ROL
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ROL::PrimalDualActiveSetStep< Real > Class Template Reference

Implements the computation of optimization steps with the Newton primal-dual active set method. More...

#include <ROL_PrimalDualActiveSetStep.hpp>

+ Inheritance diagram for ROL::PrimalDualActiveSetStep< Real >:

Classes

class  HessianPD
 
class  PrecondPD
 

Public Member Functions

 PrimalDualActiveSetStep (ROL::ParameterList &parlist)
 Constructor. More...
 
void initialize (Vector< Real > &x, const Vector< Real > &s, const Vector< Real > &g, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state)
 Initialize step with bound constraint. More...
 
void compute (Vector< Real > &s, const Vector< Real > &x, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state)
 Compute step. More...
 
void update (Vector< Real > &x, const Vector< Real > &s, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state)
 Update step, if successful. More...
 
std::string printHeader (void) const
 Print iterate header. More...
 
std::string printName (void) const
 Print step name. More...
 
virtual std::string print (AlgorithmState< Real > &algo_state, bool print_header=false) const
 Print iterate status. More...
 
- Public Member Functions inherited from ROL::Step< Real >
virtual ~Step ()
 
 Step (void)
 
virtual void initialize (Vector< Real > &x, const Vector< Real > &g, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state)
 Initialize step with bound constraint. More...
 
virtual void initialize (Vector< Real > &x, const Vector< Real > &g, Vector< Real > &l, const Vector< Real > &c, Objective< Real > &obj, Constraint< Real > &con, AlgorithmState< Real > &algo_state)
 Initialize step with equality constraint. More...
 
virtual void initialize (Vector< Real > &x, const Vector< Real > &g, Vector< Real > &l, const Vector< Real > &c, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
 Initialize step with equality constraint. More...
 
virtual void compute (Vector< Real > &s, const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, Constraint< Real > &con, AlgorithmState< Real > &algo_state)
 Compute step (equality constraints). More...
 
virtual void update (Vector< Real > &x, Vector< Real > &l, const Vector< Real > &s, Objective< Real > &obj, Constraint< Real > &con, AlgorithmState< Real > &algo_state)
 Update step, if successful (equality constraints). More...
 
virtual void compute (Vector< Real > &s, const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
 Compute step (equality constraints). More...
 
virtual void update (Vector< Real > &x, Vector< Real > &l, const Vector< Real > &s, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
 Update step, if successful (equality constraints). More...
 
const ROL::Ptr< const
StepState< Real > > 
getStepState (void) const
 Get state for step object. More...
 
void reset (const Real searchSize=1.0)
 Get state for step object. More...
 

Private Member Functions

Real computeCriticalityMeasure (Vector< Real > &x, Objective< Real > &obj, BoundConstraint< Real > &con, Real tol)
 Compute the gradient-based criticality measure. More...
 

Private Attributes

ROL::Ptr< Krylov< Real > > krylov_
 
int iterCR_
 CR iteration counter. More...
 
int flagCR_
 CR termination flag. More...
 
Real itol_
 Inexact CR tolerance. More...
 
int maxit_
 Maximum number of PDAS iterations. More...
 
int iter_
 PDAS iteration counter. More...
 
int flag_
 PDAS termination flag. More...
 
Real stol_
 PDAS minimum step size stopping tolerance. More...
 
Real gtol_
 PDAS gradient stopping tolerance. More...
 
Real scale_
 Scale for dual variables in the active set, \(c\). More...
 
Real neps_
 \(\epsilon\)-active set parameter More...
 
bool feasible_
 Flag whether the current iterate is feasible or not. More...
 
ROL::Ptr< Vector< Real > > lambda_
 Container for dual variables. More...
 
ROL::Ptr< Vector< Real > > xlam_
 Container for primal plus dual variables. More...
 
ROL::Ptr< Vector< Real > > x0_
 Container for initial priaml variables. More...
 
ROL::Ptr< Vector< Real > > xbnd_
 Container for primal variable bounds. More...
 
ROL::Ptr< Vector< Real > > As_
 Container for step projected onto active set. More...
 
ROL::Ptr< Vector< Real > > xtmp_
 Container for temporary primal storage. More...
 
ROL::Ptr< Vector< Real > > res_
 Container for optimality system residual for quadratic model. More...
 
ROL::Ptr< Vector< Real > > Ag_
 Container for gradient projected onto active set. More...
 
ROL::Ptr< Vector< Real > > rtmp_
 Container for temporary right hand side storage. More...
 
ROL::Ptr< Vector< Real > > gtmp_
 Container for temporary gradient storage. More...
 
ESecant esec_
 Enum for secant type. More...
 
ROL::Ptr< Secant< Real > > secant_
 Secant object. More...
 
bool useSecantPrecond_
 
bool useSecantHessVec_
 

Additional Inherited Members

- Protected Member Functions inherited from ROL::Step< Real >
ROL::Ptr< StepState< Real > > getState (void)
 

Detailed Description

template<class Real>
class ROL::PrimalDualActiveSetStep< Real >

Implements the computation of optimization steps with the Newton primal-dual active set method.

To describe primal-dual active set (PDAS), we consider the following abstract setting. Suppose \(\mathcal{X}\) is a Hilbert space of functions mapping \(\Xi\) to \(\mathbb{R}\). For example, \(\Xi\subset\mathbb{R}^n\) and \(\mathcal{X}=L^2(\Xi)\) or \(\Xi = \{1,\ldots,n\}\) and \(\mathcal{X}=\mathbb{R}^n\). We assume \( f:\mathcal{X}\to\mathbb{R}\) is twice-continuously Fréchet differentiable and \(a,\,b\in\mathcal{X}\) with \(a\le b\) almost everywhere in \(\Xi\). Note that the PDAS algorithm will also work with secant approximations of the Hessian.

Traditionally, PDAS is an algorithm for the minimizing quadratic objective functions subject to bound constraints. ROL implements a Newton PDAS which extends PDAS to general bound-constrained nonlinear programs, i.e.,

\[ \min_x \quad f(x) \quad \text{s.t.} \quad a \le x \le b. \]

Given the \(k\)-th iterate \(x_k\), the Newton PDAS algorithm computes steps by applying PDAS to the quadratic subproblem

\[ \min_s \quad \langle \nabla^2 f(x_k)s + \nabla f(x_k),s \rangle_{\mathcal{X}} \quad \text{s.t.} \quad a \le x_k + s \le b. \]

For the \(k\)-th quadratic subproblem, PDAS builds an approximation of the active set \(\mathcal{A}_k\) using the dual variable \(\lambda_k\) as

\[ \mathcal{A}^+_k = \{\,\xi\in\Xi\,:\,(\lambda_k + c(x_k-b))(\xi) > 0\,\}, \quad \mathcal{A}^-_k = \{\,\xi\in\Xi\,:\,(\lambda_k + c(x_k-a))(\xi) < 0\,\}, \quad\text{and}\quad \mathcal{A}_k = \mathcal{A}^-_k\cup\mathcal{A}^+_k. \]

We define the inactive set \(\mathcal{I}_k=\Xi\setminus\mathcal{A}_k\). The solution to the quadratic subproblem is then computed iteratively by solving

\[ \nabla^2 f(x_k) s_k + \lambda_{k+1} = -\nabla f(x_k), \quad x_k+s_k = a \;\text{on}\;\mathcal{A}^-_k,\quad x_k+s_k = b\;\text{on}\;\mathcal{A}^+_k, \quad\text{and}\quad \lambda_{k+1} = 0\;\text{on}\;\mathcal{I}_k \]

and updating the active and inactive sets.

One can rewrite this system by consolidating active and inactive parts, i.e.,

\[ \begin{pmatrix} \nabla^2 f(x_k)_{\mathcal{A}_k,\mathcal{A}_k} & \nabla^2 f(x_k)_{\mathcal{A}_k,\mathcal{I}_k} \\ \nabla^2 f(x_k)_{\mathcal{I}_k,\mathcal{A}_k} & \nabla^2 f(x_k)_{\mathcal{I}_k,\mathcal{I}_k} \end{pmatrix} \begin{pmatrix} (s_k)_{\mathcal{A}_k} \\ (s_k)_{\mathcal{I}_k} \end{pmatrix} + \begin{pmatrix} (\lambda_{k+1})_{\mathcal{A}_k} \\ 0 \end{pmatrix} = - \begin{pmatrix} \nabla f(x_k)_{\mathcal{A}_k}\\ \nabla f(x_k)_{\mathcal{I}_k} \end{pmatrix}. \]

Here the subscripts \(\mathcal{A}_k\) and \(\mathcal{I}_k\) denote the active and inactive components, respectively. Moreover, the active components of \(s_k\) are \(s_k(\xi) = a(\xi)-x_k(\xi)\) if \(\xi\in\mathcal{A}^-_k\) and \(s_k(\xi) = b(\xi)-x_k(\xi)\) if \(\xi\in\mathcal{A}^+_k\). Since \((s_k)_{\mathcal{A}_k}\) is fixed, we only need to solve for the inactive components of \(s_k\) which we can do this using conjugate residuals (CR) (i.e., the Hessian operator corresponding to the inactive indices may not be positive definite). Once \((s_k)_{\mathcal{I}_k}\) is computed, it is straight forward to update the dual variables.

Definition at line 98 of file ROL_PrimalDualActiveSetStep.hpp.

Constructor & Destructor Documentation

template<class Real >
ROL::PrimalDualActiveSetStep< Real >::PrimalDualActiveSetStep ( ROL::ParameterList &  parlist)
inline

Member Function Documentation

template<class Real >
Real ROL::PrimalDualActiveSetStep< Real >::computeCriticalityMeasure ( Vector< Real > &  x,
Objective< Real > &  obj,
BoundConstraint< Real > &  con,
Real  tol 
)
inlineprivate

Compute the gradient-based criticality measure.

  The criticality measure is 

\(\|x_k - P_{[a,b]}(x_k-\nabla f(x_k))\|_{\mathcal{X}}\). Here, \(P_{[a,b]}\) denotes the projection onto the bound constraints.

Parameters
[in]xis the current iteration
[in]objis the objective function
[in]conare the bound constraints
[in]tolis a tolerance for inexact evaluations of the objective function

Definition at line 227 of file ROL_PrimalDualActiveSetStep.hpp.

References ROL::Step< Real >::getState(), ROL::Objective< Real >::gradient(), ROL::BoundConstraint< Real >::project(), and ROL::PrimalDualActiveSetStep< Real >::xtmp_.

Referenced by ROL::PrimalDualActiveSetStep< Real >::initialize(), and ROL::PrimalDualActiveSetStep< Real >::update().

template<class Real >
void ROL::PrimalDualActiveSetStep< Real >::initialize ( Vector< Real > &  x,
const Vector< Real > &  s,
const Vector< Real > &  g,
Objective< Real > &  obj,
BoundConstraint< Real > &  con,
AlgorithmState< Real > &  algo_state 
)
inlinevirtual
template<class Real >
void ROL::PrimalDualActiveSetStep< Real >::compute ( Vector< Real > &  s,
const Vector< Real > &  x,
Objective< Real > &  obj,
BoundConstraint< Real > &  bnd,
AlgorithmState< Real > &  algo_state 
)
inlinevirtual

Compute step.

Reimplemented from ROL::Step< Real >.

Definition at line 345 of file ROL_PrimalDualActiveSetStep.hpp.

References ROL::PrimalDualActiveSetStep< Real >::Ag_, ROL::PrimalDualActiveSetStep< Real >::As_, ROL::PrimalDualActiveSetStep< Real >::flag_, ROL::PrimalDualActiveSetStep< Real >::flagCR_, ROL::BoundConstraint< Real >::getLowerBound(), ROL::Step< Real >::getState(), ROL::BoundConstraint< Real >::getUpperBound(), ROL::AlgorithmState< Real >::gnorm, ROL::PrimalDualActiveSetStep< Real >::gtmp_, ROL::PrimalDualActiveSetStep< Real >::gtol_, ROL::Objective< Real >::hessVec(), ROL::PrimalDualActiveSetStep< Real >::iter_, ROL::AlgorithmState< Real >::iterateVec, ROL::PrimalDualActiveSetStep< Real >::iterCR_, ROL::PrimalDualActiveSetStep< Real >::itol_, ROL::PrimalDualActiveSetStep< Real >::krylov_, ROL::PrimalDualActiveSetStep< Real >::lambda_, ROL::PrimalDualActiveSetStep< Real >::maxit_, ROL::PrimalDualActiveSetStep< Real >::neps_, ROL::Vector< Real >::norm(), ROL::Vector< Real >::plus(), ROL::BoundConstraint< Real >::project(), ROL::BoundConstraint< Real >::pruneActive(), ROL::BoundConstraint< Real >::pruneLowerActive(), ROL::BoundConstraint< Real >::pruneUpperActive(), ROL::PrimalDualActiveSetStep< Real >::res_, ROL::PrimalDualActiveSetStep< Real >::rtmp_, ROL::PrimalDualActiveSetStep< Real >::scale_, ROL::PrimalDualActiveSetStep< Real >::secant_, ROL::PrimalDualActiveSetStep< Real >::stol_, ROL::PrimalDualActiveSetStep< Real >::useSecantHessVec_, ROL::PrimalDualActiveSetStep< Real >::useSecantPrecond_, ROL::PrimalDualActiveSetStep< Real >::x0_, ROL::PrimalDualActiveSetStep< Real >::xbnd_, ROL::PrimalDualActiveSetStep< Real >::xlam_, ROL::PrimalDualActiveSetStep< Real >::xtmp_, zero, and ROL::Vector< Real >::zero().

template<class Real >
void ROL::PrimalDualActiveSetStep< Real >::update ( Vector< Real > &  x,
const Vector< Real > &  s,
Objective< Real > &  obj,
BoundConstraint< Real > &  bnd,
AlgorithmState< Real > &  algo_state 
)
inlinevirtual
template<class Real >
std::string ROL::PrimalDualActiveSetStep< Real >::printHeader ( void  ) const
inlinevirtual

Print iterate header.

This function produces a string containing header information.

Reimplemented from ROL::Step< Real >.

Definition at line 510 of file ROL_PrimalDualActiveSetStep.hpp.

References ROL::PrimalDualActiveSetStep< Real >::maxit_.

Referenced by ROL::PrimalDualActiveSetStep< Real >::print().

template<class Real >
std::string ROL::PrimalDualActiveSetStep< Real >::printName ( void  ) const
inlinevirtual

Print step name.

This function produces a string containing the algorithmic step information.

Reimplemented from ROL::Step< Real >.

Definition at line 537 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::print().

template<class Real >
virtual std::string ROL::PrimalDualActiveSetStep< Real >::print ( AlgorithmState< Real > &  algo_state,
bool  print_header = false 
) const
inlinevirtual

Member Data Documentation

template<class Real >
ROL::Ptr<Krylov<Real> > ROL::PrimalDualActiveSetStep< Real >::krylov_
private
template<class Real >
int ROL::PrimalDualActiveSetStep< Real >::iterCR_
private
template<class Real >
int ROL::PrimalDualActiveSetStep< Real >::flagCR_
private
template<class Real >
Real ROL::PrimalDualActiveSetStep< Real >::itol_
private

Inexact CR tolerance.

Definition at line 106 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute().

template<class Real >
int ROL::PrimalDualActiveSetStep< Real >::maxit_
private
template<class Real >
int ROL::PrimalDualActiveSetStep< Real >::iter_
private
template<class Real >
int ROL::PrimalDualActiveSetStep< Real >::flag_
private
template<class Real >
Real ROL::PrimalDualActiveSetStep< Real >::stol_
private

PDAS minimum step size stopping tolerance.

Definition at line 112 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::PrimalDualActiveSetStep().

template<class Real >
Real ROL::PrimalDualActiveSetStep< Real >::gtol_
private
template<class Real >
Real ROL::PrimalDualActiveSetStep< Real >::scale_
private

Scale for dual variables in the active set, \(c\).

Definition at line 114 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::PrimalDualActiveSetStep().

template<class Real >
Real ROL::PrimalDualActiveSetStep< Real >::neps_
private

\(\epsilon\)-active set parameter

Definition at line 115 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute().

template<class Real >
bool ROL::PrimalDualActiveSetStep< Real >::feasible_
private

Flag whether the current iterate is feasible or not.

Definition at line 116 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::print(), and ROL::PrimalDualActiveSetStep< Real >::update().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::lambda_
private
template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::xlam_
private

Container for primal plus dual variables.

Definition at line 120 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::x0_
private

Container for initial priaml variables.

Definition at line 121 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::xbnd_
private

Container for primal variable bounds.

Definition at line 122 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::As_
private

Container for step projected onto active set.

Definition at line 123 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::xtmp_
private
template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::res_
private

Container for optimality system residual for quadratic model.

Definition at line 125 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::Ag_
private

Container for gradient projected onto active set.

Definition at line 126 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::rtmp_
private

Container for temporary right hand side storage.

Definition at line 127 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::compute(), and ROL::PrimalDualActiveSetStep< Real >::initialize().

template<class Real >
ROL::Ptr<Vector<Real> > ROL::PrimalDualActiveSetStep< Real >::gtmp_
private
template<class Real >
ESecant ROL::PrimalDualActiveSetStep< Real >::esec_
private

Enum for secant type.

Definition at line 131 of file ROL_PrimalDualActiveSetStep.hpp.

Referenced by ROL::PrimalDualActiveSetStep< Real >::PrimalDualActiveSetStep().

template<class Real >
ROL::Ptr<Secant<Real> > ROL::PrimalDualActiveSetStep< Real >::secant_
private
template<class Real >
bool ROL::PrimalDualActiveSetStep< Real >::useSecantPrecond_
private
template<class Real >
bool ROL::PrimalDualActiveSetStep< Real >::useSecantHessVec_
private

The documentation for this class was generated from the following file: