ROL_Bundle_U_TT_Def.hpp

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// @HEADER
// *****************************************************************************
//               Rapid Optimization Library (ROL) Package
//
// Copyright 2014 NTESS and the ROL contributors.
// SPDX-License-Identifier: BSD-3-Clause
// *****************************************************************************
// @HEADER
 
#ifndef ROL_BUNDLE_U_TT_DEF_H
#define ROL_BUNDLE_U_TT_DEF_H
 
#include "ROL_Types.hpp"
#include <limits.h>
#include <stdint.h>
#include <float.h>
#include <math.h>
#include <algorithm> // TT: std::find
 
#define EXACT 1
#define TABOO_LIST 1
#define FIRST_VIOLATED 0
 
namespace ROL {
 
template<typename Real>
Bundle_U_TT<Real>::Bundle_U_TT(const unsigned maxSize,
            const Real coeff,
            const Real omega,
            const unsigned remSize)
  : Bundle_U<Real>(maxSize,coeff,omega,remSize),
    maxSize_(maxSize), isInitialized_(false) {
  maxind_ = std::numeric_limits<int>::max();
  Id_.reshape(maxSize_,maxSize_);
  for(unsigned i=0; i<maxSize_; ++i) {
    Id_(i,i) = static_cast<Real>(1);
  }
}
 
template<typename Real>
Real Bundle_U_TT<Real>::sgn(const Real x) const {
  const Real zero(0), one(1);
  return ((x < zero) ? -one :
         ((x > zero) ?  one : zero));
}
 
template<typename Real>
unsigned Bundle_U_TT<Real>::solveDual(const Real t, const unsigned maxit, const Real tol) {
  unsigned iter = 0;
  if (Bundle_U<Real>::size() == 1) {
    iter = Bundle_U<Real>::solveDual_dim1(t,maxit,tol);
  }
  else if (Bundle_U<Real>::size() == 2) {
    iter = Bundle_U<Real>::solveDual_dim2(t,maxit,tol);
  }
  else {
    iter = solveDual_arbitrary(t,maxit,tol);
  }
  return iter;
}
 
template<typename Real>
void Bundle_U_TT<Real>::swapRowsL(unsigned ind1, unsigned ind2, bool trans) {
  const Real zero(0), one(1);
  if( ind1 > ind2){
    unsigned tmp = ind1;
    ind2 = ind1;
    ind1 = tmp;
  }
  unsigned dd = ind1;
  for (unsigned n=ind1+1; n<=ind2; ++n){
    LA::Matrix<Real> Id_n(LA::Copy,Id_,currSize_,currSize_);
    Id_n(dd,dd) = zero; Id_n(dd,n) = one;
    Id_n(n,dd)  = one;  Id_n(n,n)  = zero;
    LA::Matrix<Real> prod(currSize_,currSize_);
    if( !trans ) {
      prod.multiply(LA::ETransp::NO_TRANS,LA::ETransp::NO_TRANS,one,Id_n,L_,zero);
    }
    else {
      prod.multiply(LA::ETransp::NO_TRANS,LA::ETransp::NO_TRANS,one,L_,Id_n,zero);
    }
    L_ = prod;
    dd++;
  }
}
 
template<typename Real>
void Bundle_U_TT<Real>::updateK(void) {
  if (currSize_ <= dependent_) { // L is empty
    kappa_ = static_cast<Real>(1);
  }
  else{
    Real tmpdiagMax = ROL_NINF<Real>();
    Real tmpdiagMin = ROL_INF<Real>();
    for (unsigned j=0;j<currSize_-dependent_;j++){
      if( L_(j,j) > tmpdiagMax ){
        tmpdiagMax = L_(j,j);
        LiMax_ = j;
      }
      if( L_(j,j) < tmpdiagMin ){
        tmpdiagMin = L_(j,j);
        LiMin_ = j;
      }
    }
    kappa_ = tmpdiagMax/tmpdiagMin;
  }
}
 
template<typename Real>
void Bundle_U_TT<Real>::addSubgradToBase(unsigned ind, Real delta) {
  // update z1, z2, kappa
  // swap rows if: dependent == 1 and we insert independent row (dependent row is always last)
  //               dependent == 2 and Lj has become independent (Lh still dependent)
  if(dependent_ && (ind == currSize_-1)){
      swapRowsL(currSize_-2,currSize_-1);
      int tmp;
      tmp = base_[currSize_-2];
      base_[currSize_-2] = base_[currSize_-1];
      base_[currSize_-1] = tmp;
      ind--;
  } // end if dependent
 
  L_(ind,ind) = delta;
 
  // update z1 and z2
  unsigned zsize = ind+1;
  z1_.resize(zsize); z2_.resize(zsize);
  z1_[ind] = ( static_cast<Real>(1) - lhz1_ ) / delta;
  z2_[ind] = ( Bundle_U<Real>::alpha(base_[ind]) - lhz2_ ) / delta;
  //z2[zsize-1] = ( Bundle_U<Real>::alpha(entering_) - lhz2_ ) / delta;
 
  // update kappa
  if(delta > L_(LiMax_,LiMax_)){
    LiMax_ = ind;
    kappa_ = delta/L_(LiMin_,LiMin_);
  }
  if(delta < L_(LiMin_,LiMin_)){
    LiMin_ = ind;
    kappa_ = L_(LiMax_,LiMax_)/delta;
  }
}
 
template<typename Real>
void Bundle_U_TT<Real>::deleteSubgradFromBase(unsigned ind, Real tol){
  const Real zero(0), one(1);
  // update L, currSize, base_, z1, z2, dependent, dualVariables_, kappa
  if (ind >= currSize_-dependent_){
    // if dependent > 0, the last one or two rows of L are lin. dependent
    if (ind < currSize_-1){ // eliminate currSize_-2 but keep currSize_-1
      // swap the last row with the second to last
      swapRowsL(ind,currSize_-1);
      base_[ind] = base_[currSize_-1];
#if( ! EXACT )
      lhNorm = ljNorm; // new last row is lh
#endif
    }
 
    dependent_--;
    currSize_--;
    L_.reshape(currSize_,currSize_); // the row to be eliminated is the last row
    base_.resize(currSize_);
 
    // note: z1, z2, kappa need not be updated
    return;
  } // end if dependent item
 
  /* currently L_B is lower trapezoidal
 
           | L_1  0  0   |
     L_B = | l    d  0   |
           | Z    v  L_2 |
 
     Apply Givens rotations to transform it to
 
     | L_1  0  0    |
     | l    d  0    |
     | Z    0  L_2' |
 
     then delete row and column to obtain factorization of L_B' with B' = B/{i}
 
     L_B' = | L_1  0    |
            | Z    L_2' |
 
  */
  for (unsigned j=ind+1; j<currSize_-dependent_; ++j){
    Real ai = L_(j,ind);
    if (std::abs(ai) <= tol*currSize_) { // nothing to do
      continue;
    }
    Real aj = L_(j,j);
    Real d, Gc, Gs;
    // Find Givens
    // Anderson's implementation
    if (std::abs(aj) <= tol*currSize_){ // aj is zero
      Gc = zero;
      d  = std::abs(ai);
      Gs = -sgn(ai); // Gs = -sgn(ai)
    }
    else if ( std::abs(ai) > std::abs(aj) ){
      Real t = aj/ai;
      Real sgnb = sgn(ai);
      Real u = sgnb * std::sqrt(one + t*t );
      Gs = -one/u;
      Gc = -Gs*t;
      d  = ai*u;
    }
    else{
      Real t = ai/aj;
      Real sgna = sgn(aj);
      Real u = sgna * std::sqrt(one + t*t );
      Gc = one/u;
      Gs = -Gc*t;
      d  = aj*u;
    }
 
    // // "Naive" implementation
    // d  = hypot(ai,aj);
    // Gc = aj/d;
    // Gs = -ai/d;
 
    L_(j,j) = d; L_(j,ind) = zero;
    // apply Givens to columns i,j of L
    for (unsigned h=j+1; h<currSize_; ++h){
      Real tmp1 = L_(h,ind);
      Real tmp2 = L_(h,j);
      L_(h,ind) = Gc*tmp1 + Gs*tmp2;
      L_(h,j) = Gc*tmp2 - Gs*tmp1;
    }
    // apply Givens to z1, z2
    Real tmp1 = z1_[ind];
    Real tmp2 = z1_[j];
    Real tmp3 = z2_[ind];
    Real tmp4 = z2_[j];
    z1_[ind] = Gc*tmp1 + Gs*tmp2;
    z1_[j] = Gc*tmp2 - Gs*tmp1;
    z2_[ind] = Gc*tmp3 + Gs*tmp4;
    z2_[j] = Gc*tmp4 - Gs*tmp3;
  }// end loop over j
 
  if(dependent_){
    deltaLh_ = L_(currSize_-dependent_,ind);  // h = currSize_ - dependent
    if( dependent_ > 1 )                 // j = currSize_ - 1, h = currSize_ - 2
      deltaLj_ = L_(currSize_-1,ind);
  }
 
  // shift rows and columns of L by exchanging i-th row with next row and i-th column with next column until the row to be deleted is the last, then deleting last row and column
  swapRowsL(ind,currSize_-1);
  swapRowsL(ind,currSize_-1,true);
  L_.reshape(currSize_-1,currSize_-1);
 
  // delete i-th item from z1 and z2
  // note: z1 and z2 are of size currSize_-dependent
  unsigned zsize = currSize_-dependent_;
  for( unsigned m=ind; m<zsize; m++ ){
    z1_[m] = z1_[m+1];
    z2_[m] = z2_[m+1];
  }
  z1_.resize(zsize-1);
  z2_.resize(zsize-1);
 
  // update base
  base_.erase(base_.begin()+ind);
  currSize_--; // update size
 
  // update kappa
  updateK();
 
  if(dependent_){
    // if some previously dependent item have become independent
    // recompute deltaLh
    Real ghNorm = GiGj(base_[currSize_-dependent_],base_[currSize_-dependent_]);
    Real lhnrm(0); // exact lhNorm
#if( EXACT)
    for (unsigned ii=0; ii<currSize_-dependent_; ++ii){
      lhnrm += L_(currSize_-dependent_,ii)*L_(currSize_-dependent_,ii);
    }
    deltaLh_ = std::abs(ghNorm - lhnrm);
#else
    Real sgn1 = sgn(deltaLh_);
    Real sgn2 = sgn(dletaLj);
    Real signum = sgn1 * sgn2; // sgn( deltaLh ) * sgn ( deltaLj );
    deltaLh_ = std::abs( ghNorm - lhNorm + deltaLh_ * deltaLh_);
#endif
 
    if( std::sqrt(deltaLh_) > tol*kappa_*std::max(static_cast<Real>(1),ghNorm) ){ // originally had deltaLh without sqrt
      unsigned newind = currSize_-dependent_;
      dependent_--;
      // get the last row of L
      lh_.size(newind); // initialize to zeros;
      lhz1_ = zero;
      lhz2_ = zero;
      for (unsigned ii=0; ii<newind; ++ii){
        lh_[ii] = L_(newind,ii);
        lhz1_ += lh_[ii]*z1_[ii];
        lhz2_ += lh_[ii]*z2_[ii];
      }
      deltaLh_ = std::sqrt(deltaLh_);
      addSubgradToBase(newind,deltaLh_);
 
      if(dependent_){ // dependent was 2
#if( ! EXACT )
        Real gjNorm = GiGj(base_[currSize_-1],base_[currSize_-1]);
        ljNorm     -= deltaLj_ * deltaLj_;
        lhNorm      = gjNorm;
        deltaLj_    = std::abs(gjNorm - ljNorm);
        if ( signum < 0 )
          deltaLj_ = - std::sqrt( deltaLj_ );
        else
          deltaLj_ = std::sqrt( deltaLj_ );
#else
        // recompute deltaLj
        Real gjTgh = GiGj(base_[currSize_-1],base_[currSize_-2]);
        Real ljTlh = 0.0;
        for (unsigned ii=0;ii<currSize_;ii++){
          ljTlh += L_(currSize_-1,ii)*L_(currSize_-2,ii);
        }
        deltaLj_ = (gjTgh - ljTlh) / deltaLh_;
#endif
        L_(currSize_-1,currSize_-2) = deltaLj_;
      }
    } // end if deltaLh > 0
 
    if (dependent_ > 1){ // deltaLh is 0 but deltaLj is not
      // recompute deltaLj
      Real gjNorm = GiGj(base_[currSize_-1],base_[currSize_-1]);
      Real ljnrm = zero; // exact ljNorm
#if( EXACT )
      for (unsigned ii=0; ii<currSize_; ++ii) {
        ljnrm += L_(currSize_-1,ii)*L_(currSize_-1,ii);
      }
      deltaLj_ = std::abs(gjNorm - ljnrm);
#else
      deltaLj_ = std::abs( gjNorm - ljNorm + deltaLj_ * deltaLj_);
#endif
 
      if( std::sqrt(deltaLj_) > tol*kappa_*std::max(static_cast<Real>(1),gjNorm) ){ // originally had deltaLj without sqrt
        unsigned newind = currSize_-1;
        dependent_--;
        // get the last row of L
        lj_.size(newind-1); // initialize to zeros;
        ljz1_ = zero;
        Real ljTz2 = zero;
        for (unsigned ii=0;ii<newind-1;ii++){
          lj_[ii] = L_(newind,ii);
          ljz1_ += lj_[ii]*z1_[ii];
          ljTz2 += lj_[ii]*z2_[ii];
        }
        deltaLj_ = std::sqrt(deltaLj_);
        addSubgradToBase(newind,deltaLj_);
#if( EXACT )
        deltaLh_ = GiGj(base_[currSize_-2],base_[currSize_-1]);
        for (unsigned ii=0;ii<currSize_-1;ii++){
          deltaLh_ -= L_(currSize_-2,ii)*L_(currSize_-1,ii);
        }
        deltaLh_ /= deltaLj_;
#else
        if ( signum < 0) {
          deltaLh_ = - std::sqrt( deltaLh_ );
        }
        else {
          deltaLh_ = std::sqrt ( deltaLh_ );
        }
#endif
        L_(currSize_-1,currSize_-2) = deltaLh_;
      } // end if deltaLj > 0
    } // end if ( dependent > 1 )
  } // end if(dependent)
}// end deleteSubgradFromBase()
 
template<typename Real>
void Bundle_U_TT<Real>::solveSystem(int size, char tran, LA::Matrix<Real> &L, LA::Vector<Real> &v){
  int info;
  if( L.numRows()!=size )
    std::cout << "Error: Wrong size matrix!" << std::endl;
  else if( v.numRows()!=size )
    std::cout << "Error: Wrong size vector!" << std::endl;
  else if( size==0 )
    return;
  else{
    //std::cout << L_.stride() << ' ' << size << std::endl;
    lapack_.TRTRS( 'L', tran, 'N', size, 1, L.values(), L.stride(), v.values(), v.stride(), &info );
  }
}
 
template<typename Real>
bool Bundle_U_TT<Real>::isFeasible(LA::Vector<Real> &v, const Real &tol){
  bool feas = true;
  for(int i=0;i<v.numRows();i++){
    if(v[i]<-tol){
      feas = false;
    }
  }
  return feas;
}
 
template<typename Real>
unsigned Bundle_U_TT<Real>::solveDual_TT(const Real t, const unsigned maxit, const Real tol) {
  const Real zero(0), half(0.5), one(1);
  Real z1z2(0), z1z1(0);
  QPStatus_ = 1; // normal status
  entering_ = maxind_;
 
  // cold start
  optimal_   = true;
  dependent_ = 0;
  rho_       = ROL_INF<Real>(); // value of rho = -v
  currSize_  = 1;               // current base size
  base_.clear();
  base_.push_back(0);           // initial base
  L_.reshape(1,1);
  L_(0,0) = std::sqrt(GiGj(0,0));
  Bundle_U<Real>::resetDualVariables();
  Bundle_U<Real>::setDualVariable(0,one);
  tempv_.resize(1);
  tempw1_.resize(1);
  tempw2_.resize(1);
  lh_.resize(1);
  lj_.resize(1);
  z1_.resize(1); z2_.resize(1);
  z1_[0]     = one/L_(0,0);
  z2_[0]     = Bundle_U<Real>::alpha(0)/L_(0,0);
  LiMax_     = 0;                    // index of max element of diag(L)
  LiMin_     = 0;                    // index of min element of diag(L)
  kappa_     = one;                  // condition number of matrix L ( >= max|L_ii|/min|L_ii| )
  objval_    = ROL_INF<Real>();      // value of objective
  minobjval_ = ROL_INF<Real>();      // min value of objective (ever reached)
 
  unsigned iter;
  //-------------------------- MAIN LOOP --------------------------------//
  for (iter=0;iter<maxit;iter++){
    //---------------------- INNER LOOP -----------------------//
    while( !optimal_ ){
      switch( dependent_ ){
      case(0): // KT system admits unique solution
        {
          /*
            L = L_B'
          */
          z1z2    = z1_.dot(z2_);
          z1z1    = z1_.dot(z1_);
          rho_    = ( one + z1z2/t )/z1z1;
          tempv_  = z1_; tempv_.scale(rho_);
          tempw1_ = z2_; tempw1_.scale(one/t);
          tempv_ -= tempw1_;
          solveSystem(currSize_,'T',L_,tempv_); // tempv contains solution
          optimal_ = true;
          break;
        }
      case(1):
        {
          /*
            L = | L_B'   0 | \ currSize
                | l_h^T  0 | /
          */
          LA::Matrix<Real> LBprime( LA::Copy,L_,currSize_-1,currSize_-1);
          lh_.size(currSize_-1); // initialize to zeros;
          lhz1_ = zero;
          lhz2_ = zero;
          for(unsigned i=0; i<currSize_-1; ++i){
            Real tmp = L_(currSize_-1,i);
            lhz1_ += tmp*z1_(i);
            lhz2_ += tmp*z2_(i);
            lh_[i] = tmp;
          }
          // Test for singularity
          if (std::abs(lhz1_-one) <= tol*kappa_){
            // tempv is an infinite direction
            tempv_ = lh_;
            solveSystem(currSize_-1,'T',LBprime,tempv_);
            tempv_.resize(currSize_);   // add last entry
            tempv_[currSize_-1] = -one;
            optimal_ = false;
          }
          else{
            // system has (unique) solution
            rho_ = ( (Bundle_U<Real>::alpha(base_[currSize_-1]) - lhz2_)/t ) / ( one - lhz1_ );
            z1z2 = z1_.dot(z2_);
            z1z1 = z1_.dot(z1_);
            Real tmp = ( one + z1z2 / t - rho_ * z1z1 )/( one - lhz1_ );
            tempw1_ = z1_; tempw1_.scale(rho_);
            tempw2_ = z2_; tempw2_.scale(one/t);
            tempw1_ -= tempw2_;
            tempw2_ = lh_; tempw2_.scale(tmp);
            tempw1_ -= tempw2_;
            solveSystem(currSize_-1,'T',LBprime,tempw1_);
            tempv_ = tempw1_;
            tempv_.resize(currSize_);
            tempv_[currSize_-1] = tmp;
            optimal_ = true;
          }
          break;
        } // case(1)
      case(2):
        {
          /*     | L_B'  0 0 | \
             L = | l_h^T 0 0 | | currSize
                 | l_j^T 0 0 | /
          */
          LA::Matrix<Real> LBprime( LA::Copy,L_,currSize_-2,currSize_-2 );
             lj_.size(currSize_-2); // initialize to zeros;
          lh_.size(currSize_-2); // initialize to zeros;
          ljz1_ = zero;
          lhz1_ = zero;
          for(unsigned i=0; i<currSize_-2; ++i){
            Real tmp1 = L_(currSize_-1,i);
            Real tmp2 = L_(currSize_-2,i);
            ljz1_ += tmp1*z1_(i);
            lhz1_ += tmp2*z1_(i);
            lj_[i] = tmp1;
            lh_[i] = tmp2;
          }
          if(std::abs(ljz1_-one) <= tol*kappa_){
            // tempv is an infinite direction
            tempv_ = lj_;
            solveSystem(currSize_-2,'T',LBprime,tempv_);
            tempv_.resize(currSize_);   // add two last entries
            tempv_[currSize_-2] = zero;
            tempv_[currSize_-1] = -one;
          }
          else{
            // tempv is an infinite direction
            Real mu = ( one - lhz1_ )/( one - ljz1_ );
            tempw1_ = lj_; tempw1_.scale(-mu);
            tempw1_ += lh_;
            solveSystem(currSize_-2,'T',LBprime,tempw1_);
            tempv_ = tempw1_;
            tempv_.resize(currSize_);
            tempv_[currSize_-2] = -one;
            tempv_[currSize_-1] = mu;
          }
          optimal_ = false;
        }// case(2)
      } // end switch(dependent_)
 
      // optimal is true if tempv is a solution, otherwise, tempv is an infinite direction
      if (optimal_){
        // check feasibility (inequality constraints)
        if (isFeasible(tempv_,tol*currSize_)){
          // set dual variables to values in tempv
          Bundle_U<Real>::resetDualVariables();
          for (unsigned i=0; i<currSize_; ++i){
            Bundle_U<Real>::setDualVariable(base_[i],tempv_[i]);
          }
        }
        else{
          // w_B = /bar{x}_B - x_B
          for (unsigned i=0; i<currSize_; ++i){
            tempv_[i] -= Bundle_U<Real>::getDualVariable(base_[i]);
          }
          optimal_ = false;
        }
      } // if(optimal)
      else{ // choose the direction of descent
        if ( ( entering_ < maxind_ ) && ( Bundle_U<Real>::getDualVariable(entering_) == zero ) ){
          if ( tempv_[currSize_-1] < zero ) // w_h < 0
            tempv_.scale(-one);
        }
        else{ // no i such that dualVariables_[i] == 0
          Real sp(0);
          for( unsigned kk=0; kk<currSize_; ++kk)
            sp += tempv_[kk]*Bundle_U<Real>::alpha(base_[kk]);
          if ( sp > zero )
            tempv_.scale(-one);
        }
      }// end else ( not optimal )
 
      if(!optimal_){
        // take a step in direction tempv (possibly infinite)
        Real myeps = tol*currSize_;
        Real step  = ROL_INF<Real>();
        for (unsigned h=0; h<currSize_; ++h){
          if ( (tempv_[h] < -myeps) && (-Bundle_U<Real>::getDualVariable(base_[h])/tempv_[h] < step) )
            if ( (Bundle_U<Real>::getDualVariable(base_[h]) > myeps)
              || (Bundle_U<Real>::getDualVariable(base_[h]) < -myeps) ) // otherwise, consider it 0
              step = -Bundle_U<Real>::getDualVariable(base_[h])/tempv_[h];
        }
 
        if (step <= zero || step == ROL_INF<Real>()){
          QPStatus_ = -1; // invalid step
          return iter;
        }
 
        for (unsigned i=0; i<currSize_; ++i)
          Bundle_U<Real>::setDualVariable(base_[i],Bundle_U<Real>::getDualVariable(base_[i]) + step * tempv_[i]);
      }// if(!optimal)
 
      //------------------------- ITEMS ELIMINATION ---------------------------//
 
      // Eliminate items with 0 multipliers from base
      bool deleted = optimal_;
      for (unsigned baseitem=0; baseitem<currSize_; ++baseitem){
        if (Bundle_U<Real>::getDualVariable(base_[baseitem]) <= tol){
          deleted = true;
 
#if( TABOO_LIST )
          // item that just entered shouldn't exit; if it does, mark it as taboo
          if( base_[baseitem] == entering_ ){
            taboo_.push_back(entering_);
            entering_ = maxind_;
          }
#endif
 
          // eliminate item from base;
          deleteSubgradFromBase(baseitem,tol);
 
        } // end if(dualVariables_[baseitem] < tol)
      } // end loop over baseitem
 
      if(!deleted){ // nothing deleted and not optimal
        QPStatus_ = -2; // loop
        return iter;
      }
    } // end inner loop
 
    Real newobjval(0), Lin(0), Quad(0); // new objective value
    for (unsigned i=0; i<currSize_; ++i){
      Lin += Bundle_U<Real>::alpha(base_[i])*Bundle_U<Real>::getDualVariable(base_[i]);
    }
    if (rho_ == ROL_NINF<Real>()){
      Quad = -Lin/t;
      newobjval = -half*Quad;
    }
    else{
      Quad = rho_ - Lin/t;
      newobjval = half*(rho_ + Lin/t);
    }
 
#if( TABOO_LIST )
    // -- test for strict decrease -- //
    // if item didn't provide decrease, move it to taboo list ...
    if( ( entering_ < maxind_ ) && ( objval_ < ROL_INF<Real>() ) ){
      if( newobjval >= objval_ - std::max( tol*std::abs(objval_), ROL_EPSILON<Real>() ) ){
        taboo_.push_back(entering_);
      }
    }
#endif
 
    objval_ = newobjval;
 
    // if sufficient decrease obtained
    if ( objval_ + std::max( tol*std::abs(objval_), ROL_EPSILON<Real>() ) <= minobjval_ ){
      taboo_.clear(); // reset taboo list
      minobjval_ = objval_;
    }
 
    //---------------------- OPTIMALITY TEST -------------------------//
    if ( (rho_ >= ROL_NINF<Real>()) && (objval_ <= ROL_NINF<Real>()) ) // if current x (dualVariables_) is feasible
      break;
 
    entering_  = maxind_;
    Real minro = - std::max( tol*currSize_*std::abs(objval_), ROL_EPSILON<Real>() );
#if ( ! FIRST_VIOLATED )
    Real diff  = ROL_NINF<Real>(), olddiff = ROL_NINF<Real>();
#endif
 
    for (unsigned bundleitem=0; bundleitem<Bundle_U<Real>::size(); ++bundleitem){ // loop over items in bundle
    //for (int bundleitem=size_-1;bundleitem>-1;bundleitem--){ // loop over items in bundle (in reverse order)
      if ( std::find(taboo_.begin(),taboo_.end(),bundleitem) != taboo_.end() ){
        continue; // if item is taboo, move on
      }
 
      if ( std::find(base_.begin(),base_.end(),bundleitem) == base_.end() ){
        // base does not contain bundleitem
        Real ro = zero;
        for (unsigned j=0;j<currSize_;j++){
          ro += Bundle_U<Real>::getDualVariable(base_[j]) * GiGj(bundleitem,base_[j]);
        }
        ro += Bundle_U<Real>::alpha(bundleitem)/t;
 
 
        if (rho_ >= ROL_NINF<Real>()){
          ro = ro - rho_; // note: rho = -v
        }
        else{
          ro         = ROL_NINF<Real>();
          minobjval_ = ROL_INF<Real>();
          objval_    = ROL_INF<Real>();
        }
 
        if (ro < minro){
#if ( FIRST_VIOLATED )
          entering_ = bundleitem;
          break; // skip going through rest of constraints; alternatively, could look for "most violated"
#else
          diff = minro - ro;
          if ( diff > olddiff ){
            entering_ = bundleitem;
            olddiff = diff;
          }
#endif
        }
 
      } // end if item not in base
    }// end of loop over items in bundle
 
    //----------------- INSERTING ITEM ------------------------//
    if (entering_ < maxind_){ // dual constraint is violated
      optimal_ = false;
      Bundle_U<Real>::setDualVariable(entering_,zero);
      base_.push_back(entering_);
      // construct new row of L_B
      unsigned zsize = currSize_ - dependent_; // zsize is the size of L_Bprime (current one)
      lh_.size(zsize); // initialize to zeros;
      lhz1_ = zero;
      lhz2_ = zero;
      for (unsigned i=0; i<zsize; ++i){
        lh_[i] = GiGj(entering_,base_[i]);
      }
      LA::Matrix<Real> LBprime( LA::Copy,L_,zsize,zsize);
      solveSystem(zsize,'N',LBprime,lh_); // lh = (L_B^{-1})*(G_B^T*g_h)
      for (unsigned i=0; i<zsize; ++i){
        lhz1_ += lh_[i]*z1_[i];
        lhz2_ += lh_[i]*z2_[i];
      }
 
      Real nrm = lh_.dot(lh_);
      Real delta = GiGj(entering_,entering_) - nrm; // delta squared
#if( ! EXACT )
      if(dependent_)
        ljNorm = nrm; // adding second dependent
      else
        lhNorm = nrm; // adding first dependent
#endif
 
      currSize_++; // update base size
 
      L_.reshape(currSize_,currSize_);
      zsize = currSize_ - dependent_; // zsize is the size of L_Bprime (new one)
      for (unsigned i=0; i<zsize-1; ++i){
        L_(currSize_-1,i) = lh_[i];
      }
 
      Real deltaeps = tol*kappa_*std::max(one,lh_.dot(lh_));
      if ( delta > deltaeps ){ // new row is independent
        // add subgradient to the base
        unsigned ind = currSize_-1;
        delta = std::sqrt(delta);
        addSubgradToBase(ind,delta);
      }
      else if(delta < -deltaeps){
        dependent_++;
        QPStatus_ = 0; // negative delta
        return iter;
      }
      else{ // delta zero
        dependent_++;
      }
    } // end if(entering_ < maxind_)
    else{ // no dual constraint violated
      if( objval_ - std::max( tol*std::abs( objval_ ), ROL_EPSILON<Real>() ) > minobjval_ ){ // check if f is as good as minf
        QPStatus_ = -3; // loop
        return iter;
      }
    }
 
    if(optimal_)
      break;
  } // end main loop
 
  taboo_.clear();
  return iter;
}// end solveDual_TT()
 
template<typename Real>
unsigned Bundle_U_TT<Real>::solveDual_arbitrary(const Real t, const unsigned maxit, const Real tol) {
  Real mytol = tol;
  unsigned outermaxit = 20;
  bool increase = false, decrease = false;
  unsigned iter = 0;
  for ( unsigned it=0; it < outermaxit; ++it ){
    iter += solveDual_TT(t,maxit,mytol);
    if ( QPStatus_ == 1 ) {
      break;
    }
    else if ( QPStatus_ == -2  || QPStatus_ == -3 ) {
      mytol /= static_cast<Real>(10);
      decrease = true;
    }
    else {
      mytol *= static_cast<Real>(10);
      increase = true;
    }
    if ( (mytol > static_cast<Real>(1e-4))
      || (mytol < static_cast<Real>(1e-16)) ){
      break;
    }
    if ( increase && decrease ) {
      break;
    }
  }// end outer loop
  return iter;
}
 
} // namespace ROL
 
#endif