ROL_TrustRegionStep.hpp

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#ifndef ROL_TRUSTREGIONSTEP_H
#define ROL_TRUSTREGIONSTEP_H

#include "ROL_Step.hpp"
#include "ROL_Types.hpp"
#include "ROL_Secant.hpp"
#include "ROL_TrustRegion.hpp"
#include <sstream>
#include <iomanip>

namespace ROL {

template <class Real>
class TrustRegionStep : public Step<Real> {
private:

  // ADDITIONAL VECTOR STORAGE
  Ptr<Vector<Real>> xnew_; 
  Ptr<Vector<Real>> xold_; 
  Ptr<Vector<Real>> gp_;   

  // TRUST REGION INFORMATION
  Ptr<TrustRegion<Real>>                trustRegion_; 
  Ptr<TrustRegionModel<Real>>           model_;       
  ETrustRegion                          etr_;         
  ETrustRegionModel                     TRmodel_;     
  Real                                  delMax_;      
  ETrustRegionFlag                      TRflag_;      
  int                                   SPflag_;      
  int                                   SPiter_;      
  bool                                  bndActive_;   

  // SECANT INFORMATION
  Ptr<Secant<Real>> secant_;                     
  ESecant                     esec_;             
  bool                        useSecantHessVec_; 
  bool                        useSecantPrecond_; 

  // BOUND CONSTRAINED PARAMETERS
  Real scaleEps_;         
  bool useProjectedGrad_; 

  // POST SMOOTHING PARAMETERS
  Real alpha_init_; 
  int  max_fval_;   
  Real mu_;         
  Real beta_;       

  // COLEMAN-LI PARAMETERS
  Real stepBackMax_;
  Real stepBackScale_;
  bool singleReflect_;

  // INEXACT COMPUTATION PARAMETERS
  std::vector<bool> useInexact_; 
  Real              scale0_;     
  Real              scale1_;     

  // VERBOSITY SETTING
  int verbosity_; 

  void parseParameterList(ROL::ParameterList &parlist) {
    ROL::Ptr<StepState<Real>> step_state = Step<Real>::getState();
    // Trust-Region Parameters
    ROL::ParameterList &slist = parlist.sublist("Step");
    ROL::ParameterList &list  = slist.sublist("Trust Region");
    step_state->searchSize = list.get("Initial Radius", static_cast<Real>(-1));
    delMax_                = list.get("Maximum Radius", static_cast<Real>(1.e8));
    // Inexactness Information
    ROL::ParameterList &glist = parlist.sublist("General");
    useInexact_.clear();
    useInexact_.push_back(glist.get("Inexact Objective Function",     false));
    useInexact_.push_back(glist.get("Inexact Gradient",               false));
    useInexact_.push_back(glist.get("Inexact Hessian-Times-A-Vector", false));
    // Trust-Region Inexactness Parameters
    ROL::ParameterList &ilist = list.sublist("Inexact").sublist("Gradient");
    scale0_ = ilist.get("Tolerance Scaling",  static_cast<Real>(0.1));
    scale1_ = ilist.get("Relative Tolerance", static_cast<Real>(2)); 
    // Initialize Trust Region Subproblem Solver Object
    etr_              = StringToETrustRegion(list.get("Subproblem Solver", "Dogleg"));  
    TRmodel_          = StringToETrustRegionModel(list.get("Subproblem Model", "Kelley-Sachs"));
    useProjectedGrad_ = glist.get("Projected Gradient Criticality Measure", false);
    trustRegion_      = TrustRegionFactory<Real>(parlist);
    // Scale for epsilon active sets
    scaleEps_  = glist.get("Scale for Epsilon Active Sets", static_cast<Real>(1));
    verbosity_ = glist.get("Print Verbosity",               0);
    // Post-smoothing parameters
    max_fval_    = list.sublist("Post-Smoothing").get("Function Evaluation Limit", 20);
    alpha_init_  = list.sublist("Post-Smoothing").get("Initial Step Size", static_cast<Real>(1));
    mu_          = list.sublist("Post-Smoothing").get("Tolerance",         static_cast<Real>(0.9999));
    beta_        = list.sublist("Post-Smoothing").get("Rate",              static_cast<Real>(0.01));
    // Coleman-Li parameters
    stepBackMax_   = list.sublist("Coleman-Li").get("Maximum Step Back",  static_cast<Real>(0.9999));
    stepBackScale_ = list.sublist("Coleman-Li").get("Maximum Step Scale", static_cast<Real>(1));
    singleReflect_ = list.sublist("Coleman-Li").get("Single Reflection",  true);
  }

  void updateGradient( Vector<Real> &x, Objective<Real> &obj, BoundConstraint<Real> &bnd, 
                       AlgorithmState<Real> &algo_state ) {
    Ptr<StepState<Real>> state = Step<Real>::getState();
    if ( useInexact_[1] ) {
      const Real one(1);
      //const Real oem2(1.e-2), oe4(1.e4);
      //Real c = scale0_*std::max(oem2,std::min(one,oe4*algo_state.gnorm));
      //Real gtol1  = c*std::min(algo_state.gnorm,state->searchSize);
      //Real gtol0  = scale1_*gtol1 + one;
      //while ( gtol0 > gtol1*scale1_ ) {
      //  obj.gradient(*(state->gradientVec),x,gtol1);
      //  algo_state.gnorm = computeCriticalityMeasure(*(state->gradientVec),x,bnd);
      //  gtol0 = gtol1;
      //  c = scale0_*std::max(oem2,std::min(one,oe4*algo_state.gnorm));
      //  gtol1 = c*std::min(algo_state.gnorm,state->searchSize);
      //}
      //algo_state.ngrad++;
      Real gtol1  = scale0_*state->searchSize;
      //Real gtol1  = scale0_*std::min(algo_state.gnorm,state->searchSize);
      Real gtol0  = gtol1 + one;
      while ( gtol0 > gtol1 ) {
        obj.gradient(*(state->gradientVec),x,gtol1);
        algo_state.gnorm = computeCriticalityMeasure(*(state->gradientVec),x,bnd);
        gtol0 = gtol1;
        gtol1 = scale0_*std::min(algo_state.gnorm,state->searchSize);
      }
      algo_state.ngrad++;
    }
    else {
      Real gtol = std::sqrt(ROL_EPSILON<Real>());
      obj.gradient(*(state->gradientVec),x,gtol);
      algo_state.ngrad++;
      algo_state.gnorm = computeCriticalityMeasure(*(state->gradientVec),x,bnd);
    }
  }

  Real computeCriticalityMeasure( const Vector<Real> &g, const Vector<Real> &x, BoundConstraint<Real> &bnd ) {
    if ( bnd.isActivated() ) {
      if ( useProjectedGrad_ ) {
        gp_->set(g);
        bnd.computeProjectedGradient( *gp_, x );
        return gp_->norm();
      }
      else {
        Real one(1);
        xnew_->set(x);
        xnew_->axpy(-one,g.dual());
        bnd.project(*xnew_);
        xnew_->axpy(-one,x);
        return xnew_->norm();
      }
    }
    else {
      return g.norm();
    }
  }

public:

  using Step<Real>::initialize;
  using Step<Real>::compute;
  using Step<Real>::update;

  virtual ~TrustRegionStep() {}

  TrustRegionStep( ROL::ParameterList & parlist )
    : Step<Real>(),
      xnew_(nullPtr), xold_(nullPtr), gp_(nullPtr),
      trustRegion_(nullPtr), model_(nullPtr),
      etr_(TRUSTREGION_DOGLEG), TRmodel_(TRUSTREGION_MODEL_KELLEYSACHS),
      delMax_(1e8), TRflag_(TRUSTREGION_FLAG_SUCCESS),
      SPflag_(0), SPiter_(0), bndActive_(false),
      secant_(nullPtr), esec_(SECANT_LBFGS),
      useSecantHessVec_(false), useSecantPrecond_(false),
      scaleEps_(1), useProjectedGrad_(false),
      alpha_init_(1), max_fval_(20), mu_(0.9999), beta_(0.01),
      stepBackMax_(0.9999), stepBackScale_(1), singleReflect_(true),
      scale0_(1), scale1_(1),
      verbosity_(0) {
    // Parse input parameterlist
    parseParameterList(parlist);
    // Create secant object
    ROL::ParameterList &glist = parlist.sublist("General");
    esec_             = StringToESecant(glist.sublist("Secant").get("Type","Limited-Memory BFGS"));
    useSecantPrecond_ = glist.sublist("Secant").get("Use as Preconditioner", false);
    useSecantHessVec_ = glist.sublist("Secant").get("Use as Hessian",        false);
    secant_           = SecantFactory<Real>(parlist);
  }

  TrustRegionStep( ROL::Ptr<Secant<Real> > &secant, ROL::ParameterList &parlist ) 
    : Step<Real>(),
      xnew_(nullPtr), xold_(nullPtr), gp_(nullPtr),
      trustRegion_(nullPtr), model_(nullPtr),
      etr_(TRUSTREGION_DOGLEG), TRmodel_(TRUSTREGION_MODEL_KELLEYSACHS),
      delMax_(1e8), TRflag_(TRUSTREGION_FLAG_SUCCESS),
      SPflag_(0), SPiter_(0), bndActive_(false),
      secant_(nullPtr), esec_(SECANT_LBFGS),
      useSecantHessVec_(false), useSecantPrecond_(false),
      scaleEps_(1), useProjectedGrad_(false),
      alpha_init_(1), max_fval_(20), mu_(0.9999), beta_(0.01),
      stepBackMax_(0.9999), stepBackScale_(1), singleReflect_(true),
      scale0_(1), scale1_(1),
      verbosity_(0) {
    // Parse input parameterlist
    parseParameterList(parlist);
    // Create secant object
    ROL::ParameterList &glist = parlist.sublist("General");
    useSecantPrecond_ = glist.sublist("Secant").get("Use as Preconditioner", false);
    useSecantHessVec_ = glist.sublist("Secant").get("Use as Hessian",        false);
    if ( ROL::is_nullPtr(secant_) ) {
      ROL::ParameterList Slist;
      Slist.sublist("General").sublist("Secant").set("Type","Limited-Memory BFGS");
      Slist.sublist("General").sublist("Secant").set("Maximum Storage",10);
      secant_ = SecantFactory<Real>(Slist);
    }
  }

  void initialize( Vector<Real> &x, const Vector<Real> &s, const Vector<Real> &g, 
                   Objective<Real> &obj, BoundConstraint<Real> &bnd, 
                   AlgorithmState<Real> &algo_state ) {
    if (!isValidTrustRegionSubproblem(etr_,TRmodel_,bnd.isActivated())) {
      throw Exception::NotImplemented(">>> ROL::TrustRegionStep : Invalid Trust Region Solver and Model pair!");
    }
    Real p1(0.1), oe10(1.e10), zero(0), one(1), half(0.5), three(3), two(2), six(6);
    Ptr<StepState<Real>> step_state = Step<Real>::getState();
    bndActive_ = bnd.isActivated();

    trustRegion_->initialize(x,s,g);

    Real htol = std::sqrt(ROL_EPSILON<Real>());
    Real ftol = p1*ROL_OVERFLOW<Real>(); 

    step_state->descentVec  = s.clone();
    step_state->gradientVec = g.clone();

    if ( bnd.isActivated() ) {
      // Make initial guess feasible
      if ( TRmodel_ == TRUSTREGION_MODEL_COLEMANLI ) {
        bnd.projectInterior(x);
      }
      else {
        bnd.project(x);
      }
      xnew_ = x.clone();
      xold_ = x.clone();
    }
    gp_ = g.clone();

    // Update approximate gradient and approximate objective function.
    obj.update(x,true,algo_state.iter);    
    algo_state.snorm = oe10;
    algo_state.value = obj.value(x,ftol); 
    algo_state.nfval++;
    algo_state.gnorm = ROL_INF<Real>();
    updateGradient(x,obj,bnd,algo_state);

    // Try to apply inverse Hessian
    if ( !useSecantHessVec_ &&
        (etr_ == TRUSTREGION_DOGLEG || etr_ == TRUSTREGION_DOUBLEDOGLEG) ) {
      try {
        Ptr<Vector<Real>> v  = g.clone();
        Ptr<Vector<Real>> hv = x.clone();
        obj.invHessVec(*hv,*v,x,htol);
      }
      catch (std::exception &e) {
        useSecantHessVec_ = true;
      }
    }

    // Evaluate Objective Function at Cauchy Point
    bool autoRad = false;
    if ( step_state->searchSize <= zero ) {
      autoRad = true;
      Ptr<Vector<Real>> Bg = g.clone();
      if ( useSecantHessVec_ ) {
        secant_->applyB(*Bg,(step_state->gradientVec)->dual());
      }
      else {
        obj.hessVec(*Bg,(step_state->gradientVec)->dual(),x,htol);
      }
      Real gBg = Bg->dot(*(step_state->gradientVec));
      Real alpha = one;
      if ( gBg > ROL_EPSILON<Real>() ) {
        alpha = algo_state.gnorm*algo_state.gnorm/gBg;
      }
      // Evaluate the objective function at the Cauchy point
      Ptr<Vector<Real>> cp = s.clone();
      cp->set((step_state->gradientVec)->dual()); 
      cp->scale(-alpha);
      Ptr<Vector<Real>> xcp = x.clone();
      xcp->set(x);
      xcp->plus(*cp);
      if ( bnd.isActivated() ) {
        bnd.project(*xcp);
      }
      obj.update(*xcp);
      Real fnew = obj.value(*xcp,ftol); // MUST DO SOMETHING HERE WITH FTOL
      algo_state.nfval++;
      // Perform cubic interpolation to determine initial trust region radius
      Real gs = cp->dot((step_state->gradientVec)->dual());
      Real a  = fnew - algo_state.value - gs - half*alpha*alpha*gBg;
      if ( std::abs(a) < ROL_EPSILON<Real>() ) { 
        // a = 0 implies the objective is quadratic in the negative gradient direction
        step_state->searchSize = std::min(alpha*algo_state.gnorm,delMax_);
      }
      else {
        Real b  = half*alpha*alpha*gBg;
        Real c  = gs;
        if ( b*b-three*a*c > ROL_EPSILON<Real>() ) {
          // There is at least one critical point
          Real t1 = (-b-std::sqrt(b*b-three*a*c))/(three*a);
          Real t2 = (-b+std::sqrt(b*b-three*a*c))/(three*a);
          if ( six*a*t1 + two*b > zero ) {
            // t1 is the minimizer
            step_state->searchSize = std::min(t1*alpha*algo_state.gnorm,delMax_);
          }
          else {
            // t2 is the minimizer
            step_state->searchSize = std::min(t2*alpha*algo_state.gnorm,delMax_);
          }
        }
        else {
          step_state->searchSize = std::min(alpha*algo_state.gnorm,delMax_);
        }
      }
      if (step_state->searchSize <= ROL_EPSILON<Real>()*algo_state.gnorm && autoRad) {
        step_state->searchSize = one;
      }
      obj.update(x,true,algo_state.iter);
    }
    // Build trust-region model
    if (bnd.isActivated()) { 
      if ( TRmodel_ == TRUSTREGION_MODEL_KELLEYSACHS ) {
        model_ = makePtr<KelleySachsModel<Real>>(obj,
                                                 bnd,
                                                 x,
                                                 *(step_state->gradientVec),
                                                 secant_,
                                                 useSecantPrecond_,
                                                 useSecantHessVec_);
      }
      else if ( TRmodel_ == TRUSTREGION_MODEL_COLEMANLI ) {
        model_ = makePtr<ColemanLiModel<Real>>(obj,
                                               bnd,
                                               x,
                                               *(step_state->gradientVec),
                                               stepBackMax_,
                                               stepBackScale_,
                                               singleReflect_,
                                               secant_,
                                               useSecantPrecond_,
                                               useSecantHessVec_);
      } 
      else if ( TRmodel_ == TRUSTREGION_MODEL_LINMORE ) {
        model_ = makePtr<LinMoreModel<Real>>(obj,
                                             bnd,
                                             x,
                                             *(step_state->gradientVec),
                                             secant_,
                                             useSecantPrecond_,
                                             useSecantHessVec_);
      }
      else {
        ROL_TEST_FOR_EXCEPTION( true, std::invalid_argument,
          ">>> ERROR (TrustRegionStep): Invalid trust-region model!");
      }
    }
    else {
      model_ = makePtr<TrustRegionModel<Real>>(obj,
                                               bnd,
                                               x,
                                               *(step_state->gradientVec),
                                               secant_,
                                               useSecantPrecond_,
                                               useSecantHessVec_);
    }
  }

  void compute( Vector<Real> &s, const Vector<Real> &x,
                Objective<Real> &obj, BoundConstraint<Real> &bnd, 
                AlgorithmState<Real> &algo_state ) {
    // Get step state
    Ptr<StepState<Real>> step_state = Step<Real>::getState();
    // Build trust-region model
    model_->update(obj,bnd,x,*step_state->gradientVec,secant_);
    if (bnd.isActivated()) {
      if ( TRmodel_ == TRUSTREGION_MODEL_KELLEYSACHS ) {
//      Real eps = scaleEps_*algo_state.gnorm;
        Real eps = scaleEps_ * std::min(std::pow(algo_state.gnorm,static_cast<Real>(0.75)),
                                        static_cast<Real>(0.001));
        dynamicPtrCast<KelleySachsModel<Real>>(model_)->setEpsilon(eps);
      }
      else if ( TRmodel_ == TRUSTREGION_MODEL_COLEMANLI ) {
        dynamicPtrCast<ColemanLiModel<Real>>(model_)->setRadius(step_state->searchSize);
      }
    }
    // Minimize trust-region model over trust-region constraint
    SPflag_ = 0; SPiter_ = 0;
    trustRegion_->run(s,algo_state.snorm,SPflag_,SPiter_,step_state->searchSize,*model_);
  }

  void update( Vector<Real>          &x,
               const Vector<Real>    &s,
               Objective<Real>       &obj,
               BoundConstraint<Real> &bnd,
               AlgorithmState<Real>  &algo_state ) {
    // Get step state
    Ptr<StepState<Real>> state = Step<Real>::getState();
    // Store previous step for constraint computations
    if ( bnd.isActivated() ) {
      xold_->set(x);
    }
    // Update trust-region information;
    // Performs a hard update on the objective function
    TRflag_ = TRUSTREGION_FLAG_SUCCESS;
    state->nfval = 0;
    state->ngrad = 0;
    Real fold = algo_state.value;
    Real fnew(0);
    algo_state.iter++;
    trustRegion_->update(x,fnew,state->searchSize,state->nfval,state->ngrad,TRflag_,
                         s,algo_state.snorm,fold,*(state->gradientVec),algo_state.iter,
                         obj,bnd,*model_);
    algo_state.nfval += state->nfval;
    algo_state.ngrad += state->ngrad;
    state->flag   = static_cast<int>(TRflag_);
    state->SPiter = SPiter_;
    state->SPflag = SPflag_;
    // If step is accepted ...
    // Compute new gradient and update secant storage
    if ( TRflag_ == TRUSTREGION_FLAG_SUCCESS || 
         TRflag_ == TRUSTREGION_FLAG_POSPREDNEG ) {  
      // Store previous gradient for secant update
      if ( useSecantHessVec_ || useSecantPrecond_ ) {
        gp_->set(*(state->gradientVec));
      }
      // Update objective function and approximate model
      updateGradient(x,obj,bnd,algo_state);
      // Update secant information
      if ( useSecantHessVec_ || useSecantPrecond_ ) {
        if ( bnd.isActivated() ) { // Compute new constrained step
          xnew_->set(x);
          xnew_->axpy(-static_cast<Real>(1),*xold_);
          secant_->updateStorage(x,*(state->gradientVec),*gp_,*xnew_,algo_state.snorm,algo_state.iter+1);
        }
        else {
          secant_->updateStorage(x,*(state->gradientVec),*gp_,s,algo_state.snorm,algo_state.iter+1);
        }
      }
      // Update algorithm state
      (algo_state.iterateVec)->set(x);
    }
    else {
      if ( useInexact_[1] ) {
        // Update objective function and approximate model
        updateGradient(x,obj,bnd,algo_state);
      }
    }
    // Update algorithm state
    algo_state.value = fnew;
  }

  std::string printHeader( void ) const  {
    std::stringstream hist;

    if(verbosity_>0) {
      hist << std::string(114,'-') << "\n"; 

      hist << "Trust-Region status output definitions\n\n";
       
      hist << "  iter    - Number of iterates (steps taken) \n";        
      hist << "  value   - Objective function value \n";        
      hist << "  gnorm   - Norm of the gradient\n";
      hist << "  snorm   - Norm of the step (update to optimization vector)\n";  
      hist << "  delta   - Trust-Region radius\n";
      hist << "  #fval   - Number of times the objective function was evaluated\n";
      hist << "  #grad   - Number of times the gradient was computed\n";
      
        
      
      hist << "\n";
      hist << "  tr_flag - Trust-Region flag" << "\n";
      for( int flag = TRUSTREGION_FLAG_SUCCESS; flag != TRUSTREGION_FLAG_UNDEFINED; ++flag ) {
        hist << "    " << NumberToString(flag) << " - " 
             << ETrustRegionFlagToString(static_cast<ETrustRegionFlag>(flag)) << "\n";
          
      } 

      if( etr_ == TRUSTREGION_TRUNCATEDCG ) {
        hist << "\n";
        hist << "  iterCG - Number of Truncated CG iterations\n\n";
        hist << "  flagGC - Trust-Region Truncated CG flag" << "\n";
        for( int flag = CG_FLAG_SUCCESS; flag != CG_FLAG_UNDEFINED; ++flag ) {
          hist << "    " << NumberToString(flag) << " - "
               << ECGFlagToString(static_cast<ECGFlag>(flag)) << "\n"; 
        }            
      }

      hist << std::string(114,'-') << "\n"; 
    }

    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 << "delta";
    hist << std::setw(10) << std::left << "#fval";
    hist << std::setw(10) << std::left << "#grad";
    hist << std::setw(10) << std::left << "tr_flag";
    if ( etr_ == TRUSTREGION_TRUNCATEDCG || etr_ == TRUSTREGION_LINMORE ) {
      hist << std::setw(10) << std::left << "iterCG";
      hist << std::setw(10) << std::left << "flagCG";
    }
    hist << "\n";
    return hist.str();
  }

  std::string printName( void ) const {
    std::stringstream hist;
    hist << "\n" << ETrustRegionToString(etr_) << " Trust-Region Solver";
    if ( useSecantPrecond_ || useSecantHessVec_ ) {
      if ( useSecantPrecond_ && !useSecantHessVec_ ) {
        hist << " with " << ESecantToString(esec_) << " Preconditioning\n";
      }
      else if ( !useSecantPrecond_ && useSecantHessVec_ ) {
        hist << " with " << ESecantToString(esec_) << " Hessian Approximation\n";
      }
      else {
        hist << " with " << ESecantToString(esec_) << " Preconditioning and Hessian Approximation\n";
      }
    }
    else {
      hist << "\n";
    }
    if ( bndActive_ ) {
      hist << "Trust-Region Model: " << ETrustRegionModelToString(TRmodel_) << "\n";
    }
    return hist.str();
  }

  std::string print( AlgorithmState<Real> & algo_state, bool print_header = false ) const  {
    const Ptr<const StepState<Real>>& step_state = Step<Real>::getStepState();

    std::stringstream hist;
    hist << std::scientific << std::setprecision(6);
    if ( algo_state.iter == 0 ) {
      hist << printName();
    }
    if ( print_header ) {
      hist << printHeader();
    }
    if ( algo_state.iter == 0 ) {
      hist << "  ";
      hist << std::setw(6)  << std::left << algo_state.iter;
      hist << std::setw(15) << std::left << algo_state.value;
      hist << std::setw(15) << std::left << algo_state.gnorm;
      hist << std::setw(15) << std::left << " "; 
      hist << std::setw(15) << std::left << step_state->searchSize; 
      hist << "\n";
    }
    else {
      hist << "  "; 
      hist << std::setw(6)  << std::left << algo_state.iter;  
      hist << std::setw(15) << std::left << algo_state.value; 
      hist << std::setw(15) << std::left << algo_state.gnorm; 
      hist << std::setw(15) << std::left << algo_state.snorm; 
      hist << std::setw(15) << std::left << step_state->searchSize; 
      hist << std::setw(10) << std::left << algo_state.nfval;              
      hist << std::setw(10) << std::left << algo_state.ngrad;              
      hist << std::setw(10) << std::left << TRflag_;              
      if ( etr_ == TRUSTREGION_TRUNCATEDCG || etr_ == TRUSTREGION_LINMORE ) {
        hist << std::setw(10) << std::left << SPiter_;
        hist << std::setw(10) << std::left << SPflag_;
      }
      hist << "\n";
    }
    return hist.str();
  }

}; // class Step

} // namespace ROL

#endif