bool TangentialStepWithInequStd_Step::do_step(
Algorithm& _algo, poss_type step_poss, IterationPack::EDoStepType type
,poss_type assoc_step_poss
)
{
using Teuchos::RCP;
using Teuchos::dyn_cast;
using ::fabs;
using LinAlgOpPack::Vt_S;
using LinAlgOpPack::V_VpV;
using LinAlgOpPack::V_VmV;
using LinAlgOpPack::Vp_StV;
using LinAlgOpPack::Vp_V;
using LinAlgOpPack::V_StV;
using LinAlgOpPack::V_MtV;
// using ConstrainedOptPack::min_abs;
using AbstractLinAlgPack::max_near_feas_step;
typedef VectorMutable::vec_mut_ptr_t vec_mut_ptr_t;
NLPAlgo &algo = rsqp_algo(_algo);
NLPAlgoState &s = algo.rsqp_state();
EJournalOutputLevel olevel = algo.algo_cntr().journal_output_level();
EJournalOutputLevel ns_olevel = algo.algo_cntr().null_space_journal_output_level();
std::ostream &out = algo.track().journal_out();
//const bool check_results = algo.algo_cntr().check_results();
// print step header.
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
using IterationPack::print_algorithm_step;
print_algorithm_step( algo, step_poss, type, assoc_step_poss, out );
}
// problem dimensions
const size_type
//n = algo.nlp().n(),
m = algo.nlp().m(),
r = s.equ_decomp().size();
// Get the iteration quantity container objects
IterQuantityAccess<value_type>
&alpha_iq = s.alpha(),
&zeta_iq = s.zeta(),
&eta_iq = s.eta();
IterQuantityAccess<VectorMutable>
&dl_iq = dl_iq_(s),
&du_iq = du_iq_(s),
&nu_iq = s.nu(),
*c_iq = m > 0 ? &s.c() : NULL,
*lambda_iq = m > 0 ? &s.lambda() : NULL,
&rGf_iq = s.rGf(),
&w_iq = s.w(),
&qp_grad_iq = s.qp_grad(),
&py_iq = s.py(),
&pz_iq = s.pz(),
&Ypy_iq = s.Ypy(),
&Zpz_iq = s.Zpz();
IterQuantityAccess<MatrixOp>
&Z_iq = s.Z(),
//*Uz_iq = (m > r) ? &s.Uz() : NULL,
*Uy_iq = (m > r) ? &s.Uy() : NULL;
IterQuantityAccess<MatrixSymOp>
&rHL_iq = s.rHL();
IterQuantityAccess<ActSetStats>
&act_set_stats_iq = act_set_stats_(s);
// Accessed/modified/updated (just some)
VectorMutable *Ypy_k = (m ? &Ypy_iq.get_k(0) : NULL);
const MatrixOp &Z_k = Z_iq.get_k(0);
VectorMutable &pz_k = pz_iq.set_k(0);
VectorMutable &Zpz_k = Zpz_iq.set_k(0);
// Comupte qp_grad which is an approximation to rGf + Z'*HL*Y*py
// qp_grad = rGf
VectorMutable
&qp_grad_k = ( qp_grad_iq.set_k(0) = rGf_iq.get_k(0) );
// qp_grad += zeta * w
if( w_iq.updated_k(0) ) {
if(zeta_iq.updated_k(0))
Vp_StV( &qp_grad_k, zeta_iq.get_k(0), w_iq.get_k(0) );
else
Vp_V( &qp_grad_k, w_iq.get_k(0) );
}
//
// Set the bounds for:
//
// dl <= Z*pz + Y*py <= du -> dl - Ypy <= Z*pz <= du - Ypz
vec_mut_ptr_t
bl = s.space_x().create_member(),
bu = s.space_x().create_member();
if(m) {
// bl = dl_k - Ypy_k
V_VmV( bl.get(), dl_iq.get_k(0), *Ypy_k );
// bu = du_k - Ypy_k
V_VmV( bu.get(), du_iq.get_k(0), *Ypy_k );
}
else {
*bl = dl_iq.get_k(0);
*bu = du_iq.get_k(0);
}
// Print out the QP bounds for the constraints
if( static_cast<int>(ns_olevel) >= static_cast<int>(PRINT_VECTORS) ) {
out << "\nqp_grad_k = \n" << qp_grad_k;
}
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_VECTORS) ) {
out << "\nbl = \n" << *bl;
out << "\nbu = \n" << *bu;
}
//
// Determine if we should perform a warm start or not.
//
bool do_warm_start = false;
if( act_set_stats_iq.updated_k(-1) ) {
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << "\nDetermining if the QP should use a warm start ...\n";
}
// We need to see if we should preform a warm start for the next iteration
ActSetStats &stats = act_set_stats_iq.get_k(-1);
const size_type
num_active = stats.num_active(),
num_adds = stats.num_adds(),
num_drops = stats.num_drops();
const value_type
frac_same
= ( num_adds == ActSetStats::NOT_KNOWN || num_active == 0
? 0.0
: my_max(((double)(num_active)-num_adds-num_drops) / num_active, 0.0 ) );
do_warm_start = ( num_active > 0 && frac_same >= warm_start_frac() );
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << "\nnum_active = " << num_active;
if( num_active ) {
out << "\nmax(num_active-num_adds-num_drops,0)/(num_active) = "
<< "max("<<num_active<<"-"<<num_adds<<"-"<<num_drops<<",0)/("<<num_active<<") = "
<< frac_same;
if( do_warm_start )
out << " >= ";
else
out << " < ";
out << "warm_start_frac = " << warm_start_frac();
}
if( do_warm_start )
out << "\nUse a warm start this time!\n";
else
out << "\nDon't use a warm start this time!\n";
}
}
// Use active set from last iteration as an estimate for current active set
// if we are to use a warm start.
//
// ToDo: If the selection of dependent and independent variables changes
// then you will have to adjust this or not perform a warm start at all!
if( do_warm_start ) {
nu_iq.set_k(0,-1);
}
else {
nu_iq.set_k(0) = 0.0; // No guess of the active set
}
VectorMutable &nu_k = nu_iq.get_k(0);
//
// Setup the reduced QP subproblem
//
// The call to the QP is setup for the more flexible call to the QPSolverRelaxed
// interface to deal with the three independent variabilities: using simple
// bounds for pz or not, general inequalities included or not, and extra equality
// constraints included or not.
// If this method of calling the QP solver were not used then 4 separate
// calls to solve_qp(...) would have to be included to handle the four possible
// QP formulations.
//
// The numeric arguments for the QP solver (in the nomenclatrue of QPSolverRelaxed)
const value_type qp_bnd_inf = NLP::infinite_bound();
const Vector &qp_g = qp_grad_k;
const MatrixSymOp &qp_G = rHL_iq.get_k(0);
const value_type qp_etaL = 0.0;
vec_mut_ptr_t qp_dL = Teuchos::null;
vec_mut_ptr_t qp_dU = Teuchos::null;
Teuchos::RCP<const MatrixOp>
qp_E = Teuchos::null;
BLAS_Cpp::Transp qp_trans_E = BLAS_Cpp::no_trans;
vec_mut_ptr_t qp_b = Teuchos::null;
vec_mut_ptr_t qp_eL = Teuchos::null;
vec_mut_ptr_t qp_eU = Teuchos::null;
Teuchos::RCP<const MatrixOp>
qp_F = Teuchos::null;
BLAS_Cpp::Transp qp_trans_F = BLAS_Cpp::no_trans;
vec_mut_ptr_t qp_f = Teuchos::null;
value_type qp_eta = 0.0;
VectorMutable &qp_d = pz_k; // pz_k will be updated directly!
vec_mut_ptr_t qp_nu = Teuchos::null;
vec_mut_ptr_t qp_mu = Teuchos::null;
vec_mut_ptr_t qp_Ed = Teuchos::null;
vec_mut_ptr_t qp_lambda = Teuchos::null;
//
// Determine if we can use simple bounds on pz.
//
// If we have a variable-reduction null-space matrix
// (with any choice for Y) then:
//
// d = Z*pz + (1-eta) * Y*py
//
// [ d(var_dep) ] = [ D ] * pz + (1-eta) * [ Ypy(var_dep) ]
// [ d(var_indep) ] [ I ] [ Ypy(var_indep) ]
//
// For a cooridinate decomposition (Y = [ I ; 0 ]) then Ypy(var_indep) ==
// 0.0 and in this case the bounds on d(var_indep) become simple bounds on
// pz even with the relaxation. Also, if dl(var_dep) and du(var_dep) are
// unbounded, then we can also use simple bounds since we don't need the
// relaxation and we can set eta=0. In this case we just have to subtract
// from the upper and lower bounds on pz!
//
// Otherwise, we can not use simple variable bounds and implement the
// relaxation properly.
//
const MatrixIdentConcat
*Zvr = dynamic_cast<const MatrixIdentConcat*>( &Z_k );
const Range1D
var_dep = Zvr ? Zvr->D_rng() : Range1D::Invalid,
var_indep = Zvr ? Zvr->I_rng() : Range1D();
RCP<Vector> Ypy_indep;
const value_type
Ypy_indep_norm_inf
= ( m ? (Ypy_indep=Ypy_k->sub_view(var_indep))->norm_inf() : 0.0);
if( (int)olevel >= (int)PRINT_ALGORITHM_STEPS )
out
<< "\nDetermine if we can use simple bounds on pz ...\n"
<< " m = " << m << std::endl
<< " dynamic_cast<const MatrixIdentConcat*>(&Z_k) = " << Zvr << std::endl
<< " ||Ypy_k(var_indep)||inf = " << Ypy_indep_norm_inf << std::endl;
const bool
bounded_var_dep
= (
m > 0
&&
num_bounded( *bl->sub_view(var_dep), *bu->sub_view(var_dep), qp_bnd_inf )
);
const bool
use_simple_pz_bounds
= (
m == 0
||
( Zvr != NULL && ( Ypy_indep_norm_inf == 0.0 || bounded_var_dep == 0 ) )
);
if( (int)olevel >= (int)PRINT_ALGORITHM_STEPS )
out
<< (use_simple_pz_bounds
? "\nUsing simple bounds on pz ...\n"
: "\nUsing bounds on full Z*pz ...\n")
<< (bounded_var_dep
? "\nThere are finite bounds on dependent variables. Adding extra inequality constrints for D*pz ...\n"
: "\nThere are no finite bounds on dependent variables. There will be no extra inequality constraints added on D*pz ...\n" ) ;
if( use_simple_pz_bounds ) {
// Set simple bound constraints on pz
qp_dL = bl->sub_view(var_indep);
qp_dU = bu->sub_view(var_indep);
qp_nu = nu_k.sub_view(var_indep); // nu_k(var_indep) will be updated directly!
if( m && bounded_var_dep ) {
// Set general inequality constraints for D*pz
qp_E = Teuchos::rcp(&Zvr->D(),false);
qp_b = Ypy_k->sub_view(var_dep);
qp_eL = bl->sub_view(var_dep);
qp_eU = bu->sub_view(var_dep);
qp_mu = nu_k.sub_view(var_dep); // nu_k(var_dep) will be updated directly!
qp_Ed = Zpz_k.sub_view(var_dep); // Zpz_k(var_dep) will be updated directly!
}
else {
// Leave these as NULL since there is no extra general inequality constraints
}
}
else if( !use_simple_pz_bounds ) { // ToDo: Leave out parts for unbounded dependent variables!
// There are no simple bounds! (leave qp_dL, qp_dU and qp_nu as null)
// Set general inequality constraints for Z*pz
qp_E = Teuchos::rcp(&Z_k,false);
qp_b = Teuchos::rcp(Ypy_k,false);
qp_eL = bl;
qp_eU = bu;
qp_mu = Teuchos::rcp(&nu_k,false);
qp_Ed = Teuchos::rcp(&Zpz_k,false); // Zpz_k will be updated directly!
}
else {
TEST_FOR_EXCEPT(true);
}
// Set the general equality constriants (if they exist)
Range1D equ_undecomp = s.equ_undecomp();
if( m > r && m > 0 ) {
// qp_f = Uy_k * py_k + c_k(equ_undecomp)
qp_f = s.space_c().sub_space(equ_undecomp)->create_member();
V_MtV( qp_f.get(), Uy_iq->get_k(0), BLAS_Cpp::no_trans, py_iq.get_k(0) );
Vp_V( qp_f.get(), *c_iq->get_k(0).sub_view(equ_undecomp) );
// Must resize for the undecomposed constriants if it has not already been
qp_F = Teuchos::rcp(&Uy_iq->get_k(0),false);
qp_lambda = lambda_iq->set_k(0).sub_view(equ_undecomp); // lambda_k(equ_undecomp), will be updated directly!
}
// Setup the rest of the arguments
QPSolverRelaxed::EOutputLevel
qp_olevel;
switch( olevel ) {
case PRINT_NOTHING:
qp_olevel = QPSolverRelaxed::PRINT_NONE;
break;
case PRINT_BASIC_ALGORITHM_INFO:
qp_olevel = QPSolverRelaxed::PRINT_NONE;
break;
case PRINT_ALGORITHM_STEPS:
qp_olevel = QPSolverRelaxed::PRINT_BASIC_INFO;
break;
case PRINT_ACTIVE_SET:
qp_olevel = QPSolverRelaxed::PRINT_ITER_SUMMARY;
break;
case PRINT_VECTORS:
qp_olevel = QPSolverRelaxed::PRINT_ITER_VECTORS;
break;
case PRINT_ITERATION_QUANTITIES:
qp_olevel = QPSolverRelaxed::PRINT_EVERY_THING;
break;
default:
TEST_FOR_EXCEPT(true);
}
// ToDo: Set print options so that only vectors matrices etc
// are only printed in the null space
//
// Solve the QP
//
qp_solver().infinite_bound(qp_bnd_inf);
const QPSolverStats::ESolutionType
solution_type =
qp_solver().solve_qp(
int(olevel) == int(PRINT_NOTHING) ? NULL : &out
,qp_olevel
,( algo.algo_cntr().check_results()
? QPSolverRelaxed::RUN_TESTS : QPSolverRelaxed::NO_TESTS )
,qp_g, qp_G, qp_etaL, qp_dL.get(), qp_dU.get()
,qp_E.get(), qp_trans_E, qp_E.get() ? qp_b.get() : NULL
,qp_E.get() ? qp_eL.get() : NULL, qp_E.get() ? qp_eU.get() : NULL
,qp_F.get(), qp_trans_F, qp_F.get() ? qp_f.get() : NULL
,NULL // obj_d
,&qp_eta, &qp_d
,qp_nu.get()
,qp_mu.get(), qp_E.get() ? qp_Ed.get() : NULL
,qp_F.get() ? qp_lambda.get() : NULL
,NULL // qp_Fd
);
//
// Check the optimality conditions for the QP
//
std::ostringstream omsg;
bool throw_qp_failure = false;
if( qp_testing() == QP_TEST
|| ( qp_testing() == QP_TEST_DEFAULT && algo.algo_cntr().check_results() ) )
{
if( int(olevel) >= int(PRINT_ALGORITHM_STEPS) ) {
out << "\nChecking the optimality conditions of the reduced QP subproblem ...\n";
}
if(!qp_tester().check_optimality_conditions(
solution_type,qp_solver().infinite_bound()
,int(olevel) == int(PRINT_NOTHING) ? NULL : &out
,int(olevel) >= int(PRINT_VECTORS) ? true : false
,int(olevel) >= int(PRINT_ITERATION_QUANTITIES) ? true : false
,qp_g, qp_G, qp_etaL, qp_dL.get(), qp_dU.get()
,qp_E.get(), qp_trans_E, qp_E.get() ? qp_b.get() : NULL
,qp_E.get() ? qp_eL.get() : NULL, qp_E.get() ? qp_eU.get() : NULL
,qp_F.get(), qp_trans_F, qp_F.get() ? qp_f.get() : NULL
,NULL // obj_d
,&qp_eta, &qp_d
,qp_nu.get()
,qp_mu.get(), qp_E.get() ? qp_Ed.get() : NULL
,qp_F.get() ? qp_lambda.get() : NULL
,NULL // qp_Fd
))
{
omsg << "\n*** Alert! at least one of the QP optimality conditions did not check out.\n";
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << omsg.str();
}
throw_qp_failure = true;
}
}
//
// Set the solution
//
if( !use_simple_pz_bounds ) {
// Everything is already updated!
}
else if( use_simple_pz_bounds ) {
// Just have to set Zpz_k(var_indep) = pz_k
*Zpz_k.sub_view(var_indep) = pz_k;
if( m && !bounded_var_dep ) {
// Must compute Zpz_k(var_dep) = D*pz
LinAlgOpPack::V_MtV( &*Zpz_k.sub_view(var_dep), Zvr->D(), BLAS_Cpp::no_trans, pz_k );
// ToDo: Remove the compuation of Zpz here unless you must
}
}
else {
TEST_FOR_EXCEPT(true);
}
// Set the solution statistics
qp_solver_stats_(s).set_k(0) = qp_solver().get_qp_stats();
// Cut back Ypy_k = (1-eta) * Ypy_k
const value_type eps = std::numeric_limits<value_type>::epsilon();
if( fabs(qp_eta - 0.0) > eps ) {
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out
<< "\n*** Alert! the QP was infeasible (eta = "<<qp_eta<<"). Cutting back Ypy_k = (1.0 - eta)*Ypy_k ...\n";
}
Vt_S( Ypy_k, 1.0 - qp_eta );
}
// eta_k
eta_iq.set_k(0) = qp_eta;
//
// Modify the solution if we have to!
//
switch(solution_type) {
case QPSolverStats::OPTIMAL_SOLUTION:
break; // we are good!
case QPSolverStats::PRIMAL_FEASIBLE_POINT:
{
omsg
<< "\n*** Alert! the returned QP solution is PRIMAL_FEASIBLE_POINT but not optimal!\n";
if( primal_feasible_point_error() )
omsg
<< "\n*** primal_feasible_point_error == true, this is an error!\n";
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << omsg.str();
}
throw_qp_failure = primal_feasible_point_error();
break;
}
case QPSolverStats::DUAL_FEASIBLE_POINT:
{
omsg
<< "\n*** Alert! the returned QP solution is DUAL_FEASIBLE_POINT"
<< "\n*** but not optimal so we cut back the step ...\n";
if( dual_feasible_point_error() )
omsg
<< "\n*** dual_feasible_point_error == true, this is an error!\n";
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << omsg.str();
}
// Cut back the step to fit in the bounds
//
// dl <= u*(Ypy_k+Zpz_k) <= du
//
vec_mut_ptr_t
zero = s.space_x().create_member(0.0),
d_tmp = s.space_x().create_member();
V_VpV( d_tmp.get(), *Ypy_k, Zpz_k );
const std::pair<value_type,value_type>
u_steps = max_near_feas_step( *zero, *d_tmp, dl_iq.get_k(0), du_iq.get_k(0), 0.0 );
const value_type
u = my_min( u_steps.first, 1.0 ); // largest positive step size
alpha_iq.set_k(0) = u;
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << "\nFinding u s.t. dl <= u*(Ypy_k+Zpz_k) <= du\n"
<< "max step length u = " << u << std::endl
<< "alpha_k = u = " << alpha_iq.get_k(0) << std::endl;
}
throw_qp_failure = dual_feasible_point_error();
break;
}
case QPSolverStats::SUBOPTIMAL_POINT:
{
omsg
<< "\n*** Alert!, the returned QP solution is SUBOPTIMAL_POINT!\n";
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << omsg.str();
}
throw_qp_failure = true;
break;
}
default:
TEST_FOR_EXCEPT(true); // should not happen!
}
//
// Output the final solution!
//
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << "\n||pz_k||inf = " << s.pz().get_k(0).norm_inf()
<< "\nnu_k.nz() = " << s.nu().get_k(0).nz()
<< "\nmax(|nu_k(i)|) = " << s.nu().get_k(0).norm_inf()
// << "\nmin(|nu_k(i)|) = " << min_abs( s.nu().get_k(0)() )
;
if( m > r ) out << "\n||lambda_k(undecomp)||inf = " << s.lambda().get_k(0).norm_inf();
out << "\n||Zpz_k||2 = " << s.Zpz().get_k(0).norm_2()
;
if(qp_eta > 0.0) out << "\n||Ypy||2 = " << s.Ypy().get_k(0).norm_2();
out << std::endl;
}
if( static_cast<int>(ns_olevel) >= static_cast<int>(PRINT_VECTORS) ) {
out << "\npz_k = \n" << s.pz().get_k(0);
if(var_indep.size())
out << "\nnu_k(var_indep) = \n" << *s.nu().get_k(0).sub_view(var_indep);
}
if( static_cast<int>(ns_olevel) >= static_cast<int>(PRINT_VECTORS) ) {
if(var_indep.size())
out << "\nZpz(var_indep)_k = \n" << *s.Zpz().get_k(0).sub_view(var_indep);
out << std::endl;
}
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_VECTORS) ) {
if(var_dep.size())
out << "\nZpz(var_dep)_k = \n" << *s.Zpz().get_k(0).sub_view(var_dep);
out << "\nZpz_k = \n" << s.Zpz().get_k(0);
out << std::endl;
}
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_VECTORS) ) {
out << "\nnu_k = \n" << s.nu().get_k(0);
if(var_dep.size())
out << "\nnu_k(var_dep) = \n" << *s.nu().get_k(0).sub_view(var_dep);
if( m > r )
out << "\nlambda_k(equ_undecomp) = \n" << *s.lambda().get_k(0).sub_view(equ_undecomp);
if(qp_eta > 0.0) out << "\nYpy = \n" << s.Ypy().get_k(0);
}
if( qp_eta == 1.0 ) {
omsg
<< "TangentialStepWithInequStd_Step::do_step(...) : Error, a QP relaxation parameter\n"
<< "of eta = " << qp_eta << " was calculated and therefore it must be assumed\n"
<< "that the NLP's constraints are infeasible\n"
<< "Throwing an InfeasibleConstraints exception!\n";
if( static_cast<int>(olevel) >= static_cast<int>(PRINT_ALGORITHM_STEPS) ) {
out << omsg.str();
}
throw InfeasibleConstraints(omsg.str());
}
if( throw_qp_failure )
throw QPFailure( omsg.str(), qp_solver().get_qp_stats() );
return true;
}
void MatrixSymPosDefLBFGS::V_InvMtV(
VectorMutable* y, BLAS_Cpp::Transp trans_rhs1
, const Vector& x
) const
{
using AbstractLinAlgPack::Vp_StMtV;
using DenseLinAlgPack::V_InvMtV;
using LinAlgOpPack::V_mV;
using LinAlgOpPack::V_StV;
using LinAlgOpPack::Vp_V;
using LinAlgOpPack::V_MtV;
using LinAlgOpPack::V_StMtV;
using LinAlgOpPack::Vp_MtV;
using DenseLinAlgPack::Vp_StMtV;
typedef VectorDenseEncap vde;
typedef VectorDenseMutableEncap vdme;
using Teuchos::Workspace;
Teuchos::WorkspaceStore* wss = Teuchos::get_default_workspace_store().get();
assert_initialized();
TEUCHOS_TEST_FOR_EXCEPT( !( inverse_is_updated_ ) ); // For now just always update
// y = inv(Bk) * x = Hk * x
//
// = gk*x + [S gk*Y] * [ inv(R')*(D+gk*Y'Y)*inv(R) -inv(R') ] * [ S' ] * x
// [ -inv(R) 0 ] [ gk*Y' ]
//
// Perform in the following (in order):
//
// y = gk*x
//
// t1 = [ S'*x ] <: R^(2*m)
// [ gk*Y'*x ]
//
// t2 = inv(R) * t1(1:m) <: R^(m)
//
// t3 = - inv(R') * t1(m+1,2*m) <: R^(m)
//
// t4 = gk * Y'Y * t2 <: R^(m)
//
// t4 += D*t2
//
// t5 = inv(R') * t4 <: R^(m)
//
// t5 += t3
//
// y += S*t5
//
// y += -gk*Y*t2
// y = gk*x
V_StV( y, gamma_k_, x );
const size_type
mb = m_bar_;
if( !mb )
return; // No updates have been performed.
const multi_vec_ptr_t
S = this->S(),
Y = this->Y();
// Get workspace
Workspace<value_type> t1_ws(wss,2*mb);
DVectorSlice t1(&t1_ws[0],t1_ws.size());
Workspace<value_type> t2_ws(wss,mb);
DVectorSlice t2(&t2_ws[0],t2_ws.size());
Workspace<value_type> t3_ws(wss,mb);
DVectorSlice t3(&t3_ws[0],t3_ws.size());
Workspace<value_type> t4_ws(wss,mb);
DVectorSlice t4(&t4_ws[0],t4_ws.size());
Workspace<value_type> t5_ws(wss,mb);
DVectorSlice t5(&t5_ws[0],t5_ws.size());
VectorSpace::vec_mut_ptr_t
t = S->space_rows().create_member();
const DMatrixSliceTri
&R = this->R();
const DMatrixSliceSym
&YTY = this->YTY();
// t1 = [ S'*x ]
// [ gk*Y'*x ]
V_MtV( t.get(), *S, BLAS_Cpp::trans, x );
t1(1,mb) = vde(*t)();
V_StMtV( t.get(), gamma_k_, *Y, BLAS_Cpp::trans, x );
t1(mb+1,2*mb) = vde(*t)();
// t2 = inv(R) * t1(1:m)
V_InvMtV( &t2, R, BLAS_Cpp::no_trans, t1(1,mb) );
// t3 = - inv(R') * t1(m+1,2*m)
V_mV( &t3, t1(mb+1,2*mb) );
V_InvMtV( &t3, R, BLAS_Cpp::trans, t3 );
// t4 = gk * Y'Y * t2
V_StMtV( &t4, gamma_k_, YTY, BLAS_Cpp::no_trans, t2 );
// t4 += D*t2
Vp_DtV( &t4, t2 );
// t5 = inv(R') * t4
V_InvMtV( &t5, R, BLAS_Cpp::trans, t4 );
// t5 += t3
Vp_V( &t5, t3 );
// y += S*t5
(vdme(*t)() = t5);
Vp_MtV( y, *S, BLAS_Cpp::no_trans, *t );
// y += -gk*Y*t2
(vdme(*t)() = t2);
Vp_StMtV( y, -gamma_k_, *Y, BLAS_Cpp::no_trans, *t );
}