ROL_TypeP_InexactNewtonAlgorithm_Def.hpp

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#ifndef ROL_TYPEP_QUASINEWTONALGORITHM_DEF_HPP
#define ROL_TYPEP_QUASINEWTONALGORITHM_DEF_HPP
 
#include "ROL_TypeP_ProxGradientAlgorithm.hpp"
#include "ROL_TypeP_SpectralGradientAlgorithm.hpp"
#include "ROL_TypeP_iPianoAlgorithm.hpp"
#include "ROL_TypeP_TrustRegionAlgorithm.hpp"
 
namespace ROL {
namespace TypeP {
 
template<typename Real>
InexactNewtonAlgorithm<Real>::InexactNewtonAlgorithm(ParameterList &list)
  : list_(list) {
  // Set status test
  status_->reset();
  status_->add(makePtr<StatusTest<Real>>(list));
 
  // Parse parameter list
  ParameterList &lslist = list.sublist("Step").sublist("Line Search");
  t0_           = list.sublist("Status Test").get("Gradient Scale"            , 1.0);
  initProx_     = lslist.get("Apply Prox to Initial Guess",                   false);
  maxit_        = lslist.get("Function Evaluation Limit",                        20);
  c1_           = lslist.get("Sufficient Decrease Tolerance",                  1e-4);
  rhodec_       = lslist.sublist("Line-Search Method").get("Backtracking Rate", 0.5);
  sigma1_       = lslist.sublist("Inexact Newton").get("Lower Step Size Safeguard",        0.1);
  sigma2_       = lslist.sublist("Inexact Newton").get("Upper Step Size Safeguard",        0.9);
  algoName_     = lslist.sublist("Inexact Newton").get("Subproblem Solver","Spectral Gradient");
  int sp_maxit  = lslist.sublist("Inexact Newton").get("Subproblem Iteration Limit",      1000);
  sp_tol1_      = lslist.sublist("Inexact Newton").get("Subproblem Absolute Tolerance",   1e-4);
  sp_tol2_      = lslist.sublist("Inexact Newton").get("Subproblem Relative Tolerance",   1e-2);
  sp_exp_       = lslist.sublist("Inexact Newton").get("Subproblem Tolerance Exponent",    1.0);
  Real opt_tol  = lslist.sublist("Status Test").get("Gradient Tolerance",      1e-8);
  sp_tol_min_   = static_cast<Real>(1e-4)*opt_tol;
  verbosity_    = list.sublist("General").get("Output Level",                     0);
  writeHeader_  = verbosity_ > 2;
 
  list_.sublist("Status Test").set("Iteration Limit",    sp_maxit);
  list_.sublist("General").set("Output Level", verbosity_>0 ? verbosity_-1 : 0);
}
 
 
template<typename Real>
void InexactNewtonAlgorithm<Real>::initialize(Vector<Real>          &x,
                                              const Vector<Real>    &g,
                                              Objective<Real>       &sobj,
                                              Objective<Real>       &nobj,
                                              Vector<Real>          &dg,
                                              Vector<Real>          &px,
                                              std::ostream &outStream) {
  const Real one(1);
  Real tol(std::sqrt(ROL_EPSILON<Real>()));
  // Initialize data
  TypeP::Algorithm<Real>::initialize(x,g);
  // Update approximate gradient and approximate objective function.
  Real ftol = std::sqrt(ROL_EPSILON<Real>());
  if (initProx_) {
    state_->iterateVec->set(x);
    nobj.prox(x,*state_->iterateVec,one,tol); state_->nprox++;
  }
  sobj.update(x,UpdateType::Initial,state_->iter);    
  nobj.update(x,UpdateType::Initial,state_->iter);    
  state_->svalue = sobj.value(x,ftol); state_->nsval++;
  state_->nvalue = nobj.value(x,ftol); state_->nnval++;
  state_->value  = state_->svalue + state_->nvalue;
  sobj.gradient(*state_->gradientVec,x,ftol); state_->ngrad++;
  dg.set(state_->gradientVec->dual());
  pgstep(*state_->iterateVec,px,nobj,x,dg,t0_,tol);
  state_->gnorm = px.norm() / t0_;
  state_->snorm = ROL_INF<Real>();
  nhess_ = 0;
}
 
template<typename Real>
void InexactNewtonAlgorithm<Real>::run( Vector<Real>          &x,
                                        const Vector<Real>    &g, 
                                        Objective<Real>       &sobj,
                                        Objective<Real>       &nobj,
                                        std::ostream          &outStream ) {
  const Real half(0.5), one(1), eps(ROL_EPSILON<Real>());
  // Initialize trust-region data
  Ptr<Vector<Real>> s = x.clone(), gp = x.clone(), xs = x.clone(), px = x.clone();
  initialize(x,g,sobj,nobj,*gp,*px,outStream);
  Real strial(0), ntrial(0), ftrial(0), gs(0), Qk(0), rhoTmp(0);
  Real tol(std::sqrt(ROL_EPSILON<Real>())), gtol(1);
 
  Ptr<TypeP::Algorithm<Real>> algo;
  Ptr<NewtonObj>              qobj = makePtr<NewtonObj>(makePtrFromRef(sobj),x,g);
 
  // Output
  if (verbosity_ > 0) writeOutput(outStream,true);
 
  // Compute steepest descent step
  xs->set(*state_->iterateVec);
  state_->iterateVec->set(x);
  while (status_->check(*state_)) {
    qobj->setData(x,*state_->gradientVec);
    // Compute step
    gtol = std::max(sp_tol_min_,std::min(sp_tol1_,sp_tol2_*std::pow(state_->gnorm,sp_exp_)));
    list_.sublist("Status Test").set("Gradient Tolerance",gtol);
    if (algoName_ == "Line Search")       algo = makePtr<TypeP::ProxGradientAlgorithm<Real>>(list_);
    else if (algoName_ == "iPiano")       algo = makePtr<TypeP::iPianoAlgorithm<Real>>(list_);
    else if (algoName_ == "Trust Region") algo = makePtr<TypeP::TrustRegionAlgorithm<Real>>(list_);
    else                                  algo = makePtr<TypeP::SpectralGradientAlgorithm<Real>>(list_);
    algo->run(*xs,*qobj,nobj,outStream);
    s->set(*xs); s->axpy(-one,x);
    spgIter_ = algo->getState()->iter;
    nhess_  += qobj->numHessVec();
    state_->nprox += staticPtrCast<const TypeP::AlgorithmState<Real>>(algo->getState())->nprox;
 
    // Perform backtracking line search 
    state_->searchSize = one;
    x.set(*state_->iterateVec);
    x.axpy(state_->searchSize,*s);
    sobj.update(x,UpdateType::Trial);
    nobj.update(x,UpdateType::Trial);
    strial = sobj.value(x,tol);
    ntrial = nobj.value(x,tol);
    ftrial = strial + ntrial;
    ls_nfval_ = 1;
    gs = state_->gradientVec->apply(*s);
    Qk = gs + ntrial - state_->nvalue;
    if (verbosity_ > 1) {
      outStream << "  In TypeP::InexactNewtonAlgorithm: Line Search"  << std::endl;
      outStream << "    Step size:                        " << state_->searchSize   << std::endl;
      outStream << "    Trial objective value:            " << ftrial               << std::endl;
      outStream << "    Computed reduction:               " << state_->value-ftrial << std::endl;
      outStream << "    Dot product of gradient and step: " << gs                   << std::endl;
      outStream << "    Sufficient decrease bound:        " << -Qk*c1_              << std::endl;
      outStream << "    Number of function evaluations:   " << ls_nfval_            << std::endl;
    }
    if (Qk > -eps) {
      s->set(*px);
      x.set(*state_->iterateVec);
      x.axpy(state_->searchSize,*s);
      sobj.update(x,UpdateType::Trial);
      nobj.update(x,UpdateType::Trial);
      strial = sobj.value(x,tol);
      ntrial = nobj.value(x,tol);
      ftrial = strial + ntrial;
      ls_nfval_++;
      gs = state_->gradientVec->apply(*s);
      Qk = gs + ntrial - state_->nvalue;
    }
    while ( ftrial > state_->value + c1_*Qk && ls_nfval_ < maxit_ ) {
      rhoTmp = -half * Qk / (strial-state_->svalue-state_->searchSize*gs);
      state_->searchSize = ((sigma1_ <= rhoTmp && rhoTmp <= sigma2_) ? rhoTmp : rhodec_) * state_->searchSize;
      x.set(*state_->iterateVec);
      x.axpy(state_->searchSize,*s);
      sobj.update(x,UpdateType::Trial);
      nobj.update(x,UpdateType::Trial);
      strial = sobj.value(x,tol);
      ntrial = nobj.value(x,tol);
      ftrial = strial + ntrial;
      Qk     = state_->searchSize * gs + ntrial - state_->nvalue;
      ls_nfval_++;
      if (verbosity_ > 1) {
        outStream << std::endl;
        outStream << "    Step size:                        " << state_->searchSize   << std::endl;
        outStream << "    Trial objective value:            " << ftrial               << std::endl;
        outStream << "    Computed reduction:               " << state_->value-ftrial << std::endl;
        outStream << "    Dot product of gradient and step: " << gs                   << std::endl;
        outStream << "    Sufficient decrease bound:        " << -Qk*c1_              << std::endl;
        outStream << "    Number of function evaluations:   " << ls_nfval_            << std::endl;
      }
    }
    state_->nsval += ls_nfval_;
    state_->nnval += ls_nfval_;
 
    // Compute norm of step
    state_->stepVec->set(*s);
    state_->stepVec->scale(state_->searchSize);
    state_->snorm = state_->stepVec->norm();
 
    // Update iterate
    state_->iterateVec->set(x);
 
    // Compute new value and gradient
    state_->iter++;
    state_->value  = ftrial;
    state_->svalue = strial;
    state_->nvalue = ntrial;
    sobj.update(x,UpdateType::Accept,state_->iter);
    nobj.update(x,UpdateType::Accept,state_->iter);
    sobj.gradient(*state_->gradientVec,x,tol); state_->ngrad++;
    gp->set(state_->gradientVec->dual());
 
    // Compute projected gradient norm
    pgstep(*xs,*px,nobj,x,*gp,t0_,tol);
    state_->gnorm = s->norm() / t0_;
 
    // Update Output
    if (verbosity_ > 0) writeOutput(outStream,writeHeader_);
  }
  if (verbosity_ > 0) TypeP::Algorithm<Real>::writeExitStatus(outStream);
}
 
template<typename Real>
void InexactNewtonAlgorithm<Real>::writeHeader( std::ostream& os ) const {
  std::stringstream hist;
  if (verbosity_ > 1) {
    hist << std::string(114,'-') << std::endl;
    hist << "Line-Search Inexact Proximal Newton";
    hist << " status output definitions" << std::endl << std::endl;
    hist << "  iter     - Number of iterates (steps taken)" << std::endl;
    hist << "  value    - Objective function value" << std::endl;
    hist << "  gnorm    - Norm of the gradient" << std::endl;
    hist << "  snorm    - Norm of the step (update to optimization vector)" << std::endl;
    hist << "  alpha    - Line search step length" << std::endl;
    hist << "  #sval    - Cumulative number of times the smooth objective function was evaluated" << std::endl;
    hist << "  #nval    - Cumulative number of times the nonsmooth objective function was evaluated" << std::endl;
    hist << "  #grad    - Cumulative number of times the gradient was computed" << std::endl;
    hist << "  #hess    - Cumulative number of times the Hessian was applied" << std::endl;
    hist << "  #prox    - Cumulative number of times the projection was computed" << std::endl;
    hist << "  ls_#fval - Number of the times the objective function was evaluated during the line search" << std::endl;
    hist << "  sp_iter  - Number iterations to compute quasi-Newton step" << std::endl;
    hist << std::string(114,'-') << std::endl;
  }
 
  hist << "  ";
  hist << std::setw(6)  << std::left << "iter";
  hist << std::setw(15) << std::left << "value";
  hist << std::setw(15) << std::left << "gnorm";
  hist << std::setw(15) << std::left << "snorm";
  hist << std::setw(15) << std::left << "alpha";
  hist << std::setw(10) << std::left << "#sval";
  hist << std::setw(10) << std::left << "#nval";
  hist << std::setw(10) << std::left << "#grad";
  hist << std::setw(10) << std::left << "#hess";
  hist << std::setw(10) << std::left << "#prox";
  hist << std::setw(10) << std::left << "#ls_fval";
  hist << std::setw(10) << std::left << "sp_iter";
  hist << std::endl;
  os << hist.str();
}
 
template<typename Real>
void InexactNewtonAlgorithm<Real>::writeName( std::ostream& os ) const {
  std::stringstream hist;
  hist << std::endl << "Line-Search Inexact Proximal Newton (Type P)" << std::endl;
  os << hist.str();
}
 
template<typename Real>
void InexactNewtonAlgorithm<Real>::writeOutput( std::ostream& os, bool write_header ) const {
  std::stringstream hist;
  hist << std::scientific << std::setprecision(6);
  if ( state_->iter == 0 ) writeName(os);
  if ( write_header )      writeHeader(os);
  if ( state_->iter == 0 ) {
    hist << "  ";
    hist << std::setw(6)  << std::left << state_->iter;
    hist << std::setw(15) << std::left << state_->value;
    hist << std::setw(15) << std::left << state_->gnorm;
    hist << std::setw(15) << std::left << "---";
    hist << std::setw(15) << std::left << "---";
    hist << std::setw(10) << std::left << state_->nsval;
    hist << std::setw(10) << std::left << state_->nnval;
    hist << std::setw(10) << std::left << state_->ngrad;
    hist << std::setw(10) << std::left << nhess_;
    hist << std::setw(10) << std::left << state_->nprox;
    hist << std::setw(10) << std::left << "---";
    hist << std::setw(10) << std::left << "---";
    hist << std::endl;
  }
  else {
    hist << "  ";
    hist << std::setw(6)  << std::left << state_->iter;
    hist << std::setw(15) << std::left << state_->value;
    hist << std::setw(15) << std::left << state_->gnorm;
    hist << std::setw(15) << std::left << state_->snorm;
    hist << std::setw(15) << std::left << state_->searchSize;
    hist << std::setw(10) << std::left << state_->nsval;
    hist << std::setw(10) << std::left << state_->nnval;
    hist << std::setw(10) << std::left << state_->ngrad;
    hist << std::setw(10) << std::left << nhess_;
    hist << std::setw(10) << std::left << state_->nprox;
    hist << std::setw(10) << std::left << ls_nfval_;
    hist << std::setw(10) << std::left << spgIter_;
    hist << std::endl;
  }
  os << hist.str();
}
 
} // namespace TypeP
} // namespace ROL
 
#endif