Active Set method. More...

#include <active_set.h>

Public Types
typedef TAlgebra	algebra_type
	Type of algebra. More...

typedef GridFunction< TDomain, TAlgebra >	function_type
	Type of grid function. More...

typedef algebra_type::matrix_type	matrix_type
	Type of algebra matrix. More...

typedef domain_traits< TDomain::dim >::grid_base_object	TBaseElem
	base element type of associated domain More...

typedef vector_type::value_type	value_type
	Type of algebra value. More...

typedef algebra_type::vector_type	vector_type
	Type of algebra vector. More...

Public Member Functions
bool	active_index (function_type &u, function_type &rhs, function_type &lagrangeMult, function_type &gap)
	determines the active indices, stores them in a vector and sets dirichlet values in rhs for active indices More...

template<typename TElem , typename TIterator >
void	active_index_elem (TIterator iterBegin, TIterator iterEnd, function_type &u, function_type &rhs, function_type &lagrangeMult)
	checks the distance to the prescribed obstacle/constraint More...

	ActiveSet ()
	constructor More...

bool	check_conv (function_type &u, const function_type &lambda, const size_t step)
	checks if all constraints are fulfilled & the activeSet remained unchanged More...

template<typename TElem , typename TIterator >
bool	check_conv_elem (TIterator iterBegin, TIterator iterEnd, function_type &u, const function_type &lambda)

bool	check_inequ (const matrix_type &mat, const vector_type &u, const vector_type &lagrangeMult, const vector_type &rhs)
	checks if all inequalities are fulfilled More...

void	lagrange_mat_inv (matrix_type &lagrangeMatInv)

template<typename TElem , typename TIterator >
void	lagrange_mat_inv_elem (TIterator iterBegin, TIterator iterEnd, matrix_type &lagrangeMatInv)

void	lagrange_multiplier (function_type &lagrangeMult, const function_type &u)
	computes the lagrange multiplier for a given disc More...

void	prepare (function_type &u)

void	residual_lagrange_mult (vector_type &lagMult, const matrix_type &mat, const vector_type &u, vector_type &rhs)

void	set_dirichlet_rows (matrix_type &mat)

void	set_lagrange_multiplier_disc (SmartPtr< ILagrangeMultiplierDisc< TDomain, function_type > > lagMultDisc)
	sets a discretization in order to compute the lagrange multiplier More...

void	set_obstacle (ConstSmartPtr< function_type > obs)
	sets obstacle gridfunction, which limits the solution u More...

Static Public Attributes
static const int	dim = TDomain::dim
	domain dimension More...

Private Attributes
bool	m_bObs

SmartPtr< DoFDistribution >	m_spDD
	pointer to the DofDistribution on the whole domain More...

SmartPtr< TDomain >	m_spDom

SmartPtr< ILagrangeMultiplierDisc< TDomain, function_type > >	m_spLagMultDisc
	pointer to a lagrangeMultiplier-Disc More...

ConstSmartPtr< function_type >	m_spObs
	number of functions More...

vector< DoFIndex >	m_vActiveSetGlob
	vector of the current active set of global DoFIndices More...

vector< DoFIndex >	m_vActiveSetGlobOld
	vector remembering the active set of global DoFIndices More...

vector< int >	m_vActiveSubsets
	vector of possible active subsets More...

Detailed Description

template<typename TDomain, typename TAlgebra>
class ug::ActiveSet< TDomain, TAlgebra >

Active Set method.

The active Set method is a well-known method in constrained optimization theory. A general formulation for these problems reads

\begin{eqnarray*} min_{x \in \mathbb{R}^n} f(x) \end{eqnarray*}

s.t.

\begin{eqnarray*} c_i(x) = 0, \qquad i \in E \\ c_i(x) \ge 0 \qquad i \in I, \end{eqnarray*}

where \( f \), \( c_i \) are smooth, real-valued functions on a subset of \( \mathbb{R}^n \). \( I \) (set of inequality constraints) and \( E \) (set of equality constraints) are two finite sets of indices. \( f \) is called objective function.

The active Set \( A \) is defined as:

\begin{eqnarray*} A(x) := E \cup \{ i \in I | c_i(x) = 0 \}, \end{eqnarray*}

i.e. for \( i \in I \) the inequality constraint is said to be active, if \( c_i(x) = 0 \). Otherwise ( \( c_i(x) > 0 \)) it is called inactive.

A common approach to treat the inequality constraints is its reformulation as equations by using so called complementarity functions, see e.g. C.Hager und B. I. Wohlmuth: "Hindernis- und Kontaktprobleme" for a simple introduction into this topic. By means of complementarity functions, constraints of the form

\begin{eqnarray*} a \ge 0, \, b \ge 0, \, a b = 0 \end{eqnarray*}

with \( a, b \in \mathbb{R}^n \) can be reformulated as

\begin{eqnarray*} C(a,b) = 0, \end{eqnarray*}

with \( C: \mathbb{R}^n x \mathbb{R}^n \to \mathbb{R}^n \) being an appropriate complementarity function. The value of \( C \) indicates, whether the index is active or inactive (see method 'active_index'). Thus, it determines in every step the set of active indices, for which the original system of equations needs to be adapted. The influence of these active indices on the original system of equations ( \( K u = f \), with \( K \): system-matrix; \( u, f \) vectors) can be modelled by means of a lagrange multiplier ' \( \lambda \)'. \( \lambda \) can either be defined by the residual ( \( \lambda := f - K u \), see method 'residual_lagrange_mult') or by a problem-dependent computation (see method 'lagrange_multiplier').

In every Active Set step a linear or linearized system is solved. The algorithm stops when the active and inactive Set remains unchanged (see method 'check_conv').

References:

J. Nocedal and S. J. Wright. Numerical optimization.(2000)

Template Parameters

TDomain	Domain type
TAlgebra	Algebra type