NewtonWithTrustRegion.cxx

// Copyright 2023, UChicago Argonne, LLC
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// Software Name: NEML2 -- the New Engineering material Model Library, version 2
// By: Argonne National Laboratory
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#include "neml2/solvers/NewtonWithTrustRegion.h"
#include "neml2/misc/math.h"
 
#include <iomanip>
 
namespace neml2
{
register_NEML2_object(NewtonWithTrustRegion);
 
OptionSet
NewtonWithTrustRegion::expected_options()
{
  OptionSet options = Newton::expected_options();
  options.doc() =
      "A trust-region Newton solver. The step size and direction are modified by solving a "
      "constrained minimization problem using the local quadratic approximation. The default "
      "solver parameters are chosen based on a limited set of problems we tested on and are "
      "expected to be tuned.";
 
  options.set<Real>("delta_0") = 1.0;
  options.set("delta_0").doc() = "Initial trust region radius";
 
  options.set<Real>("delta_max") = 10.0;
  options.set("delta_max").doc() = "Maximum trust region radius";
 
  options.set<Real>("reduce_criteria") = 0.25;
  options.set("reduce_criteria").doc() = "The trust region radius is reduced when the merit "
                                         "function reduction is below this threshold";
 
  options.set<Real>("expand_criteria") = 0.75;
  options.set("expand_criteria").doc() = "The trust region radius is increased when the merit "
                                         "function reduction is above this threshold";
 
  options.set<Real>("reduce_factor") = 0.25;
  options.set("reduce_factor").doc() = "Factor to apply when reducing the trust region radius";
 
  options.set<Real>("expand_factor") = 2.0;
  options.set("expand_factor").doc() = "Factor to apply when increasing the trust region radius";
 
  options.set<Real>("accept_criteria") = 0.1;
  options.set("accept_criteria").doc() =
      "Reject the current step when the merit function reduction is below this threshold";
 
  options.set<Real>("subproblem_rel_tol") = 1e-6;
  options.set("subproblem_rel_tol").doc() = "Relative tolerance used for the quadratic sub-problem";
 
  options.set<Real>("subproblem_abs_tol") = 1e-8;
  options.set("subproblem_abs_tol").doc() = "Absolute tolerance used for the quadratic sub-problem";
 
  options.set<unsigned int>("subproblem_max_its") = 10;
  options.set("subproblem_max_its").doc() =
      "Maximum number of allowable iterations when solving the quadratic sub-problem";
 
  return options;
}
 
NewtonWithTrustRegion::NewtonWithTrustRegion(const OptionSet & options)
  : Newton(options),
    _subproblem(subproblem_options(options)),
    _subproblem_solver(subproblem_solver_options(options)),
    _delta_0(options.get<Real>("delta_0")),
    _delta_max(options.get<Real>("delta_max")),
    _reduce_criteria(options.get<Real>("reduce_criteria")),
    _expand_criteria(options.get<Real>("expand_criteria")),
    _reduce_factor(options.get<Real>("reduce_factor")),
    _expand_factor(options.get<Real>("expand_factor")),
    _accept_criteria(options.get<Real>("accept_criteria"))
{
}
 
OptionSet
NewtonWithTrustRegion::subproblem_options(const OptionSet & /*options*/) const
{
  // By default the nonlinear system turns off automatic scaling (which is what we want here)
  return TrustRegionSubProblem::expected_options();
}
 
OptionSet
NewtonWithTrustRegion::subproblem_solver_options(const OptionSet & options) const
{
  auto solver_options = Newton::expected_options();
  solver_options.set<Real>("abs_tol") = options.get<Real>("subproblem_abs_tol");
  solver_options.set<Real>("rel_tol") = options.get<Real>("subproblem_rel_tol");
  solver_options.set<unsigned int>("max_its") = options.get<unsigned int>("subproblem_max_its");
  return solver_options;
}
 
void
NewtonWithTrustRegion::prepare(const NonlinearSystem & /*system*/, const BatchTensor & x)
{
  _delta = Scalar::full(x.batch_sizes(), _delta_0, x.options());
}
 
void
NewtonWithTrustRegion::update(NonlinearSystem & system, BatchTensor & x)
{
  auto p = solve_direction(system);
 
  // Predicted reduction in the merit function
  auto nR = system.residual_norm();
  auto red_b = merit_function_reduction(system, p);
 
  // Actual reduction in the objective function
  auto xp = x + system.scale_direction(p);
  auto [Rp, Jp] = system.residual_and_Jacobian(xp);
  auto nRp = system.residual_norm();
  auto red_a = 0.5 * torch::pow(nR, 2.0) - 0.5 * torch::pow(nRp, 2.0);
 
  // Quality of the subproblem solution compared to the quadratic model
  auto rho = red_a / red_b;
 
  // Adjust the trust region based on the quality of the subproblem
  _delta.batch_index_put({rho < _reduce_criteria},
                         _reduce_factor * _delta.batch_index({rho < _reduce_criteria}));
  _delta.batch_index_put({rho > _expand_criteria},
                         torch::clamp(_expand_factor * _delta.batch_index({rho > _expand_criteria}),
                                      c10::nullopt,
                                      _delta_max));
 
  // Accept or reject the current step
  auto accept = (rho >= _accept_criteria).unsqueeze(-1);
 
  // Do some printing if verbose
  if (verbose)
  {
    std::cout << "     RHO MIN/MAX            : " << std::scientific << torch::min(rho).item<Real>()
              << "/" << std::scientific << torch::max(rho).item<Real>() << std::endl;
    std::cout << "     ACCEPTANCE RATE        : " << torch::sum(accept).item<TorchSize>() << "/"
              << utils::storage_size(_delta.batch_sizes()) << std::endl;
    std::cout << "     ADJUSTED DELTA MIN/MAX : " << std::scientific
              << torch::min(_delta).item<Real>() << "/" << std::scientific
              << torch::max(_delta).item<Real>() << std::endl;
  }
 
  x.variable_data().copy_(torch::where(accept, xp, x));
  system.set_solution(x);
}
 
BatchTensor
NewtonWithTrustRegion::solve_direction(const NonlinearSystem & system)
{
  // The full Newton step
  auto p_newton = Newton::solve_direction(system);
 
  // The trust region step (obtained by solving the bound constrained subproblem)
  _subproblem.reinit(system, _delta);
  auto s = _subproblem.solution().clone();
  auto [succeeded, iters] = _subproblem_solver.solve(_subproblem, s);
  s = BatchTensor(torch::clamp(s, 0.0), s.batch_dim());
  auto p_trust = -_subproblem.preconditioned_direction(s);
 
  // Now select between the two... Basically take the full Newton step whenever possible
  auto newton_inside_trust_region =
      (math::linalg::vector_norm(p_newton) <= math::sqrt(2.0 * _delta)).unsqueeze(-1);
 
  // Do some printing if verbose
  if (verbose)
  {
    std::cout << "     TRUST-REGION ITERATIONS: " << iters << std::endl;
    std::cout << "     ACTIVE CONSTRAINTS     : " << torch::sum(s > 0).item<TorchSize>() << "/"
              << utils::storage_size(s.batch_sizes()) << std::endl;
  }
 
  return BatchTensor(torch::where(newton_inside_trust_region, p_newton, p_trust),
                     p_newton.batch_dim());
}
 
Scalar
NewtonWithTrustRegion::merit_function_reduction(const NonlinearSystem & system,
                                                const BatchTensor & p) const
{
  auto Jp = math::bmv(system.Jacobian_view(), p);
  return -math::bvv(system.residual_view(), Jp) - 0.5 * math::bvv(Jp, Jp);
}
 
} // namespace neml2