tao_optimization_solver.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
 
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 
 
 
#include "libmesh/libmesh_common.h"
 
#if defined(LIBMESH_HAVE_PETSC_TAO) && !defined(LIBMESH_USE_COMPLEX_NUMBERS)
 
 
// C++ includes
 
// Local Includes
#include "libmesh/libmesh_logging.h"
#include "libmesh/petsc_vector.h"
#include "libmesh/petsc_matrix.h"
#include "libmesh/dof_map.h"
#include "libmesh/tao_optimization_solver.h"
#include "libmesh/equation_systems.h"
 
namespace libMesh
{
 
//--------------------------------------------------------------------
// Functions with C linkage to pass to Tao. Tao will call these
// methods as needed.
//
// Since they must have C linkage they have no knowledge of a namespace.
// Give them an obscure name to avoid namespace pollution.
extern "C"
{
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate objective function at x
  PetscErrorCode
  __libmesh_tao_objective (Tao /*tao*/, Vec x, PetscReal * objective, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("objective()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(objective);
    libmesh_assert(ctx);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // Enforce constraints (if any) exactly on the
    // current_local_solution.  This is the solution vector that is
    // actually used in the computation of the objective function
    // below, and is not locked by debug-enabled PETSc the way that
    // the solution vector is.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->objective_object != nullptr)
      (*objective) = PS(solver->objective_object->objective(*(sys.current_local_solution), sys));
    else
      libmesh_error_msg("Objective function not defined in __libmesh_tao_objective");
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
 
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate the gradient at x
  PetscErrorCode
  __libmesh_tao_gradient(Tao /*tao*/, Vec x, Vec g, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("gradient()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(g);
    libmesh_assert(ctx);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // We'll also pass the gradient in to the assembly routine
    // so let's make a PETSc vector for that too.
    PetscVector<Number> gradient(g, sys.comm());
 
    // Clear the gradient prior to assembly
    gradient.zero();
 
    // Enforce constraints exactly on the current_local_solution.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->gradient_object != nullptr)
      solver->gradient_object->gradient(*(sys.current_local_solution), gradient, sys);
    else
      libmesh_error_msg("Gradient function not defined in __libmesh_tao_gradient");
 
    gradient.close();
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate the Hessian at x
  PetscErrorCode
  __libmesh_tao_hessian(Tao /*tao*/, Vec x, Mat h, Mat pc, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("hessian()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(h);
    libmesh_assert(pc);
    libmesh_assert(ctx);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // Let's also wrap pc and h in PetscMatrix objects for convenience
    PetscMatrix<Number> PC(pc, sys.comm());
    PetscMatrix<Number> hessian(h, sys.comm());
    PC.attach_dof_map(sys.get_dof_map());
    hessian.attach_dof_map(sys.get_dof_map());
 
    // Enforce constraints exactly on the current_local_solution.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->hessian_object != nullptr)
      {
        // Following PetscNonlinearSolver by passing in PC. It's not clear
        // why we pass in PC and not hessian though?
        solver->hessian_object->hessian(*(sys.current_local_solution), PC, sys);
      }
    else
      libmesh_error_msg("Hessian function not defined in __libmesh_tao_hessian");
 
    PC.close();
    hessian.close();
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate the equality constraints at x
  PetscErrorCode
  __libmesh_tao_equality_constraints(Tao /*tao*/, Vec x, Vec ce, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("equality_constraints()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(ce);
    libmesh_assert(ctx);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // We'll also pass the constraints vector ce into the assembly routine
    // so let's make a PETSc vector for that too.
    PetscVector<Number> eq_constraints(ce, sys.comm());
 
    // Clear the gradient prior to assembly
    eq_constraints.zero();
 
    // Enforce constraints exactly on the current_local_solution.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->equality_constraints_object != nullptr)
      solver->equality_constraints_object->equality_constraints(*(sys.current_local_solution), eq_constraints, sys);
    else
      libmesh_error_msg("Constraints function not defined in __libmesh_tao_equality_constraints");
 
    eq_constraints.close();
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate the Jacobian of the
  // equality constraints at x
  PetscErrorCode
  __libmesh_tao_equality_constraints_jacobian(Tao /*tao*/, Vec x, Mat J, Mat Jpre, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("equality_constraints_jacobian()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(J);
    libmesh_assert(Jpre);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // Let's also wrap J and Jpre in PetscMatrix objects for convenience
    PetscMatrix<Number> J_petsc(J, sys.comm());
    PetscMatrix<Number> Jpre_petsc(Jpre, sys.comm());
 
    // Enforce constraints exactly on the current_local_solution.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->equality_constraints_jacobian_object != nullptr)
      solver->equality_constraints_jacobian_object->equality_constraints_jacobian(*(sys.current_local_solution), J_petsc, sys);
    else
      libmesh_error_msg("Constraints function not defined in __libmesh_tao_equality_constraints_jacobian");
 
    J_petsc.close();
    Jpre_petsc.close();
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate the inequality constraints at x
  PetscErrorCode
  __libmesh_tao_inequality_constraints(Tao /*tao*/, Vec x, Vec cineq, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("inequality_constraints()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(cineq);
    libmesh_assert(ctx);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // We'll also pass the constraints vector ce into the assembly routine
    // so let's make a PETSc vector for that too.
    PetscVector<Number> ineq_constraints(cineq, sys.comm());
 
    // Clear the gradient prior to assembly
    ineq_constraints.zero();
 
    // Enforce constraints exactly on the current_local_solution.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->inequality_constraints_object != nullptr)
      solver->inequality_constraints_object->inequality_constraints(*(sys.current_local_solution), ineq_constraints, sys);
    else
      libmesh_error_msg("Constraints function not defined in __libmesh_tao_inequality_constraints");
 
    ineq_constraints.close();
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
  //---------------------------------------------------------------
  // This function is called by Tao to evaluate the Jacobian of the
  // equality constraints at x
  PetscErrorCode
  __libmesh_tao_inequality_constraints_jacobian(Tao /*tao*/, Vec x, Mat J, Mat Jpre, void * ctx)
  {
    PetscFunctionBegin;
 
    LOG_SCOPE("inequality_constraints_jacobian()", "TaoOptimizationSolver");
 
    libmesh_assert(x);
    libmesh_assert(J);
    libmesh_assert(Jpre);
 
    // ctx should be a pointer to the solver (it was passed in as void *)
    TaoOptimizationSolver<Number> * solver =
      static_cast<TaoOptimizationSolver<Number> *> (ctx);
 
    OptimizationSystem & sys = solver->system();
 
    // We'll use current_local_solution below, so let's ensure that it's consistent
    // with the vector x that was passed in.
    PetscVector<Number> & X_sys = *cast_ptr<PetscVector<Number> *>(sys.solution.get());
    PetscVector<Number> X(x, sys.comm());
 
    // Perform a swap so that sys.solution points to the input vector
    // "x", update sys.current_local_solution based on "x", then swap
    // back.
    X.swap(X_sys);
    sys.update();
    X.swap(X_sys);
 
    // Let's also wrap J and Jpre in PetscMatrix objects for convenience
    PetscMatrix<Number> J_petsc(J, sys.comm());
    PetscMatrix<Number> Jpre_petsc(Jpre, sys.comm());
 
    // Enforce constraints exactly on the current_local_solution.
    sys.get_dof_map().enforce_constraints_exactly(sys, sys.current_local_solution.get());
 
    if (solver->inequality_constraints_jacobian_object != nullptr)
      solver->inequality_constraints_jacobian_object->inequality_constraints_jacobian(*(sys.current_local_solution), J_petsc, sys);
    else
      libmesh_error_msg("Constraints function not defined in __libmesh_tao_inequality_constraints_jacobian");
 
    J_petsc.close();
    Jpre_petsc.close();
 
    PetscFunctionReturn(LIBMESH_PETSC_SUCCESS);
  }
 
} // end extern "C"
//---------------------------------------------------------------------
 
 
 
//---------------------------------------------------------------------
// TaoOptimizationSolver<> methods
template <typename T>
TaoOptimizationSolver<T>::TaoOptimizationSolver (OptimizationSystem & system_in) :
  OptimizationSolver<T>(system_in),
  _reason(TAO_CONVERGED_USER) // Arbitrary initial value...
{
}
 
 
 
template <typename T>
TaoOptimizationSolver<T>::~TaoOptimizationSolver ()
{
  this->TaoOptimizationSolver::clear ();
}
 
 
 
template <typename T>
void TaoOptimizationSolver<T>::clear () noexcept
{
  if (this->initialized())
    {
      this->_is_initialized = false;
 
      PetscErrorCode ierr = TaoDestroy(&_tao);
      if (ierr)
        libmesh_warning("Warning: TaoDestroy returned a non-zero error code which we ignored.");
    }
}
 
 
 
template <typename T>
void TaoOptimizationSolver<T>::init ()
{
  // Initialize the data structures if not done so already.
  if (!this->initialized())
    {
      this->_is_initialized = true;
 
      LibmeshPetscCall(TaoCreate(this->comm().get(),&_tao));
    }
}
 
template <typename T>
void TaoOptimizationSolver<T>::solve ()
{
  LOG_SCOPE("solve()", "TaoOptimizationSolver");
 
  this->init ();
 
  this->system().solution->zero();
 
  PetscMatrixBase<T> * hessian  = cast_ptr<PetscMatrixBase<T> *>(this->system().matrix);
  // PetscVector<T> * gradient = cast_ptr<PetscVector<T> *>(this->system().rhs);
  PetscVector<T> * x         = cast_ptr<PetscVector<T> *>(this->system().solution.get());
  PetscVector<T> * ceq       = cast_ptr<PetscVector<T> *>(this->system().C_eq.get());
  PetscMatrixBase<T> * ceq_jac   = cast_ptr<PetscMatrixBase<T> *>(this->system().C_eq_jac.get());
  PetscVector<T> * cineq     = cast_ptr<PetscVector<T> *>(this->system().C_ineq.get());
  PetscMatrixBase<T> * cineq_jac = cast_ptr<PetscMatrixBase<T> *>(this->system().C_ineq_jac.get());
  PetscVector<T> * lb        = cast_ptr<PetscVector<T> *>(&this->system().get_vector("lower_bounds"));
  PetscVector<T> * ub        = cast_ptr<PetscVector<T> *>(&this->system().get_vector("upper_bounds"));
 
  // Set the starting guess to zero.
  x->zero();
 
  // Workaround for bug where TaoSetFromOptions *reset*
  // programmatically set tolerance and max. function evaluation
  // values when "-tao_type ipm" was specified on the command line: we
  // call TaoSetFromOptions twice (both before and after setting
  // custom options programmatically)
  LibmeshPetscCall(TaoSetFromOptions(_tao));
 
  // Set convergence tolerances
  // f(X) - f(X*) (estimated)            <= fatol
  // |f(X) - f(X*)| (estimated) / |f(X)| <= frtol
  // ||g(X)||                            <= gatol
  // ||g(X)|| / |f(X)|                   <= grtol
  // ||g(X)|| / ||g(X0)||                <= gttol
  // Command line equivalents: -tao_fatol, -tao_frtol, -tao_gatol, -tao_grtol, -tao_gttol
  // Releases up to 3.7.0 had fatol and frtol, after that they were removed.
#if PETSC_VERSION_LESS_THAN(3,7,0)
  LibmeshPetscCall(TaoSetTolerances(_tao,
                                    /*fatol=*/PETSC_DEFAULT,
                                    /*frtol=*/PETSC_DEFAULT,
                                    /*gatol=*/PETSC_DEFAULT,
                                    /*grtol=*/this->objective_function_relative_tolerance,
                                    /*gttol=*/PETSC_DEFAULT));
#else
  LibmeshPetscCall(TaoSetTolerances(_tao,
                                    /*gatol=*/PETSC_DEFAULT,
                                    /*grtol=*/this->objective_function_relative_tolerance,
                                    /*gttol=*/PETSC_DEFAULT));
#endif
 
  // Set the max-allowed number of objective function evaluations
  // Command line equivalent: -tao_max_funcs
  LibmeshPetscCall(TaoSetMaximumFunctionEvaluations(_tao, this->max_objective_function_evaluations));
 
  // Set the max-allowed number of optimization iterations.
  // Command line equivalent: -tao_max_it
  // Not implemented for now as it seems fairly similar to
  // LibmeshPetscCall(TaoSetMaximumIterations(_tao, 4));
 
  // Set solution vec and an initial guess
#if PETSC_VERSION_LESS_THAN(3,17,0)
  LibmeshPetscCall(TaoSetInitialVector(_tao, x->vec()));
#else
  LibmeshPetscCall(TaoSetSolution(_tao, x->vec()));
#endif
 
  // We have to have an objective function
  libmesh_assert( this->objective_object );
 
  // Set routines for objective, gradient, hessian evaluation
#if PETSC_VERSION_LESS_THAN(3,17,0)
  LibmeshPetscCall(TaoSetObjectiveRoutine(_tao, __libmesh_tao_objective, this));
#else
  LibmeshPetscCall(TaoSetObjective(_tao, __libmesh_tao_objective, this));
#endif
 
  if (this->gradient_object)
#if PETSC_VERSION_LESS_THAN(3,17,0)
    LibmeshPetscCall(TaoSetGradientRoutine(_tao, __libmesh_tao_gradient, this));
#else
    LibmeshPetscCall(TaoSetGradient(_tao, NULL, __libmesh_tao_gradient, this));
#endif
 
  if (this->hessian_object)
#if PETSC_VERSION_LESS_THAN(3,17,0)
    LibmeshPetscCall(TaoSetHessianRoutine(_tao, hessian->mat(), hessian->mat(), __libmesh_tao_hessian, this));
#else
    LibmeshPetscCall(TaoSetHessian(_tao, hessian->mat(), hessian->mat(), __libmesh_tao_hessian, this));
#endif
 
  if (this->lower_and_upper_bounds_object)
    {
      // Need to actually compute the bounds vectors first
      this->lower_and_upper_bounds_object->lower_and_upper_bounds(this->system());
 
      LibmeshPetscCall(TaoSetVariableBounds(_tao,
                                            lb->vec(),
                                            ub->vec()));
    }
 
  if (this->equality_constraints_object)
    LibmeshPetscCall(TaoSetEqualityConstraintsRoutine(_tao, ceq->vec(), __libmesh_tao_equality_constraints, this));
 
  if (this->equality_constraints_jacobian_object)
    LibmeshPetscCall(TaoSetJacobianEqualityRoutine(_tao,
                                                   ceq_jac->mat(),
                                                   ceq_jac->mat(),
                                                   __libmesh_tao_equality_constraints_jacobian,
                                                   this));
 
  // Optionally set inequality constraints
  if (this->inequality_constraints_object)
    LibmeshPetscCall(TaoSetInequalityConstraintsRoutine(_tao, cineq->vec(), __libmesh_tao_inequality_constraints, this));
 
  // Optionally set inequality constraints Jacobian
  if (this->inequality_constraints_jacobian_object)
    LibmeshPetscCall(TaoSetJacobianInequalityRoutine(_tao,
                                                     cineq_jac->mat(),
                                                     cineq_jac->mat(),
                                                     __libmesh_tao_inequality_constraints_jacobian,
                                                     this));
 
  // Check for Tao command line options
  LibmeshPetscCall(TaoSetFromOptions(_tao));
 
  // Perform the optimization
  LibmeshPetscCall(TaoSolve(_tao));
 
  // Enforce constraints exactly now that the solve is done.  We have
  // been enforcing them on the current_local_solution during the
  // solve, but now need to be sure they are enforced on the parallel
  // solution vector as well.
  this->system().get_dof_map().enforce_constraints_exactly(this->system());
 
  // Store the convergence/divergence reason
  LibmeshPetscCall(TaoGetConvergedReason(_tao, &_reason));
}
 
 
template <typename T>
void TaoOptimizationSolver<T>::get_dual_variables()
{
  LOG_SCOPE("get_dual_variables()", "TaoOptimizationSolver");
 
  PetscVector<T> * lambda_eq_petsc =
    cast_ptr<PetscVector<T> *>(this->system().lambda_eq.get());
  PetscVector<T> * lambda_ineq_petsc =
    cast_ptr<PetscVector<T> *>(this->system().lambda_ineq.get());
 
  Vec lambda_eq_petsc_vec = lambda_eq_petsc->vec();
  Vec lambda_ineq_petsc_vec = lambda_ineq_petsc->vec();
 
  LibmeshPetscCall(TaoGetDualVariables(_tao,
                                       &lambda_eq_petsc_vec,
                                       &lambda_ineq_petsc_vec));
}
 
 
template <typename T>
void TaoOptimizationSolver<T>::print_converged_reason()
{
  libMesh::out << "Tao optimization solver convergence/divergence reason: "
               << TaoConvergedReasons[this->get_converged_reason()] << std::endl;
}
 
 
 
template <typename T>
int TaoOptimizationSolver<T>::get_converged_reason()
{
  if (this->initialized())
    LibmeshPetscCall(TaoGetConvergedReason(_tao, &_reason));
 
  return static_cast<int>(_reason);
}
 
 
//------------------------------------------------------------------
// Explicit instantiations
template class LIBMESH_EXPORT TaoOptimizationSolver<Number>;
 
} // namespace libMesh
 
 
 
#endif // #if defined(LIBMESH_HAVE_PETSC_TAO) && !defined(LIBMESH_USE_COMPLEX_NUMBERS)