VecGetArrayRead(tao->solution, &xarr));
  PetscCall(PetscMemcpy(Xarr, xarr, pdipm->nx * sizeof(PetscScalar)));
  PetscCall(VecRestoreArrayRead(tao->solution, &xarr));

  /* Initialize X.lambdae = 0.0 */
  if (pdipm->lambdae) PetscCall(VecSet(pdipm->lambdae, 0.0));

  /* Initialize X.lambdai = push_init_lambdai, X.z = push_init_slack */
  if (pdipm->Nci) {
    PetscCall(VecSet(pdipm->lambdai, pdipm->push_init_lambdai));
    PetscCall(VecSet(pdipm->z, pdipm->push_init_slack));

    /* Additional modification for X.lambdai and X.z */
    PetscCall(VecGetArrayWrite(pdipm->lambdai, &lambdai));
    PetscCall(VecGetArrayWrite(pdipm->z, &z));
    if (pdipm->Nh) {
      PetscCall(VecGetArrayRead(tao->constraints_inequality, &h));
      for (i = 0; i < pdipm->nh; i++) {
        if (h[i] < -pdipm->push_init_slack) z[i] = -h[i];
        if (pdipm->mu / z[i] > pdipm->push_init_lambdai) lambdai[i] = pdipm->mu / z[i];
      }
      PetscCall(VecRestoreArrayRead(tao->constraints_inequality, &h));
    }
    PetscCall(VecRestoreArrayWrite(pdipm->lambdai, &lambdai));
    PetscCall(VecRestoreArrayWrite(pdipm->z, &z));
  }

  PetscCall(VecRestoreArrayWrite(pdipm->X, &Xarr));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   TaoSNESJacobian_PDIPM - Evaluate the Hessian matrix at X

   Input Parameter:
   snes - SNES context
   X - KKT Vector
   *ctx - pdipm context

   Output Parameter:
   J - Hessian matrix
   Jpre - matrix to build the preconditioner from
*/
static PetscErrorCode TaoSNESJacobian_PDIPM(SNES snes, Vec X, Mat J, Mat Jpre, void *ctx)
{
  Tao                tao   = (Tao)ctx;
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  PetscInt           i, row, cols[2], Jrstart, rjstart, nc, j;
  const PetscInt    *aj, *ranges, *Jranges, *rranges, *cranges;
  const PetscScalar *Xarr, *aa;
  PetscScalar        vals[2];
  PetscInt           proc, nx_all, *nce_all = pdipm->nce_all;
  MPI_Comm           comm;
  PetscMPIInt        rank, size;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)snes, &comm));
  PetscCallMPI(MPI_Comm_rank(comm, &rank));
  PetscCallMPI(MPI_Comm_rank(comm, &size));

  PetscCall(MatGetOwnershipRanges(Jpre, &Jranges));
  PetscCall(MatGetOwnershipRange(Jpre, &Jrstart, NULL));
  PetscCall(MatGetOwnershipRangesColumn(tao->hessian, &rranges));
  PetscCall(MatGetOwnershipRangesColumn(tao->hessian, &cranges));

  PetscCall(VecGetArrayRead(X, &Xarr));

  /* (1) insert Z and Ci to the 4th block of Jpre -- overwrite existing values */
  if (pdipm->solve_symmetric_kkt) { /* 1 for eq 17 revised pdipm doc 0 for eq 18 (symmetric KKT) */
    vals[0] = 1.0;
    for (i = 0; i < pdipm->nci; i++) {
      row     = Jrstart + pdipm->off_z + i;
      cols[0] = Jrstart + pdipm->off_lambdai + i;
      cols[1] = row;
      vals[1] = Xarr[pdipm->off_lambdai + i] / Xarr[pdipm->off_z + i];
      PetscCall(MatSetValues(Jpre, 1, &row, 2, cols, vals, INSERT_VALUES));
    }
  } else {
    for (i = 0; i < pdipm->nci; i++) {
      row     = Jrstart + pdipm->off_z + i;
      cols[0] = Jrstart + pdipm->off_lambdai + i;
      cols[1] = row;
      vals[0] = Xarr[pdipm->off_z + i];
      vals[1] = Xarr[pdipm->off_lambdai + i];
      PetscCall(MatSetValues(Jpre, 1, &row, 2, cols, vals, INSERT_VALUES));
    }
  }

  /* (2) insert 2nd row block of Jpre: [ grad g, 0, 0, 0] */
  if (pdipm->Ng) {
    PetscCall(MatGetOwnershipRange(tao->jacobian_equality, &rjstart, NULL));
    for (i = 0; i < pdipm->ng; i++) {
      row = Jrstart + pdipm->off_lambdae + i;

      PetscCall(MatGetRow(tao->jacobian_equality, i + rjstart, &nc, &aj, &aa));
      proc = 0;
      for (j = 0; j < nc; j++) {
        while (aj[j] >= cranges[proc + 1]) proc++;
        cols[0] = aj[j] - cranges[proc] + Jranges[proc];
        PetscCall(MatSetValue(Jpre, row, cols[0], aa[j], INSERT_VALUES));
      }
      PetscCall(MatRestoreRow(tao->jacobian_equality, i + rjstart, &nc, &aj, &aa));
      if (pdipm->kkt_pd) {
        /* add shift \delta_c */
        PetscCall(MatSetValue(Jpre, row, row, -pdipm->deltac, INSERT_VALUES));
      }
    }
  }

  /* (3) insert 3rd row block of Jpre: [ -grad h, 0, deltac, I] */
  if (pdipm->Nh) {
    PetscCall(MatGetOwnershipRange(tao->jacobian_inequality, &rjstart, NULL));
    for (i = 0; i < pdipm->nh; i++) {
      row = Jrstart + pdipm->off_lambdai + i;
      PetscCall(MatGetRow(tao->jacobian_inequality, i + rjstart, &nc, &aj, &aa));
      proc = 0;
      for (j = 0; j < nc; j++) {
        while (aj[j] >= cranges[proc + 1]) proc++;
        cols[0] = aj[j] - cranges[proc] + Jranges[proc];
        PetscCall(MatSetValue(Jpre, row, cols[0], -aa[j], INSERT_VALUES));
      }
      PetscCall(MatRestoreRow(tao->jacobian_inequality, i + rjstart, &nc, &aj, &aa));
      if (pdipm->kkt_pd) {
        /* add shift \delta_c */
        PetscCall(MatSetValue(Jpre, row, row, -pdipm->deltac, INSERT_VALUES));
      }
    }
  }

  /* (4) insert 1st row block of Jpre: [Wxx, grad g', -grad h', 0] */
  if (pdipm->Ng) { /* grad g' */
    PetscCall(MatTranspose(tao->jacobian_equality, MAT_REUSE_MATRIX, &pdipm->jac_equality_trans));
  }
  if (pdipm->Nh) { /* grad h' */
    PetscCall(MatTranspose(tao->jacobian_inequality, MAT_REUSE_MATRIX, &pdipm->jac_inequality_trans));
  }

  PetscCall(VecPlaceArray(pdipm->x, Xarr));
  PetscCall(TaoComputeHessian(tao, pdipm->x, tao->hessian, tao->hessian_pre));
  PetscCall(VecResetArray(pdipm->x));

  PetscCall(MatGetOwnershipRange(tao->hessian, &rjstart, NULL));
  for (i = 0; i < pdipm->nx; i++) {
    row = Jrstart + i;

    /* insert Wxx = fxx + ... -- provided by user */
    PetscCall(MatGetRow(tao->hessian, i + rjstart, &nc, &aj, &aa));
    proc = 0;
    for (j = 0; j < nc; j++) {
      while (aj[j] >= cranges[proc + 1]) proc++;
      cols[0] = aj[j] - cranges[proc] + Jranges[proc];
      if (row == cols[0] && pdipm->kkt_pd) {
        /* add shift deltaw to Wxx component */
        PetscCall(MatSetValue(Jpre, row, cols[0], aa[j] + pdipm->deltaw, INSERT_VALUES));
      } else {
        PetscCall(MatSetValue(Jpre, row, cols[0], aa[j], INSERT_VALUES));
      }
    }
    PetscCall(MatRestoreRow(tao->hessian, i + rjstart, &nc, &aj, &aa));

    /* insert grad g' */
    if (pdipm->ng) {
      PetscCall(MatGetRow(pdipm->jac_equality_trans, i + rjstart, &nc, &aj, &aa));
      PetscCall(MatGetOwnershipRanges(tao->jacobian_equality, &ranges));
      proc = 0;
      for (j = 0; j < nc; j++) {
        /* find row ownership of */
        while (aj[j] >= ranges[proc + 1]) proc++;
        nx_all  = rranges[proc + 1] - rranges[proc];
        cols[0] = aj[j] - ranges[proc] + Jranges[proc] + nx_all;
        PetscCall(MatSetValue(Jpre, row, cols[0], aa[j], INSERT_VALUES));
      }
      PetscCall(MatRestoreRow(pdipm->jac_equality_trans, i + rjstart, &nc, &aj, &aa));
    }

    /* insert -grad h' */
    if (pdipm->nh) {
      PetscCall(MatGetRow(pdipm->jac_inequality_trans, i + rjstart, &nc, &aj, &aa));
      PetscCall(MatGetOwnershipRanges(tao->jacobian_inequality, &ranges));
      proc = 0;
      for (j = 0; j < nc; j++) {
        /* find row ownership of */
        while (aj[j] >= ranges[proc + 1]) proc++;
        nx_all  = rranges[proc + 1] - rranges[proc];
        cols[0] = aj[j] - ranges[proc] + Jranges[proc] + nx_all + nce_all[proc];
        PetscCall(MatSetValue(Jpre, row, cols[0], -aa[j], INSERT_VALUES));
      }
      PetscCall(MatRestoreRow(pdipm->jac_inequality_trans, i + rjstart, &nc, &aj, &aa));
    }
  }
  PetscCall(VecRestoreArrayRead(X, &Xarr));

  /* (6) assemble Jpre and J */
  PetscCall(MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY));

  if (J != Jpre) {
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   TaoSnesFunction_PDIPM - Evaluate KKT function at X

   Input Parameter:
   snes - SNES context
   X - KKT Vector
   *ctx - pdipm

   Output Parameter:
   F - Updated Lagrangian vector
*/
static PetscErrorCode TaoSNESFunction_PDIPM(SNES snes, Vec X, Vec F, void *ctx)
{
  Tao                tao   = (Tao)ctx;
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  PetscScalar       *Farr;
  Vec                x, L1;
  PetscInt           i;
  const PetscScalar *Xarr, *carr, *zarr, *larr;

  PetscFunctionBegin;
  PetscCall(VecSet(F, 0.0));

  PetscCall(VecGetArrayRead(X, &Xarr));
  PetscCall(VecGetArrayWrite(F, &Farr));

  /* (0) Evaluate f, fx, gradG, gradH at X.x Note: pdipm->x is not changed below */
  x = pdipm->x;
  PetscCall(VecPlaceArray(x, Xarr));
  PetscCall(TaoPDIPMEvaluateFunctionsAndJacobians(tao, x));

  /* Update ce, ci, and Jci at X.x */
  PetscCall(TaoPDIPMUpdateConstraints(tao, x));
  PetscCall(VecResetArray(x));

  /* (1) L1 = fx + (gradG'*DE + Jce_xfixed'*lambdae_xfixed) - (gradH'*DI + Jci_xb'*lambdai_xb) */
  L1 = pdipm->x;
  PetscCall(VecPlaceArray(L1, Farr)); /* L1 = 0.0 */
  if (pdipm->Nci) {
    if (pdipm->Nh) {
      /* L1 += gradH'*DI. Note: tao->DI is not changed below */
      PetscCall(VecPlaceArray(tao->DI, Xarr + pdipm->off_lambdai));
      PetscCall(MatMultTransposeAdd(tao->jacobian_inequality, tao->DI, L1, L1));
      PetscCall(VecResetArray(tao->DI));
    }

    /* L1 += Jci_xb'*lambdai_xb */
    PetscCall(VecPlaceArray(pdipm->lambdai_xb, Xarr + pdipm->off_lambdai + pdipm->nh));
    PetscCall(MatMultTransposeAdd(pdipm->Jci_xb, pdipm->lambdai_xb, L1, L1));
    PetscCall(VecResetArray(pdipm->lambdai_xb));

    /* L1 = - (gradH'*DI + Jci_xb'*lambdai_xb) */
    PetscCall(VecScale(L1, -1.0));
  }

  /* L1 += fx */
  PetscCall(VecAXPY(L1, 1.0, tao->gradient));

  if (pdipm->Nce) {
    if (pdipm->Ng) {
      /* L1 += gradG'*DE. Note: tao->DE is not changed below */
      PetscCall(VecPlaceArray(tao->DE, Xarr + pdipm->off_lambdae));
      PetscCall(MatMultTransposeAdd(tao->jacobian_equality, tao->DE, L1, L1));
      PetscCall(VecResetArray(tao->DE));
    }
    if (pdipm->Nxfixed) {
      /* L1 += Jce_xfixed'*lambdae_xfixed */
      PetscCall(VecPlaceArray(pdipm->lambdae_xfixed, Xarr + pdipm->off_lambdae + pdipm->ng));
      PetscCall(MatMultTransposeAdd(pdipm->Jce_xfixed, pdipm->lambdae_xfixed, L1, L1));
      PetscCall(VecResetArray(pdipm->lambdae_xfixed));
    }
  }
  PetscCall(VecResetArray(L1));

  /* (2) L2 = ce(x) */
  if (pdipm->Nce) {
    PetscCall(VecGetArrayRead(pdipm->ce, &carr));
    for (i = 0; i < pdipm->nce; i++) Farr[pdipm->off_lambdae + i] = carr[i];
    PetscCall(VecRestoreArrayRead(pdipm->ce, &carr));
  }

  if (pdipm->Nci) {
    if (pdipm->solve_symmetric_kkt) {
      /* (3) L3 = z - ci(x);
         (4) L4 = Lambdai * e - mu/z *e  */
      PetscCall(VecGetArrayRead(pdipm->ci, &carr));
      larr = Xarr + pdipm->off_lambdai;
      zarr = Xarr + pdipm->off_z;
      for (i = 0; i < pdipm->nci; i++) {
        Farr[pdipm->off_lambdai + i] = zarr[i] - carr[i];
        Farr[pdipm->off_z + i]       = larr[i] - pdipm->mu / zarr[i];
      }
      PetscCall(VecRestoreArrayRead(pdipm->ci, &carr));
    } else {
      /* (3) L3 = z - ci(x);
         (4) L4 = Z * Lambdai * e - mu * e  */
      PetscCall(VecGetArrayRead(pdipm->ci, &carr));
      larr = Xarr + pdipm->off_lambdai;
      zarr = Xarr + pdipm->off_z;
      for (i = 0; i < pdipm->nci; i++) {
        Farr[pdipm->off_lambdai + i] = zarr[i] - carr[i];
        Farr[pdipm->off_z + i]       = zarr[i] * larr[i] - pdipm->mu;
      }
      PetscCall(VecRestoreArrayRead(pdipm->ci, &carr));
    }
  }

  PetscCall(VecRestoreArrayRead(X, &Xarr));
  PetscCall(VecRestoreArrayWrite(F, &Farr));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  Evaluate F(X); then update tao->gnorm0, tao->step = mu,
  tao->residual = norm2(F_x,F_z) and tao->cnorm = norm2(F_ce,F_ci).
*/
static PetscErrorCode TaoSNESFunction_PDIPM_residual(SNES snes, Vec X, Vec F, void *ctx)
{
  Tao                tao   = (Tao)ctx;
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  PetscScalar       *Farr, *tmparr;
  Vec                L1;
  PetscInt           i;
  PetscReal          res[2], cnorm[2];
  const PetscScalar *Xarr = NULL;

  PetscFunctionBegin;
  PetscCall(TaoSNESFunction_PDIPM(snes, X, F, (void *)tao));
  PetscCall(VecGetArrayWrite(F, &Farr));
  PetscCall(VecGetArrayRead(X, &Xarr));

  /* compute res[0] = norm2(F_x) */
  L1 = pdipm->x;
  PetscCall(VecPlaceArray(L1, Farr));
  PetscCall(VecNorm(L1, NORM_2, &res[0]));
  PetscCall(VecResetArray(L1));

  /* compute res[1] = norm2(F_z), cnorm[1] = norm2(F_ci) */
  if (pdipm->z) {
    if (pdipm->solve_symmetric_kkt) {
      PetscCall(VecPlaceArray(pdipm->z, Farr + pdipm->off_z));
      if (pdipm->Nci) {
        PetscCall(VecGetArrayWrite(pdipm->z, &tmparr));
        for (i = 0; i < pdipm->nci; i++) tmparr[i] *= Xarr[pdipm->off_z + i];
        PetscCall(VecRestoreArrayWrite(pdipm->z, &tmparr));
      }

      PetscCall(VecNorm(pdipm->z, NORM_2, &res[1]));

      if (pdipm->Nci) {
        PetscCall(VecGetArrayWrite(pdipm->z, &tmparr));
        for (i = 0; i < pdipm->nci; i++) tmparr[i] /= Xarr[pdipm->off_z + i];
        PetscCall(VecRestoreArrayWrite(pdipm->z, &tmparr));
      }
      PetscCall(VecResetArray(pdipm->z));
    } else { /* !solve_symmetric_kkt */
      PetscCall(VecPlaceArray(pdipm->z, Farr + pdipm->off_z));
      PetscCall(VecNorm(pdipm->z, NORM_2, &res[1]));
      PetscCall(VecResetArray(pdipm->z));
    }

    PetscCall(VecPlaceArray(pdipm->ci, Farr + pdipm->off_lambdai));
    PetscCall(VecNorm(pdipm->ci, NORM_2, &cnorm[1]));
    PetscCall(VecResetArray(pdipm->ci));
  } else {
    res[1]   = 0.0;
    cnorm[1] = 0.0;
  }

  /* compute cnorm[0] = norm2(F_ce) */
  if (pdipm->Nce) {
    PetscCall(VecPlaceArray(pdipm->ce, Farr + pdipm->off_lambdae));
    PetscCall(VecNorm(pdipm->ce, NORM_2, &cnorm[0]));
    PetscCall(VecResetArray(pdipm->ce));
  } else cnorm[0] = 0.0;

  PetscCall(VecRestoreArrayWrite(F, &Farr));
  PetscCall(VecRestoreArrayRead(X, &Xarr));

  tao->gnorm0   = tao->residual;
  tao->residual = PetscSqrtReal(res[0] * res[0] + res[1] * res[1]);
  tao->cnorm    = PetscSqrtReal(cnorm[0] * cnorm[0] + cnorm[1] * cnorm[1]);
  tao->step     = pdipm->mu;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  KKTAddShifts - Check the inertia of Cholesky factor of KKT matrix.
  If it does not match the numbers of prime and dual variables, add shifts to the KKT matrix.
*/
static PetscErrorCode KKTAddShifts(Tao tao, SNES snes, Vec X)
{
  TAO_PDIPM *pdipm = (TAO_PDIPM *)tao->data;
  KSP        ksp;
  PC         pc;
  Mat        Factor;
  PetscBool  isCHOL;
  PetscInt   nneg, nzero, npos;

  PetscFunctionBegin;
  /* Get the inertia of Cholesky factor */
  PetscCall(SNESGetKSP(snes, &ksp));
  PetscCall(KSPGetPC(ksp, &pc));
  PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCCHOLESKY, &isCHOL));
  if (!isCHOL) PetscFunctionReturn(PETSC_SUCCESS);

  PetscCall(PCFactorGetMatrix(pc, &Factor));
  PetscCall(MatGetInertia(Factor, &nneg, &nzero, &npos));

  if (npos < pdipm->Nx + pdipm->Nci) {
    pdipm->deltaw = PetscMax(pdipm->lastdeltaw / 3, 1.e-4 * PETSC_MACHINE_EPSILON);
    PetscCall(PetscInfo(tao, "Test reduced deltaw=%g; previous MatInertia: nneg %" PetscInt_FMT ", nzero %" PetscInt_FMT ", npos %" PetscInt_FMT "(<%" PetscInt_FMT ")\n", (double)pdipm->deltaw, nneg, nzero, npos, pdipm->Nx + pdipm->Nci));
    PetscCall(TaoSNESJacobian_PDIPM(snes, X, pdipm->K, pdipm->K, tao));
    PetscCall(PCSetUp(pc));
    PetscCall(MatGetInertia(Factor, &nneg, &nzero, &npos));

    if (npos < pdipm->Nx + pdipm->Nci) {
      pdipm->deltaw = pdipm->lastdeltaw;                                           /* in case reduction update does not help, this prevents that step from impacting increasing update */
      while (npos < pdipm->Nx + pdipm->Nci && pdipm->deltaw <= 1. / PETSC_SMALL) { /* increase deltaw */
        PetscCall(PetscInfo(tao, "  deltaw=%g fails, MatInertia: nneg %" PetscInt_FMT ", nzero %" PetscInt_FMT ", npos %" PetscInt_FMT "(<%" PetscInt_FMT ")\n", (double)pdipm->deltaw, nneg, nzero, npos, pdipm->Nx + pdipm->Nci));
        pdipm->deltaw = PetscMin(8 * pdipm->deltaw, PetscPowReal(10, 20));
        PetscCall(TaoSNESJacobian_PDIPM(snes, X, pdipm->K, pdipm->K, tao));
        PetscCall(PCSetUp(pc));
        PetscCall(MatGetInertia(Factor, &nneg, &nzero, &npos));
      }

      PetscCheck(pdipm->deltaw < 1. / PETSC_SMALL, PetscObjectComm((PetscObject)tao), PETSC_ERR_CONV_FAILED, "Reached maximum delta w will not converge, try different initial x0");

      PetscCall(PetscInfo(tao, "Updated deltaw %g\n", (double)pdipm->deltaw));
      pdipm->lastdeltaw = pdipm->deltaw;
      pdipm->deltaw     = 0.0;
    }
  }

  if (nzero) { /* Jacobian is singular */
    if (pdipm->deltac == 0.0) {
      pdipm->deltac = PETSC_SQRT_MACHINE_EPSILON;
    } else {
      pdipm->deltac = pdipm->deltac * PetscPowReal(pdipm->mu, .25);
    }
    PetscCall(PetscInfo(tao, "Updated deltac=%g, MatInertia: nneg %" PetscInt_FMT ", nzero %" PetscInt_FMT "(!=0), npos %" PetscInt_FMT "\n", (double)pdipm->deltac, nneg, nzero, npos));
    PetscCall(TaoSNESJacobian_PDIPM(snes, X, pdipm->K, pdipm->K, tao));
    PetscCall(PCSetUp(pc));
    PetscCall(MatGetInertia(Factor, &nneg, &nzero, &npos));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  PCPreSolve_PDIPM -- called between MatFactorNumeric() and MatSolve()
*/
static PetscErrorCode PCPreSolve_PDIPM(PC pc, KSP ksp)
{
  Tao        tao;
  TAO_PDIPM *pdipm;

  PetscFunctionBegin;
  PetscCall(KSPGetApplicationContext(ksp, &tao));
  pdipm = (TAO_PDIPM *)tao->data;
  PetscCall(KKTAddShifts(tao, pdipm->snes, pdipm->X));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   SNESLineSearch_PDIPM - Custom line search used with PDIPM.

   Collective

   Notes:
   This routine employs a simple backtracking line-search to keep
   the slack variables (z) and inequality constraints Lagrange multipliers
   (lambdai) positive, i.e., z,lambdai >=0. It does this by calculating scalars
   alpha_p and alpha_d to keep z,lambdai non-negative. The decision (x), and the
   slack variables are updated as X = X - alpha_d*dx. The constraint multipliers
   are updated as Lambdai = Lambdai + alpha_p*dLambdai. The barrier parameter mu
   is also updated as mu = mu + z'lambdai/Nci
*/
static PetscErrorCode SNESLineSearch_PDIPM(SNESLineSearch linesearch, void *ctx)
{
  Tao                tao   = (Tao)ctx;
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  SNES               snes;
  Vec                X, F, Y;
  PetscInt           i, iter;
  PetscReal          alpha_p = 1.0, alpha_d = 1.0, alpha[4];
  PetscScalar       *Xarr, *z, *lambdai, dot, *taosolarr;
  const PetscScalar *dXarr, *dz, *dlambdai;

  PetscFunctionBegin;
  PetscCall(SNESLineSearchGetSNES(linesearch, &snes));
  PetscCall(SNESGetIterationNumber(snes, &iter));

  PetscCall(SNESLineSearchSetReason(linesearch, SNES_LINESEARCH_SUCCEEDED));
  PetscCall(SNESLineSearchGetVecs(linesearch, &X, &F, &Y, NULL, NULL));

  PetscCall(VecGetArrayWrite(X, &Xarr));
  PetscCall(VecGetArrayRead(Y, &dXarr));
  z  = Xarr + pdipm->off_z;
  dz = dXarr + pdipm->off_z;
  for (i = 0; i < pdipm->nci; i++) {
    if (z[i] - dz[i] < 0.0) alpha_p = PetscMin(alpha_p, 0.9999 * z[i] / dz[i]);
  }

  lambdai  = Xarr + pdipm->off_lambdai;
  dlambdai = dXarr + pdipm->off_lambdai;

  for (i = 0; i < pdipm->nci; i++) {
    if (lambdai[i] - dlambdai[i] < 0.0) alpha_d = PetscMin(0.9999 * lambdai[i] / dlambdai[i], alpha_d);
  }

  alpha[0] = alpha_p;
  alpha[1] = alpha_d;
  PetscCall(VecRestoreArrayRead(Y, &dXarr));
  PetscCall(VecRestoreArrayWrite(X, &Xarr));

  /* alpha = min(alpha) over all processes */
  PetscCall(MPIU_Allreduce(alpha, alpha + 2, 2, MPIU_REAL, MPIU_MIN, PetscObjectComm((PetscObject)tao)));

  alpha_p = alpha[2];
  alpha_d = alpha[3];

  /* X = X - alpha * Y */
  PetscCall(VecGetArrayWrite(X, &Xarr));
  PetscCall(VecGetArrayRead(Y, &dXarr));
  for (i = 0; i < pdipm->nx; i++) Xarr[i] -= alpha_p * dXarr[i];
  for (i = 0; i < pdipm->nce; i++) Xarr[i + pdipm->off_lambdae] -= alpha_d * dXarr[i + pdipm->off_lambdae];

  for (i = 0; i < pdipm->nci; i++) {
    Xarr[i + pdipm->off_lambdai] -= alpha_d * dXarr[i + pdipm->off_lambdai];
    Xarr[i + pdipm->off_z] -= alpha_p * dXarr[i + pdipm->off_z];
  }
  PetscCall(VecGetArrayWrite(tao->solution, &taosolarr));
  PetscCall(PetscMemcpy(taosolarr, Xarr, pdipm->nx * sizeof(PetscScalar)));
  PetscCall(VecRestoreArrayWrite(tao->solution, &taosolarr));

  PetscCall(VecRestoreArrayWrite(X, &Xarr));
  PetscCall(VecRestoreArrayRead(Y, &dXarr));

  /* Update mu = mu_update_factor * dot(z,lambdai)/pdipm->nci at updated X */
  if (pdipm->z) {
    PetscCall(VecDot(pdipm->z, pdipm->lambdai, &dot));
  } else dot = 0.0;

  /* if (PetscAbsReal(pdipm->gradL) < 0.9*pdipm->mu)  */
  pdipm->mu = pdipm->mu_update_factor * dot / pdipm->Nci;

  /* Update F; get tao->residual and tao->cnorm */
  PetscCall(TaoSNESFunction_PDIPM_residual(snes, X, F, (void *)tao));

  tao->niter++;
  PetscCall(TaoLogConvergenceHistory(tao, pdipm->obj, tao->residual, tao->cnorm, tao->niter));
  PetscCall(TaoMonitor(tao, tao->niter, pdipm->obj, tao->residual, tao->cnorm, pdipm->mu));

  PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
  if (tao->reason) PetscCall(SNESSetConvergedReason(snes, SNES_CONVERGED_FNORM_ABS));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TaoSolve_PDIPM(Tao tao)
{
  TAO_PDIPM     *pdipm = (TAO_PDIPM *)tao->data;
  SNESLineSearch linesearch; /* SNESLineSearch context */
  Vec            dummy;

  PetscFunctionBegin;
  PetscCheck(tao->constraints_equality || tao->constraints_inequality, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_NULL, "Equality and inequality constraints are not set. Either set them or switch to a different algorithm");

  /* Initialize all variables */
  PetscCall(TaoPDIPMInitializeSolution(tao));

  /* Set linesearch */
  PetscCall(SNESGetLineSearch(pdipm->snes, &linesearch));
  PetscCall(SNESLineSearchSetType(linesearch, SNESLINESEARCHSHELL));
  PetscCall(SNESLineSearchShellSetApply(linesearch, SNESLineSearch_PDIPM, tao));
  PetscCall(SNESLineSearchSetFromOptions(linesearch));

  tao->reason = TAO_CONTINUE_ITERATING;

  /* -tao_monitor for iteration 0 and check convergence */
  PetscCall(VecDuplicate(pdipm->X, &dummy));
  PetscCall(TaoSNESFunction_PDIPM_residual(pdipm->snes, pdipm->X, dummy, (void *)tao));

  PetscCall(TaoLogConvergenceHistory(tao, pdipm->obj, tao->residual, tao->cnorm, tao->niter));
  PetscCall(TaoMonitor(tao, tao->niter, pdipm->obj, tao->residual, tao->cnorm, pdipm->mu));
  PetscCall(VecDestroy(&dummy));
  PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
  if (tao->reason) PetscCall(SNESSetConvergedReason(pdipm->snes, SNES_CONVERGED_FNORM_ABS));

  while (tao->reason == TAO_CONTINUE_ITERATING) {
    SNESConvergedReason reason;
    PetscCall(SNESSolve(pdipm->snes, NULL, pdipm->X));

    /* Check SNES convergence */
    PetscCall(SNESGetConvergedReason(pdipm->snes, &reason));
    if (reason < 0) PetscCall(PetscPrintf(PetscObjectComm((PetscObject)pdipm->snes), "SNES solve did not converged due to reason %s\n", SNESConvergedReasons[reason]));

    /* Check TAO convergence */
    PetscCheck(!PetscIsInfOrNanReal(pdipm->obj), PETSC_COMM_SELF, PETSC_ERR_SUP, "User-provided compute function generated Inf or NaN");
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TaoView_PDIPM(Tao tao, PetscViewer viewer)
{
  TAO_PDIPM *pdipm = (TAO_PDIPM *)tao->data;

  PetscFunctionBegin;
  tao->constrained = PETSC_TRUE;
  PetscCall(PetscViewerASCIIPushTab(viewer));
  PetscCall(PetscViewerASCIIPrintf(viewer, "Number of prime=%" PetscInt_FMT ", Number of dual=%" PetscInt_FMT "\n", pdipm->Nx + pdipm->Nci, pdipm->Nce + pdipm->Nci));
  if (pdipm->kkt_pd) PetscCall(PetscViewerASCIIPrintf(viewer, "KKT shifts deltaw=%g, deltac=%g\n", (double)pdipm->deltaw, (double)pdipm->deltac));
  PetscCall(PetscViewerASCIIPopTab(viewer));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TaoSetup_PDIPM(Tao tao)
{
  TAO_PDIPM         *pdipm = (TAO_PDIPM *)tao->data;
  MPI_Comm           comm;
  PetscMPIInt        size;
  PetscInt           row, col, Jcrstart, Jcrend, k, tmp, nc, proc, *nh_all, *ng_all;
  PetscInt           offset, *xa, *xb, i, j, rstart, rend;
  PetscScalar        one = 1.0, neg_one = -1.0;
  const PetscInt    *cols, *rranges, *cranges, *aj, *ranges;
  const PetscScalar *aa, *Xarr;
  Mat                J;
  Mat                Jce_xfixed_trans, Jci_xb_trans;
  PetscInt          *dnz, *onz, rjstart, nx_all, *nce_all, *Jranges, cols1[2];

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)tao, &comm));
  PetscCallMPI(MPI_Comm_size(comm, &size));

  /* (1) Setup Bounds and create Tao vectors */
  PetscCall(TaoPDIPMSetUpBounds(tao));

  if (!tao->gradient) {
    PetscCall(VecDuplicate(tao->solution, &tao->gradient));
    PetscCall(VecDuplicate(tao->solution, &tao->stepdirection));
  }

  /* (2) Get sizes */
  /* Size of vector x - This is set by TaoSetSolution */
  PetscCall(VecGetSize(tao->solution, &pdipm->Nx));
  PetscCall(VecGetLocalSize(tao->solution, &pdipm->nx));

  /* Size of equality constraints and vectors */
  if (tao->constraints_equality) {
    PetscCall(VecGetSize(tao->constraints_equality, &pdipm->Ng));
    PetscCall(VecGetLocalSize(tao->constraints_equality, &pdipm->ng));
  } else {
    pdipm->ng = pdipm->Ng = 0;
  }

  pdipm->nce = pdipm->ng + pdipm->nxfixed;
  pdipm->Nce = pdipm->Ng + pdipm->Nxfixed;

  /* Size of inequality constraints and vectors */
  if (tao->constraints_inequality) {
    PetscCall(VecGetSize(tao->constraints_inequality, &pdipm->Nh));
    PetscCall(VecGetLocalSize(tao->constraints_inequality, &pdipm->nh));
  } else {
    pdipm->nh = pdipm->Nh = 0;
  }

  pdipm->nci = pdipm->nh + pdipm->nxlb + pdipm->nxub + 2 * pdipm->nxbox;
  pdipm->Nci = pdipm->Nh + pdipm->Nxlb + pdipm->Nxub + 2 * pdipm->Nxbox;

  /* Full size of the KKT system to be solved */
  pdipm->n = pdipm->nx + pdipm->nce + 2 * pdipm->nci;
  pdipm->N = pdipm->Nx + pdipm->Nce + 2 * pdipm->Nci;

  /* (3) Offsets for subvectors */
  pdipm->off_lambdae = pdipm->nx;
  pdipm->off_lambdai = pdipm->off_lambdae + pdipm->nce;
  pdipm->off_z       = pdipm->off_lambdai + pdipm->nci;

  /* (4) Create vectors and subvectors */
  /* Ce and Ci vectors */
  PetscCall(VecCreate(comm, &pdipm->ce));
  PetscCall(VecSetSizes(pdipm->ce, pdipm->nce, pdipm->Nce));
  PetscCall(VecSetFromOptions(pdipm->ce));

  PetscCall(VecCreate(comm, &pdipm->ci));
  PetscCall(VecSetSizes(pdipm->ci, pdipm->nci, pdipm->Nci));
  PetscCall(VecSetFromOptions(pdipm->ci));

  /* X=[x; lambdae; lambdai; z] for the big KKT system */
  PetscCall(VecCreate(comm, &pdipm->X));
  PetscCall(VecSetSizes(pdipm->X, pdipm->n, pdipm->N));
  PetscCall(VecSetFromOptions(pdipm->X));

  /* Subvectors; they share local arrays with X */
  PetscCall(VecGetArrayRead(pdipm->X, &Xarr));
  /* x shares local array with X.x */
  if (pdipm->Nx) PetscCall(VecCreateMPIWithArray(comm, 1, pdipm->nx, pdipm->Nx, Xarr, &pdipm->x));

  /* lambdae shares local array with X.lambdae */
  if (pdipm->Nce) PetscCall(VecCreateMPIWithArray(comm, 1, pdipm->nce, pdipm->Nce, Xarr + pdipm->off_lambdae, &pdipm->lambdae));

  /* tao->DE shares local array with X.lambdae_g */
  if (pdipm->Ng) {
    PetscCall(VecCreateMPIWithArray(comm, 1, pdipm->ng, pdipm->Ng, Xarr + pdipm->off_lambdae, &tao->DE));

    PetscCall(VecCreate(comm, &pdipm->lambdae_xfixed));
    PetscCall(VecSetSizes(pdipm->lambdae_xfixed, pdipm->nxfixed, PETSC_DECIDE));
    PetscCall(VecSetFromOptions(pdipm->lambdae_xfixed));
  }

  if (pdipm->Nci) {
    /* lambdai shares local array with X.lambdai */
    PetscCall(VecCreateMPIWithArray(comm, 1, pdipm->nci, pdipm->Nci, Xarr + pdipm->off_lambdai, &pdipm->lambdai));

    /* z for slack variables; it shares local array with X.z */
    PetscCall(VecCreateMPIWithArray(comm, 1, pdipm->nci, pdipm->Nci, Xarr + pdipm->off_z, &pdipm->z));
  }

  /* tao->DI which shares local array with X.lambdai_h */
  if (pdipm->Nh) PetscCall(VecCreateMPIWithArray(comm, 1, pdipm->nh, pdipm->Nh, Xarr + pdipm->off_lambdai, &tao->DI));
  PetscCall(VecCreate(comm, &pdipm->lambdai_xb));
  PetscCall(VecSetSizes(pdipm->lambdai_xb, (pdipm->nci - pdipm->nh), PETSC_DECIDE));
  PetscCall(VecSetFromOptions(pdipm->lambdai_xb));

  PetscCall(VecRestoreArrayRead(pdipm->X, &Xarr));

  /* (5) Create Jacobians Jce_xfixed and Jci */
  /* (5.1) PDIPM Jacobian of equality bounds cebound(x) = J_nxfixed */
  if (pdipm->Nxfixed) {
    /* Create Jce_xfixed */
    PetscCall(MatCreate(comm, &pdipm->Jce_xfixed));
    PetscCall(MatSetSizes(pdipm->Jce_xfixed, pdipm->nxfixed, pdipm->nx, PETSC_DECIDE, pdipm->Nx));
    PetscCall(MatSetFromOptions(pdipm->Jce_xfixed));
    PetscCall(MatSeqAIJSetPreallocation(pdipm->Jce_xfixed, 1, NULL));
    PetscCall(MatMPIAIJSetPreallocation(pdipm->Jce_xfixed, 1, NULL, 1, NULL));

    PetscCall(MatGetOwnershipRange(pdipm->Jce_xfixed, &Jcrstart, &Jcrend));
    PetscCall(ISGetIndices(pdipm->isxfixed, &cols));
    k = 0;
    for (row = Jcrstart; row < Jcrend; row++) {
      PetscCall(MatSetValues(pdipm->Jce_xfixed, 1, &row, 1, cols + k, &one, INSERT_VALUES));
      k++;
    }
    PetscCall(ISRestoreIndices(pdipm->isxfixed, &cols));
    PetscCall(MatAssemblyBegin(pdipm->Jce_xfixed, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(pdipm->Jce_xfixed, MAT_FINAL_ASSEMBLY));
  }

  /* (5.2) PDIPM inequality Jacobian Jci = [tao->jacobian_inequality; ...] */
  PetscCall(MatCreate(comm, &pdipm->Jci_xb));
  PetscCall(MatSetSizes(pdipm->Jci_xb, pdipm->nci - pdipm->nh, pdipm->nx, PETSC_DECIDE, pdipm->Nx));
  PetscCall(MatSetFromOptions(pdipm->Jci_xb));
  PetscCall(MatSeqAIJSetPreallocation(pdipm->Jci_xb, 1, NULL));
  PetscCall(MatMPIAIJSetPreallocation(pdipm->Jci_xb, 1, NULL, 1, NULL));

  PetscCall(MatGetOwnershipRange(pdipm->Jci_xb, &Jcrstart, &Jcrend));
  offset = Jcrstart;
  if (pdipm->Nxub) {
    /* Add xub to Jci_xb */
    PetscCall(ISGetIndices(pdipm->isxub, &cols));
    k = 0;
    for (row = offset; row < offset + pdipm->nxub; row++) {
      PetscCall(MatSetValues(pdipm->Jci_xb, 1, &row, 1, cols + k, &neg_one, INSERT_VALUES));
      k++;
    }
    PetscCall(ISRestoreIndices(pdipm->isxub, &cols));
  }

  if (pdipm->Nxlb) {
    /* Add xlb to Jci_xb */
    PetscCall(ISGetIndices(pdipm->isxlb, &cols));
    k = 0;
    offset += pdipm->nxub;
    for (row = offset; row < offset + pdipm->nxlb; row++) {
      PetscCall(MatSetValues(pdipm->Jci_xb, 1, &row, 1, cols + k, &one, INSERT_VALUES));
      k++;
    }
    PetscCall(ISRestoreIndices(pdipm->isxlb, &cols));
  }

  /* Add xbox to Jci_xb */
  if (pdipm->Nxbox) {
    PetscCall(ISGetIndices(pdipm->isxbox, &cols));
    k = 0;
    offset += pdipm->nxlb;
    for (row = offset; row < offset + pdipm->nxbox; row++) {
      PetscCall(MatSetValues(pdipm->Jci_xb, 1, &row, 1, cols + k, &neg_one, INSERT_VALUES));
      tmp = row + pdipm->nxbox;
      PetscCall(MatSetValues(pdipm->Jci_xb, 1, &tmp, 1, cols + k, &one, INSERT_VALUES));
      k++;
    }
    PetscCall(ISRestoreIndices(pdipm->isxbox, &cols));
  }

  PetscCall(MatAssemblyBegin(pdipm->Jci_xb, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(pdipm->Jci_xb, MAT_FINAL_ASSEMBLY));
  /* PetscCall(MatView(pdipm->Jci_xb,PETSC_VIEWER_STDOUT_WORLD)); */

  /* (6) Set up ISs for PC Fieldsplit */
  if (pdipm->solve_reduced_kkt) {
    PetscCall(PetscMalloc2(pdipm->nx + pdipm->nce, &xa, 2 * pdipm->nci, &xb));
    for (i = 0; i < pdipm->nx + pdipm->nce; i++) xa[i] = i;
    for (i = 0; i < 2 * pdipm->nci; i++) xb[i] = pdipm->off_lambdai + i;

    PetscCall(ISCreateGeneral(comm, pdipm->nx + pdipm->nce, xa, PETSC_OWN_POINTER, &pdipm->is1));
    PetscCall(ISCreateGeneral(comm, 2 * pdipm->nci, xb, PETSC_OWN_POINTER, &pdipm->is2));
  }

  /* (7) Gather offsets from all processes */
  PetscCall(PetscMalloc1(size, &pdipm->nce_all));

  /* Get rstart of KKT matrix */
  PetscCallMPI(MPI_Scan(&pdipm->n, &rstart, 1, MPIU_INT, MPI_SUM, comm));
  rstart -= pdipm->n;

  PetscCallMPI(MPI_Allgather(&pdipm->nce, 1, MPIU_INT, pdipm->nce_all, 1, MPIU_INT, comm));

  PetscCall(PetscMalloc3(size, &ng_all, size, &nh_all, size, &Jranges));
  PetscCallMPI(MPI_Allgather(&rstart, 1, MPIU_INT, Jranges, 1, MPIU_INT, comm));
  PetscCallMPI(MPI_Allgather(&pdipm->nh, 1, MPIU_INT, nh_all, 1, MPIU_INT, comm));
  PetscCallMPI(MPI_Allgather(&pdipm->ng, 1, MPIU_INT, ng_all, 1, MPIU_INT, comm));

  PetscCall(MatGetOwnershipRanges(tao->hessian, &rranges));
  PetscCall(MatGetOwnershipRangesColumn(tao->hessian, &cranges));

  if (pdipm->Ng) {
    PetscCall(TaoComputeJacobianEquality(tao, tao->solution, tao->jacobian_equality, tao->jacobian_equality_pre));
    PetscCall(MatTranspose(tao->jacobian_equality, MAT_INITIAL_MATRIX, &pdipm->jac_equality_trans));
  }
  if (pdipm->Nh) {
    PetscCall(TaoComputeJacobianInequality(tao, tao->solution, tao->jacobian_inequality, tao->jacobian_inequality_pre));
    PetscCall(MatTranspose(tao->jacobian_inequality, MAT_INITIAL_MATRIX, &pdipm->jac_inequality_trans));
  }

  /* Count dnz,onz for preallocation of KKT matrix */
  nce_all = pdipm->nce_all;

  if (pdipm->Nxfixed) PetscCall(MatTranspose(pdipm->Jce_xfixed, MAT_INITIAL_MATRIX, &Jce_xfixed_trans));
  PetscCall(MatTranspose(pdipm->Jci_xb, MAT_INITIAL_MATRIX, &Jci_xb_trans));

  MatPreallocateBegin(comm, pdipm->n, pdipm->n, dnz, onz);

  /* 1st row block of KKT matrix: [Wxx; gradCe'; -gradCi'; 0] */
  PetscCall(TaoPDIPMEvaluateFunctionsAndJacobians(tao, pdipm->x));
  PetscCall(TaoComputeHessian(tao, tao->solution, tao->hessian, tao->hessian_pre));

  /* Insert tao->hessian */
  PetscCall(MatGetOwnershipRange(tao->hessian, &rjstart, NULL));
  for (i = 0; i < pdipm->nx; i++) {
    row = rstart + i;

    PetscCall(MatGetRow(tao->hessian, i + rjstart, &nc, &aj, NULL));
    proc = 0;
    for (j = 0; j < nc; j++) {
      while (aj[j] >= cranges[proc + 1]) proc++;
      col = aj[j] - cranges[proc] + Jranges[proc];
      PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
    }
    PetscCall(MatRestoreRow(tao->hessian, i + rjstart, &nc, &aj, NULL));

    if (pdipm->ng) {
      /* Insert grad g' */
      PetscCall(MatGetRow(pdipm->jac_equality_trans, i + rjstart, &nc, &aj, NULL));
      PetscCall(MatGetOwnershipRanges(tao->jacobian_equality, &ranges));
      proc = 0;
      for (j = 0; j < nc; j++) {
        /* find row ownership of */
        while (aj[j] >= ranges[proc + 1]) proc++;
        nx_all = rranges[proc + 1] - rranges[proc];
        col    = aj[j] - ranges[proc] + Jranges[proc] + nx_all;
        PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
      }
      PetscCall(MatRestoreRow(pdipm->jac_equality_trans, i + rjstart, &nc, &aj, NULL));
    }

    /* Insert Jce_xfixed^T' */
    if (pdipm->nxfixed) {
      PetscCall(MatGetRow(Jce_xfixed_trans, i + rjstart, &nc, &aj, NULL));
      PetscCall(MatGetOwnershipRanges(pdipm->Jce_xfixed, &ranges));
      proc = 0;
      for (j = 0; j < nc; j++) {
        /* find row ownership of */
        while (aj[j] >= ranges[proc + 1]) proc++;
        nx_all = rranges[proc + 1] - rranges[proc];
        col    = aj[j] - ranges[proc] + Jranges[proc] + nx_all + ng_all[proc];
        PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
      }
      PetscCall(MatRestoreRow(Jce_xfixed_trans, i + rjstart, &nc, &aj, NULL));
    }

    if (pdipm->nh) {
      /* Insert -grad h' */
      PetscCall(MatGetRow(pdipm->jac_inequality_trans, i + rjstart, &nc, &aj, NULL));
      PetscCall(MatGetOwnershipRanges(tao->jacobian_inequality, &ranges));
      proc = 0;
      for (j = 0; j < nc; j++) {
        /* find row ownership of */
        while (aj[j] >= ranges[proc + 1]) proc++;
        nx_all = rranges[proc + 1] - rranges[proc];
        col    = aj[j] - ranges[proc] + Jranges[proc] + nx_all + nce_all[proc];
        PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
      }
      PetscCall(MatRestoreRow(pdipm->jac_inequality_trans, i + rjstart, &nc, &aj, NULL));
    }

    /* Insert Jci_xb^T' */
    PetscCall(MatGetRow(Jci_xb_trans, i + rjstart, &nc, &aj, NULL));
    PetscCall(MatGetOwnershipRanges(pdipm->Jci_xb, &ranges));
    proc = 0;
    for (j = 0; j < nc; j++) {
      /* find row ownership of */
      while (aj[j] >= ranges[proc + 1]) proc++;
      nx_all = rranges[proc + 1] - rranges[proc];
      col    = aj[j] - ranges[proc] + Jranges[proc] + nx_all + nce_all[proc] + nh_all[proc];
      PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
    }
    PetscCall(MatRestoreRow(Jci_xb_trans, i + rjstart, &nc, &aj, NULL));
  }

  /* 2nd Row block of KKT matrix: [grad Ce, deltac*I, 0, 0] */
  if (pdipm->Ng) {
    PetscCall(MatGetOwnershipRange(tao->jacobian_equality, &rjstart, NULL));
    for (i = 0; i < pdipm->ng; i++) {
      row = rstart + pdipm->off_lambdae + i;

      PetscCall(MatGetRow(tao->jacobian_equality, i + rjstart, &nc, &aj, NULL));
      proc = 0;
      for (j = 0; j < nc; j++) {
        while (aj[j] >= cranges[proc + 1]) proc++;
        col = aj[j] - cranges[proc] + Jranges[proc];
        PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz)); /* grad g */
      }
      PetscCall(MatRestoreRow(tao->jacobian_equality, i + rjstart, &nc, &aj, NULL));
    }
  }
  /* Jce_xfixed */
  if (pdipm->Nxfixed) {
    PetscCall(MatGetOwnershipRange(pdipm->Jce_xfixed, &Jcrstart, NULL));
    for (i = 0; i < (pdipm->nce - pdipm->ng); i++) {
      row = rstart + pdipm->off_lambdae + pdipm->ng + i;

      PetscCall(MatGetRow(pdipm->Jce_xfixed, i + Jcrstart, &nc, &cols, NULL));
      PetscCheck(nc == 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "nc != 1");

      proc = 0;
      j    = 0;
      while (cols[j] >= cranges[proc + 1]) proc++;
      col = cols[j] - cranges[proc] + Jranges[proc];
      PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
      PetscCall(MatRestoreRow(pdipm->Jce_xfixed, i + Jcrstart, &nc, &cols, NULL));
    }
  }

  /* 3rd Row block of KKT matrix: [ gradCi, 0, deltac*I, -I] */
  if (pdipm->Nh) {
    PetscCall(MatGetOwnershipRange(tao->jacobian_inequality, &rjstart, NULL));
    for (i = 0; i < pdipm->nh; i++) {
      row = rstart + pdipm->off_lambdai + i;

      PetscCall(MatGetRow(tao->jacobian_inequality, i + rjstart, &nc, &aj, NULL));
      proc = 0;
      for (j = 0; j < nc; j++) {
        while (aj[j] >= cranges[proc + 1]) proc++;
        col = aj[j] - cranges[proc] + Jranges[proc];
        PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz)); /* grad h */
      }
      PetscCall(MatRestoreRow(tao->jacobian_inequality, i + rjstart, &nc, &aj, NULL));
    }
    /* I */
    for (i = 0; i < pdipm->nh; i++) {
      row = rstart + pdipm->off_lambdai + i;
      col = rstart + pdipm->off_z + i;
      PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
    }
  }

  /* Jci_xb */
  PetscCall(MatGetOwnershipRange(pdipm->Jci_xb, &Jcrstart, NULL));
  for (i = 0; i < (pdipm->nci - pdipm->nh); i++) {
    row = rstart + pdipm->off_lambdai + pdipm->nh + i;

    PetscCall(MatGetRow(pdipm->Jci_xb, i + Jcrstart, &nc, &cols, NULL));
    PetscCheck(nc == 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "nc != 1");
    proc = 0;
    for (j = 0; j < nc; j++) {
      while (cols[j] >= cranges[proc + 1]) proc++;
      col = cols[j] - cranges[proc] + Jranges[proc];
      PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
    }
    PetscCall(MatRestoreRow(pdipm->Jci_xb, i + Jcrstart, &nc, &cols, NULL));
    /* I */
    col = rstart + pdipm->off_z + pdipm->nh + i;
    PetscCall(MatPreallocateSet(row, 1, &col, dnz, onz));
  }

  /* 4-th Row block of KKT matrix: Z and Ci */
  for (i = 0; i < pdipm->nci; i++) {
    row      = rstart + pdipm->off_z + i;
    cols1[0] = rstart + pdipm->off_lambdai + i;
    cols1[1] = row;
    PetscCall(MatPreallocateSet(row, 2, cols1, dnz, onz));
  }

  /* diagonal entry */
  for (i = 0; i < pdipm->n; i++) dnz[i]++; /* diagonal entry */

  /* Create KKT matrix */
  PetscCall(MatCreate(comm, &J));
  PetscCall(MatSetSizes(J, pdipm->n, pdipm->n, PETSC_DECIDE, PETSC_DECIDE));
  PetscCall(MatSetFromOptions(J));
  PetscCall(MatSeqAIJSetPreallocation(J, 0, dnz));
  PetscCall(MatMPIAIJSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);
  pdipm->K = J;

  /* (8) Insert constant entries to  K */
  /* Set 0.0 to diagonal of K, so that the solver does not complain *about missing diagonal value */
  PetscCall(MatGetOwnershipRange(J, &rstart, &rend));
  for (i = rstart; i < rend; i++) PetscCall(MatSetValue(J, i, i, 0.0, INSERT_VALUES));
  /* In case Wxx has no diagonal entries preset set diagonal to deltaw given */
  if (pdipm->kkt_pd) {
    for (i = 0; i < pdipm->nh; i++) {
      row = rstart + i;
      PetscCall(MatSetValue(J, row, row, pdipm->deltaw, INSERT_VALUES));
    }
  }

  /* Row block of K: [ grad Ce, 0, 0, 0] */
  if (pdipm->Nxfixed) {
    PetscCall(MatGetOwnershipRange(pdipm->Jce_xfixed, &Jcrstart, NULL));
    for (i = 0; i < (pdipm->nce - pdipm->ng); i++) {
      row = rstart + pdipm->off_lambdae + pdipm->ng + i;

      PetscCall(MatGetRow(pdipm->Jce_xfixed, i + Jcrstart, &nc, &cols, &aa));
      proc = 0;
      for (j = 0; j < nc; j++) {
        while (cols[j] >= cranges[proc + 1]) proc++;
        col = cols[j] - cranges[proc] + Jranges[proc];
        PetscCall(MatSetValue(J, row, col, aa[j], INSERT_VALUES)); /* grad Ce */
        PetscCall(MatSetValue(J, col, row, aa[j], INSERT_VALUES)); /* grad Ce' */
      }
      PetscCall(MatRestoreRow(pdipm->Jce_xfixed, i + Jcrstart, &nc, &cols, &aa));
    }
  }

  /* Row block of K: [ -grad Ci, 0, 0, I] */
  PetscCall(MatGetOwnershipRange(pdipm->Jci_xb, &Jcrstart, NULL));
  for (i = 0; i < pdipm->nci - pdipm->nh; i++) {
    row = rstart + pdipm->off_lambdai + pdipm->nh + i;

    PetscCall(MatGetRow(pdipm->Jci_xb, i + Jcrstart, &nc, &cols, &aa));
    proc = 0;
    for (j = 0; j < nc; j++) {
      while (cols[j] >= cranges[proc + 1]) proc++;
      col = cols[j] - cranges[proc] + Jranges[proc];
      PetscCall(MatSetValue(J, col, row, -aa[j], INSERT_VALUES));
      PetscCall(MatSetValue(J, row, col, -aa[j], INSERT_VALUES));
    }
    PetscCall(MatRestoreRow(pdipm->Jci_xb, i + Jcrstart, &nc, &cols, &aa));

    col = rstart + pdipm->off_z + pdipm->nh + i;
    PetscCall(MatSetValue(J, row, col, 1, INSERT_VALUES));
  }

  for (i = 0; i < pdipm->nh; i++) {
    row = rstart + pdipm->off_lambdai + i;
    col = rstart + pdipm->off_z + i;
    PetscCall(MatSetValue(J, row, col, 1, INSERT_VALUES));
  }

  /* Row block of K: [ 0, 0, I, ...] */
  for (i = 0; i < pdipm->nci; i++) {
    row = rstart + pdipm->off_z + i;
    col = rstart + pdipm->off_lambdai + i;
    PetscCall(MatSetValue(J, row, col, 1, INSERT_VALUES));
  }

  if (pdipm->Nxfixed) PetscCall(MatDestroy(&Jce_xfixed_trans));
  PetscCall(MatDestroy(&Jci_xb_trans));
  PetscCall(PetscFree3(ng_all, nh_all, Jranges));

  /* (9) Set up nonlinear solver SNES */
  PetscCall(SNESSetFunction(pdipm->snes, NULL, TaoSNESFunction_PDIPM, (void *)tao));
  PetscCall(SNESSetJacobian(pdipm->snes, J, J, TaoSNESJacobian_PDIPM, (void *)tao));

  if (pdipm->solve_reduced_kkt) {
    PC pc;
    PetscCall(KSPGetPC(tao->ksp, &pc));
    PetscCall(PCSetType(pc, PCFIELDSPLIT));
    PetscCall(PCFieldSplitSetType(pc, PC_COMPOSITE_SCHUR));
    PetscCall(PCFieldSplitSetIS(pc, "2", pdipm->is2));
    PetscCall(PCFieldSplitSetIS(pc, "1", pdipm->is1));
  }
  PetscCall(SNESSetFromOptions(pdipm->snes));

  /* (10) Setup PCPreSolve() for pdipm->solve_symmetric_kkt */
  if (pdipm->solve_symmetric_kkt) {
    KSP       ksp;
    PC        pc;
    PetscBool isCHOL;
    PetscCall(SNESGetKSP(pdipm->snes, &ksp));
    PetscCall(KSPGetPC(ksp, &pc));
    PetscCall(PCSetPreSolve(pc, PCPreSolve_PDIPM));

    PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCCHOLESKY, &isCHOL));
    if (isCHOL) {
      Mat       Factor;
      PetscBool isMUMPS;
      PetscCall(PCFactorGetMatrix(pc, &Factor));
      PetscCall(PetscObjectTypeCompare((PetscObject)Factor, "mumps", &isMUMPS));
      if (isMUMPS) { /* must set mumps ICNTL(13)=1 and ICNTL(24)=1 to call MatGetInertia() */
#if defined(PETSC_HAVE_MUMPS)
        PetscCall(MatMumpsSetIcntl(Factor, 24, 1)); /* detection of null pivot rows */
        if (size > 1) { PetscCall(MatMumpsSetIcntl(Factor, 13, 1)); /* parallelism of the root node (enable ScaLAPACK) and its splitting */ }
#else
        SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_SUP, "Requires external package MUMPS");
#endif
      }
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TaoDestroy_PDIPM(Tao tao)
{
  TAO_PDIPM *pdipm = (TAO_PDIPM *)tao->data;

  PetscFunctionBegin;
  /* Freeing Vectors assocaiated with KKT (X) */
  PetscCall(VecDestroy(&pdipm->x));       /* Solution x */
  PetscCall(VecDestroy(&pdipm->lambdae)); /* Equality constraints lagrangian multiplier*/
  PetscCall(VecDestroy(&pdipm->lambdai)); /* Inequality constraints lagrangian multiplier*/
  PetscCall(VecDestroy(&pdipm->z));       /* Slack variables */
  PetscCall(VecDestroy(&pdipm->X));       /* Big KKT system vector [x; lambdae; lambdai; z] */

  /* work vectors */
  PetscCall(VecDestroy(&pdipm->lambdae_xfixed));
  PetscCall(VecDestroy(&pdipm->lambdai_xb));

  /* Legrangian equality and inequality Vec */
  PetscCall(VecDestroy(&pdipm->ce)); /* Vec of equality constraints */
  PetscCall(VecDestroy(&pdipm->ci)); /* Vec of inequality constraints */

  /* Matrices */
  PetscCall(MatDestroy(&pdipm->Jce_xfixed));
  PetscCall(MatDestroy(&pdipm->Jci_xb)); /* Jacobian of inequality constraints Jci = [tao->jacobian_inequality ; J(nxub); J(nxlb); J(nxbx)] */
  PetscCall(MatDestroy(&pdipm->K));

  /* Index Sets */
  if (pdipm->Nxub) { PetscCall(ISDestroy(&pdipm->isxub)); /* Finite upper bound only -inf < x < ub */ }

  if (pdipm->Nxlb) { PetscCall(ISDestroy(&pdipm->isxlb)); /* Finite lower bound only  lb <= x < inf */ }

  if (pdipm->Nxfixed) { PetscCall(ISDestroy(&pdipm->isxfixed)); /* Fixed variables         lb =  x = ub */ }

  if (pdipm->Nxbox) { PetscCall(ISDestroy(&pdipm->isxbox)); /* Boxed variables         lb <= x <= ub */ }

  if (pdipm->Nxfree) { PetscCall(ISDestroy(&pdipm->isxfree)); /* Free variables        -inf <= x <= inf */ }

  if (pdipm->solve_reduced_kkt) {
    PetscCall(ISDestroy(&pdipm->is1));
    PetscCall(ISDestroy(&pdipm->is2));
  }

  /* SNES */
  PetscCall(SNESDestroy(&pdipm->snes)); /* Nonlinear solver */
  PetscCall(PetscFree(pdipm->nce_all));
  PetscCall(MatDestroy(&pdipm->jac_equality_trans));
  PetscCall(MatDestroy(&pdipm->jac_inequality_trans));

  /* Destroy pdipm */
  PetscCall(PetscFree(tao->data)); /* Holding locations of pdipm */

  /* Destroy Dual */
  PetscCall(VecDestroy(&tao->DE)); /* equality dual */
  PetscCall(VecDestroy(&tao->DI)); /* dinequality dual */
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TaoSetFromOptions_PDIPM(Tao tao, PetscOptionItems *PetscOptionsObject)
{
  TAO_PDIPM *pdipm = (TAO_PDIPM *)tao->data;

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "PDIPM method for constrained optimization");
  PetscCall(PetscOptionsReal("-tao_pdipm_push_init_slack", "parameter to push initial slack variables away from bounds", NULL, pdipm->push_init_slack, &pdipm->push_init_slack, NULL));
  PetscCall(PetscOptionsReal("-tao_pdipm_push_init_lambdai", "parameter to push initial (inequality) dual variables away from bounds", NULL, pdipm->push_init_lambdai, &pdipm->push_init_lambdai, NULL));
  PetscCall(PetscOptionsBool("-tao_pdipm_solve_reduced_kkt", "Solve reduced KKT system using Schur-complement", NULL, pdipm->solve_reduced_kkt, &pdipm->solve_reduced_kkt, NULL));
  PetscCall(PetscOptionsReal("-tao_pdipm_mu_update_factor", "Update scalar for barrier parameter (mu) update", NULL, pdipm->mu_update_factor, &pdipm->mu_update_factor, NULL));
  PetscCall(PetscOptionsBool("-tao_pdipm_symmetric_kkt", "Solve non reduced symmetric KKT system", NULL, pdipm->solve_symmetric_kkt, &pdipm->solve_symmetric_kkt, NULL));
  PetscCall(PetscOptionsBool("-tao_pdipm_kkt_shift_pd", "Add shifts to make KKT matrix positive definite", NULL, pdipm->kkt_pd, &pdipm->kkt_pd, NULL));
  PetscOptionsHeadEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*MC
  TAOPDIPM - Barrier-based primal-dual interior point algorithm for generally constrained optimization.

  Options Database Keys:
+   -tao_pdipm_push_init_lambdai - parameter to push initial dual variables away from bounds (> 0)
.   -tao_pdipm_push_init_slack - parameter to push initial slack variables away from bounds (> 0)
.   -tao_pdipm_mu_update_factor - update scalar for barrier parameter (mu) update (> 0)
.   -tao_pdipm_symmetric_kkt - Solve non-reduced symmetric KKT system
-   -tao_pdipm_kkt_shift_pd - Add shifts to make KKT matrix positive definite

  Level: beginner

.seealso: `TAOPDIPM`, `Tao`, `TaoType`
M*/

PETSC_EXTERN PetscErrorCode TaoCreate_PDIPM(Tao tao)
{
  TAO_PDIPM *pdipm;

  PetscFunctionBegin;
  tao->ops->setup          = TaoSetup_PDIPM;
  tao->ops->solve          = TaoSolve_PDIPM;
  tao->ops->setfromoptions = TaoSetFromOptions_PDIPM;
  tao->ops->view           = TaoView_PDIPM;
  tao->ops->destroy        = TaoDestroy_PDIPM;

  PetscCall(PetscNew(&pdipm));
  tao->data = (void *)pdipm;

  pdipm->nx = pdipm->Nx = 0;
  pdipm->nxfixed = pdipm->Nxfixed = 0;
  pdipm->nxlb = pdipm->Nxlb = 0;
  pdipm->nxub = pdipm->Nxub = 0;
  pdipm->nxbox = pdipm->Nxbox = 0;
  pdipm->nxfree = pdipm->Nxfree = 0;

  pdipm->ng = pdipm->Ng = pdipm->nce = pdipm->Nce = 0;
  pdipm->nh = pdipm->Nh = pdipm->nci = pdipm->Nci = 0;
  pdipm->n = pdipm->N     = 0;
  pdipm->mu               = 1.0;
  pdipm->mu_update_factor = 0.1;

  pdipm->deltaw     = 0.0;
  pdipm->lastdeltaw = 3 * 1.e-4;
  pdipm->deltac     = 0.0;
  pdipm->kkt_pd     = PETSC_FALSE;

  pdipm->push_init_slack     = 1.0;
  pdipm->push_init_lambdai   = 1.0;
  pdipm->solve_reduced_kkt   = PETSC_FALSE;
  pdipm->solve_symmetric_kkt = PETSC_TRUE;

  /* Override default settings (unless already changed) */
  if (!tao->max_it_changed) tao->max_it = 200;
  if (!tao->max_funcs_changed) tao->max_funcs = 500;

  PetscCall(SNESCreate(((PetscObject)tao)->comm, &pdipm->snes));
  PetscCall(SNESSetOptionsPrefix(pdipm->snes, tao->hdr.prefix));
  PetscCall(SNESGetKSP(pdipm->snes, &tao->ksp));
  PetscCall(PetscObjectReference((PetscObject)tao->ksp));
  PetscCall(KSPSetApplicationContext(tao->ksp, (void *)tao));
  PetscFunctionReturn(PETSC_SUCCESS);
}