ROL_DoubleDogLeg_U.hpp

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#ifndef ROL_DOUBLEDOGLEG_U_H
#define ROL_DOUBLEDOGLEG_U_H

#include "ROL_TrustRegion_U.hpp"
#include "ROL_Types.hpp"

namespace ROL { 

template<class Real>
class DoubleDogLeg_U : public TrustRegion_U<Real> {
private:

  Ptr<Vector<Real>> primal_, dual_;

public:

  DoubleDogLeg_U() {}

  void initialize(const Vector<Real> &x, const Vector<Real> &g) {
    primal_ = x.clone();
    dual_   = g.clone();
  }

  void solve( Vector<Real>             &s,
              Real                     &snorm,
              Real                     &pRed,
              int                      &iflag,
              int                      &iter,
              const Real                del,
              TrustRegionModel_U<Real> &model ) {
    Real tol = std::sqrt(ROL_EPSILON<Real>());
    const Real one(1), zero(0), half(0.5), p2(0.2), p8(0.8), two(2);
    // Set s to be the (projected) gradient
    s.set(model.getGradient()->dual());
    // Compute (quasi-)Newton step
    model.invHessVec(*primal_,*model.getGradient(),s,tol);
    Real sNnorm  = primal_->norm();
    Real tmp     = -primal_->dot(s);
    Real gsN     = std::abs(tmp);
    // Check if (quasi-)Newton step is feasible
    if ( tmp >= zero ) {
      // Use the Cauchy point
      model.hessVec(*dual_,s,s,tol);
      //Real gBg   = dual_->dot(s.dual());
      Real gBg   = dual_->apply(s);
      Real gnorm = s.dual().norm();
      Real gg    = gnorm*gnorm;
      Real alpha = del/gnorm;
      if ( gBg > ROL_EPSILON<Real>() ) {
        alpha = std::min(gg/gBg, del/gnorm);
      }
      s.scale(-alpha);
      snorm = alpha*gnorm;
      iflag = 2;
      pRed  = alpha*(gg - half*alpha*gBg);
    }
    else {
      // Approximately solve trust region subproblem using double dogleg curve
      if (sNnorm <= del) { // Use the (quasi-)Newton step
        s.set(*primal_); 
        s.scale(-one);
        snorm = sNnorm;
        pRed  = half*gsN;
        iflag = 0;
      }
      else { // The (quasi-)Newton step is outside of trust region
        model.hessVec(*dual_,s,s,tol);
        Real alpha  = zero;
        Real beta   = zero;
        Real gnorm  = s.norm();
        Real gnorm2 = gnorm*gnorm;
        //Real gBg    = dual_->dot(s.dual());
        Real gBg    = dual_->apply(s);
        Real gamma1 = gnorm/gBg;
        Real gamma2 = gnorm/gsN;
        Real eta    = p8*gamma1*gamma2 + p2;
        if (eta*sNnorm <= del || gBg <= zero) { // Dogleg Point is inside trust region
          alpha = del/sNnorm;
          beta  = zero;
          s.set(*primal_);
          s.scale(-alpha);
          snorm = del;
          iflag = 1;
        }
        else {
          if (gnorm2*gamma1 >= del) { // Cauchy Point is outside trust region
            alpha = zero;
            beta  = -del/gnorm;
            s.scale(beta); 
            snorm = del;
            iflag = 2;
          }
          else {              // Find convex combination of Cauchy and Dogleg point
            s.scale(-gamma1*gnorm);
            primal_->scale(eta);
            primal_->plus(s);
            primal_->scale(-one);
            Real wNorm = primal_->dot(*primal_);
            Real sigma = del*del-std::pow(gamma1*gnorm,two);
            Real phi   = s.dot(*primal_);
            Real theta = (-phi + std::sqrt(phi*phi+wNorm*sigma))/wNorm;
            s.axpy(theta,*primal_);
            snorm = del;
            alpha = theta*eta;
            beta  = (one-theta)*(-gamma1*gnorm);
            iflag = 3;
          }
        }
        pRed = -(alpha*(half*alpha-one)*gsN + half*beta*beta*gBg + beta*(one-alpha)*gnorm2);
      }
    }
  }
};

} // namespace ROL

#endif