Implement a gradient descent optimizer. More...

#include <itkRegularStepGradientDescentOptimizer.h>

Inheritance diagram for itk::RegularStepGradientDescentOptimizer:

Collaboration diagram for itk::RegularStepGradientDescentOptimizer:

Public Types
typedef SmartPointer< const Self >	ConstPointer
typedef CostFunctionType::Pointer	CostFunctionPointer
typedef Superclass::CostFunctionType	CostFunctionType
typedef CostFunctionType::DerivativeType	DerivativeType
typedef CostFunctionType::MeasureType	MeasureType
typedef Superclass::ParametersType	ParametersType
typedef SmartPointer< Self >	Pointer
typedef Superclass::ScalesType	ScalesType
typedef RegularStepGradientDescentOptimizer	Self
enum	StopConditionType { GradientMagnitudeTolerance = 1, StepTooSmall = 2, ImageNotAvailable = 3, CostFunctionError = 4, MaximumNumberOfIterations = 5, Unknown = 6 }
typedef RegularStepGradientDescentBaseOptimizer	Superclass
Public Member Functions
virtual LightObject::Pointer	CreateAnother () const
virtual void	DebugOff () const
virtual void	DebugOn () const
virtual void	Delete ()
Command *	GetCommand (unsigned long tag)
virtual const CostFunctionType *	GetCostFunction ()
virtual const ParametersType &	GetCurrentPosition ()
bool	GetDebug () const
virtual const ParametersType &	GetInitialPosition ()
const MetaDataDictionary &	GetMetaDataDictionary (void) const
MetaDataDictionary &	GetMetaDataDictionary (void)
virtual unsigned long	GetMTime () const
virtual const char *	GetNameOfClass () const
virtual int	GetReferenceCount () const
virtual const ScalesType &	GetScales ()
virtual const std::string	GetStopConditionDescription () const
MeasureType	GetValue (const ParametersType &parameters) const
bool	HasObserver (const EventObject &event) const
void	InvokeEvent (const EventObject &)
void	InvokeEvent (const EventObject &) const
virtual void	Modified () const
void	Print (std::ostream &os, Indent indent=0) const
virtual void	Register () const
void	RemoveAllObservers ()
void	RemoveObserver (unsigned long tag)
void	ResumeOptimization (void)
virtual void	SetCostFunction (CostFunctionType *costFunction)
void	SetDebug (bool debugFlag) const
virtual void	SetInitialPosition (const ParametersType &param)
void	SetMetaDataDictionary (const MetaDataDictionary &rhs)
virtual void	SetReferenceCount (int)
void	SetScales (const ScalesType &scales)
void	StartOptimization (void)
void	StopOptimization (void)
virtual void	UnRegister () const


virtual void	SetMaximize (bool _arg)
virtual const bool &	GetMaximize ()
virtual void	MaximizeOn ()
virtual void	MaximizeOff ()
bool	GetMinimize () const
void	SetMinimize (bool v)
void	MinimizeOn (void)
void	MinimizeOff (void)


virtual void	SetMaximumStepLength (double _arg)
virtual void	SetMinimumStepLength (double _arg)
virtual void	SetRelaxationFactor (double _arg)
virtual void	SetNumberOfIterations (unsigned long _arg)
virtual void	SetGradientMagnitudeTolerance (double _arg)
virtual const double &	GetCurrentStepLength ()
virtual const double &	GetMaximumStepLength ()
virtual const double &	GetMinimumStepLength ()
virtual const double &	GetRelaxationFactor ()
virtual const unsigned long &	GetNumberOfIterations ()
virtual const double &	GetGradientMagnitudeTolerance ()
virtual unsigned int	GetCurrentIteration () const
virtual const StopConditionType &	GetStopCondition ()
virtual const MeasureType &	GetValue ()
virtual const DerivativeType &	GetGradient ()


unsigned long	AddObserver (const EventObject &event, Command *)
unsigned long	AddObserver (const EventObject &event, Command *) const
Static Public Member Functions
static void	BreakOnError ()
static Pointer	New ()


static void	SetGlobalWarningDisplay (bool flag)
static bool	GetGlobalWarningDisplay ()
static void	GlobalWarningDisplayOn ()
static void	GlobalWarningDisplayOff ()
Protected Types


typedef int	InternalReferenceCountType
Protected Member Functions
virtual void	AdvanceOneStep (void)
bool	PrintObservers (std::ostream &os, Indent indent) const
void	PrintSelf (std::ostream &os, Indent indent) const
	RegularStepGradientDescentOptimizer ()
virtual void	SetCurrentPosition (const ParametersType &param)
virtual void	StepAlongGradient (double factor, const DerivativeType &transformedGradient)
virtual	~RegularStepGradientDescentOptimizer ()


virtual void	StepAlongGradient (double, const DerivativeType &)


virtual void	PrintHeader (std::ostream &os, Indent indent) const
virtual void	PrintTrailer (std::ostream &os, Indent indent) const
Protected Attributes
CostFunctionPointer	m_CostFunction
unsigned long	m_CurrentIteration
ParametersType	m_CurrentPosition
double	m_CurrentStepLength
DerivativeType	m_Gradient
double	m_GradientMagnitudeTolerance
bool	m_Maximize
double	m_MaximumStepLength
double	m_MinimumStepLength
unsigned long	m_NumberOfIterations
DerivativeType	m_PreviousGradient
InternalReferenceCountType	m_ReferenceCount
SimpleFastMutexLock	m_ReferenceCountLock
double	m_RelaxationFactor
bool	m_ScalesInitialized
bool	m_Stop
StopConditionType	m_StopCondition
OStringStream	m_StopConditionDescription
MeasureType	m_Value

Detailed Description

Implement a gradient descent optimizer.

Definition at line 31 of file itkRegularStepGradientDescentOptimizer.h.

Member Typedef Documentation

typedef SmartPointer<const Self> itk::RegularStepGradientDescentOptimizer::ConstPointer

Reimplemented from itk::RegularStepGradientDescentBaseOptimizer.

Definition at line 39 of file itkRegularStepGradientDescentOptimizer.h.

typedef CostFunctionType::Pointer itk::RegularStepGradientDescentOptimizer::CostFunctionPointer

Reimplemented from itk::SingleValuedNonLinearOptimizer.

Definition at line 50 of file itkRegularStepGradientDescentOptimizer.h.

typedef Superclass::CostFunctionType itk::RegularStepGradientDescentOptimizer::CostFunctionType

Cost function typedefs.

Reimplemented from itk::SingleValuedNonLinearOptimizer.

Definition at line 46 of file itkRegularStepGradientDescentOptimizer.h.

typedef CostFunctionType::DerivativeType itk::SingleValuedNonLinearOptimizer::DerivativeType [inherited]

Derivative type. It defines a type used to return the cost function derivative.

Definition at line 64 of file itkSingleValuedNonLinearOptimizer.h.

typedef int itk::LightObject::InternalReferenceCountType [protected, inherited]

Define the type of the reference count according to the target. This allows the use of atomic operations

Definition at line 139 of file itkLightObject.h.

typedef CostFunctionType::MeasureType itk::SingleValuedNonLinearOptimizer::MeasureType [inherited]

Measure type. It defines a type used to return the cost function value.

Definition at line 60 of file itkSingleValuedNonLinearOptimizer.h.

typedef Superclass::ParametersType itk::SingleValuedNonLinearOptimizer::ParametersType [inherited]

Parameters type. It defines a position in the optimization search space.

Reimplemented from itk::NonLinearOptimizer.

Reimplemented in itk::AmoebaOptimizer, itk::FRPROptimizer, itk::PowellOptimizer, and itk::QuaternionRigidTransformGradientDescentOptimizer.

Definition at line 48 of file itkSingleValuedNonLinearOptimizer.h.

typedef SmartPointer<Self> itk::RegularStepGradientDescentOptimizer::Pointer

Reimplemented from itk::RegularStepGradientDescentBaseOptimizer.

Definition at line 38 of file itkRegularStepGradientDescentOptimizer.h.

typedef Superclass::ScalesType itk::NonLinearOptimizer::ScalesType [inherited]

Scale type. This array defines scale to be applied to parameters before being evaluated in the cost function. This allows to map to a more convenient space. In particular this is used to normalize parameter spaces in which some parameters have a different dynamic range.

Reimplemented from itk::Optimizer.

Definition at line 52 of file itkNonLinearOptimizer.h.

typedef RegularStepGradientDescentOptimizer itk::RegularStepGradientDescentOptimizer::Self

Standard class typedefs.

Reimplemented from itk::RegularStepGradientDescentBaseOptimizer.

Definition at line 36 of file itkRegularStepGradientDescentOptimizer.h.

typedef RegularStepGradientDescentBaseOptimizer itk::RegularStepGradientDescentOptimizer::Superclass

Reimplemented from itk::RegularStepGradientDescentBaseOptimizer.

Definition at line 37 of file itkRegularStepGradientDescentOptimizer.h.

Member Enumeration Documentation

enum itk::RegularStepGradientDescentBaseOptimizer::StopConditionType [inherited]

Codes of stopping conditions.

Enumerator:

GradientMagnitudeTolerance
StepTooSmall
ImageNotAvailable
CostFunctionError
MaximumNumberOfIterations
Unknown

Definition at line 49 of file itkRegularStepGradientDescentBaseOptimizer.h.

Constructor & Destructor Documentation

itk::RegularStepGradientDescentOptimizer::RegularStepGradientDescentOptimizer ( ) [inline, protected]

Definition at line 54 of file itkRegularStepGradientDescentOptimizer.h.

virtual itk::RegularStepGradientDescentOptimizer::~RegularStepGradientDescentOptimizer ( ) [inline, protected, virtual]

Definition at line 55 of file itkRegularStepGradientDescentOptimizer.h.

Member Function Documentation

unsigned long itk::Object::AddObserver	(	const EventObject &	event,
		Command *
	)			`[inherited]`

Allow people to add/remove/invoke observers (callbacks) to any ITK object. This is an implementation of the subject/observer design pattern. An observer is added by specifying an event to respond to and an itk::Command to execute. It returns an unsigned long tag which can be used later to remove the event or retrieve the command. The memory for the Command becomes the responsibility of this object, so don't pass the same instance of a command to two different objects

unsigned long itk::Object::AddObserver	(	const EventObject &	event,
		Command *
	)			const `[inherited]`

Allow people to add/remove/invoke observers (callbacks) to any ITK object. This is an implementation of the subject/observer design pattern. An observer is added by specifying an event to respond to and an itk::Command to execute. It returns an unsigned long tag which can be used later to remove the event or retrieve the command. The memory for the Command becomes the responsibility of this object, so don't pass the same instance of a command to two different objects

virtual void itk::RegularStepGradientDescentBaseOptimizer::AdvanceOneStep ( void ) [protected, virtual, inherited]

Advance one step following the gradient direction This method verifies if a change in direction is required and if a reduction in steplength is required.

static void itk::LightObject::BreakOnError ( ) [static, inherited]

This method is called when itkExceptionMacro executes. It allows the debugger to break on error.

virtual LightObject::Pointer itk::Object::CreateAnother ( ) const [virtual, inherited]

Create an object from an instance, potentially deferring to a factory. This method allows you to create an instance of an object that is exactly the same type as the referring object. This is useful in cases where an object has been cast back to a base class.

Reimplemented from itk::LightObject.

virtual void itk::Object::DebugOff ( ) const [virtual, inherited]

Turn debugging output off.

virtual void itk::Object::DebugOn ( ) const [virtual, inherited]

Turn debugging output on.

virtual void itk::LightObject::Delete ( ) [virtual, inherited]

Delete an itk object. This method should always be used to delete an object when the new operator was used to create it. Using the C delete method will not work with reference counting.

Command* itk::Object::GetCommand ( unsigned long tag ) [inherited]

Get the command associated with the given tag. NOTE: This returns a pointer to a Command, but it is safe to asign this to a Command::Pointer. Since Command inherits from LightObject, at this point in the code, only a pointer or a reference to the Command can be used.

virtual const CostFunctionType* itk::SingleValuedNonLinearOptimizer::GetCostFunction ( ) [virtual, inherited]

Get the cost function.

virtual unsigned int itk::RegularStepGradientDescentBaseOptimizer::GetCurrentIteration ( ) const [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual const ParametersType& itk::Optimizer::GetCurrentPosition ( ) [virtual, inherited]

Get current position of the optimization.

virtual const double& itk::RegularStepGradientDescentBaseOptimizer::GetCurrentStepLength ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

bool itk::Object::GetDebug ( ) const [inherited]

Get the value of the debug flag.

static bool itk::Object::GetGlobalWarningDisplay ( ) [static, inherited]

This is a global flag that controls whether any debug, warning or error messages are displayed.

virtual const DerivativeType& itk::RegularStepGradientDescentBaseOptimizer::GetGradient ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual const double& itk::RegularStepGradientDescentBaseOptimizer::GetGradientMagnitudeTolerance ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual const ParametersType& itk::Optimizer::GetInitialPosition ( ) [virtual, inherited]

Get the position to initialize the optimization.

virtual const bool& itk::RegularStepGradientDescentBaseOptimizer::GetMaximize ( ) [virtual, inherited]

Specify whether to minimize or maximize the cost function.

virtual const double& itk::RegularStepGradientDescentBaseOptimizer::GetMaximumStepLength ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

const MetaDataDictionary& itk::Object::GetMetaDataDictionary ( void ) const [inherited]

Returns:: A constant reference to this objects MetaDataDictionary.

MetaDataDictionary& itk::Object::GetMetaDataDictionary ( void ) [inherited]

Returns:: A reference to this objects MetaDataDictionary.

Warning:: This reference may be changed.

bool itk::RegularStepGradientDescentBaseOptimizer::GetMinimize ( ) const [inline, inherited]

Specify whether to minimize or maximize the cost function.

Definition at line 62 of file itkRegularStepGradientDescentBaseOptimizer.h.

virtual const double& itk::RegularStepGradientDescentBaseOptimizer::GetMinimumStepLength ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual unsigned long itk::Object::GetMTime ( ) const [virtual, inherited]

Return this objects modified time.

Referenced by itk::SpatialObject< ::itk::GetMeshDimension< TMesh >::PointDimension >::GetObjectMTime().

virtual const char* itk::RegularStepGradientDescentOptimizer::GetNameOfClass ( ) const [virtual]

Run-time type information (and related methods).

Reimplemented from itk::RegularStepGradientDescentBaseOptimizer.

virtual const unsigned long& itk::RegularStepGradientDescentBaseOptimizer::GetNumberOfIterations ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual int itk::LightObject::GetReferenceCount ( ) const [inline, virtual, inherited]

Gets the reference count on this object.

Definition at line 106 of file itkLightObject.h.

virtual const double& itk::RegularStepGradientDescentBaseOptimizer::GetRelaxationFactor ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual const ScalesType& itk::Optimizer::GetScales ( ) [virtual, inherited]

Get current parameters scaling.

virtual const StopConditionType& itk::RegularStepGradientDescentBaseOptimizer::GetStopCondition ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual const std::string itk::RegularStepGradientDescentBaseOptimizer::GetStopConditionDescription ( ) const [virtual, inherited]

Get the reason for termination

Reimplemented from itk::Optimizer.

MeasureType itk::SingleValuedNonLinearOptimizer::GetValue ( const ParametersType & parameters ) const [inherited]

Get the cost function value at the given parameters.

Reimplemented in itk::SPSAOptimizer.

virtual const MeasureType& itk::RegularStepGradientDescentBaseOptimizer::GetValue ( ) [virtual, inherited]

Set/Get parameters to control the optimization process.

static void itk::Object::GlobalWarningDisplayOff ( ) [inline, static, inherited]

This is a global flag that controls whether any debug, warning or error messages are displayed.

Definition at line 100 of file itkObject.h.

References itk::Object::SetGlobalWarningDisplay().

static void itk::Object::GlobalWarningDisplayOn ( ) [inline, static, inherited]

This is a global flag that controls whether any debug, warning or error messages are displayed.

Definition at line 98 of file itkObject.h.

References itk::Object::SetGlobalWarningDisplay().

bool itk::Object::HasObserver ( const EventObject & event ) const [inherited]

Return true if an observer is registered for this event.

void itk::Object::InvokeEvent ( const EventObject & ) [inherited]

Call Execute on all the Commands observing this event id.

void itk::Object::InvokeEvent ( const EventObject & ) const [inherited]

Call Execute on all the Commands observing this event id. The actions triggered by this call doesn't modify this object.

virtual void itk::RegularStepGradientDescentBaseOptimizer::MaximizeOff ( ) [virtual, inherited]

Specify whether to minimize or maximize the cost function.

virtual void itk::RegularStepGradientDescentBaseOptimizer::MaximizeOn ( ) [virtual, inherited]

Specify whether to minimize or maximize the cost function.

void itk::RegularStepGradientDescentBaseOptimizer::MinimizeOff ( void ) [inline, inherited]

Specify whether to minimize or maximize the cost function.

Definition at line 68 of file itkRegularStepGradientDescentBaseOptimizer.h.

void itk::RegularStepGradientDescentBaseOptimizer::MinimizeOn ( void ) [inline, inherited]

Specify whether to minimize or maximize the cost function.

Definition at line 66 of file itkRegularStepGradientDescentBaseOptimizer.h.

virtual void itk::Object::Modified ( ) const [virtual, inherited]

Update the modification time for this object. Many filters rely on the modification time to determine if they need to recompute their data.

Referenced by itk::NarrowBandImageFilterBase< TInputImage, Image< TOutputPixelType,::itk::GetImageDimension< TInputImage >::ImageDimension > >::InsertNarrowBandNode(), itk::MatrixOffsetTransformBase< TScalarType, 3, 3 >::SetCenter(), itk::MatrixOffsetTransformBase< TScalarType, 3, 3 >::SetMatrix(), itk::NarrowBandImageFilterBase< TInputImage, Image< TOutputPixelType,::itk::GetImageDimension< TInputImage >::ImageDimension > >::SetNarrowBand(), itk::NarrowBandImageFilterBase< TInputImage, Image< TOutputPixelType,::itk::GetImageDimension< TInputImage >::ImageDimension > >::SetNarrowBandInnerRadius(), itk::NarrowBandImageFilterBase< TInputImage, Image< TOutputPixelType,::itk::GetImageDimension< TInputImage >::ImageDimension > >::SetNarrowBandTotalRadius(), itk::MatrixOffsetTransformBase< TScalarType, 3, 3 >::SetOffset(), itk::ThresholdLabelerImageFilter< TInputImage, TOutputImage >::SetRealThresholds(), itk::ThresholdLabelerImageFilter< TInputImage, TOutputImage >::SetThresholds(), itk::Statistics::GoodnessOfFitFunctionBase< TInputHistogram >::SetTotalObservedScale(), and itk::MatrixOffsetTransformBase< TScalarType, 3, 3 >::SetTranslation().

static Pointer itk::RegularStepGradientDescentOptimizer::New ( ) [static]

Method for creation through the object factory.

Reimplemented from itk::RegularStepGradientDescentBaseOptimizer.

void itk::LightObject::Print	(	std::ostream &	os,
		Indent	indent = `0`
	)			const `[inherited]`

Cause the object to print itself out.

Referenced by itk::WeakPointer< ProcessObject >::Print().

virtual void itk::LightObject::PrintHeader	(	std::ostream &	os,
		Indent	indent
	)			const `[protected, virtual, inherited]`

Methods invoked by Print() to print information about the object including superclasses. Typically not called by the user (use Print() instead) but used in the hierarchical print process to combine the output of several classes.

bool itk::Object::PrintObservers	(	std::ostream &	os,
		Indent	indent
	)			const `[protected, inherited]`

void itk::RegularStepGradientDescentBaseOptimizer::PrintSelf	(	std::ostream &	os,
		Indent	indent
	)			const `[protected, virtual, inherited]`

Methods invoked by Print() to print information about the object including superclasses. Typically not called by the user (use Print() instead) but used in the hierarchical print process to combine the output of several classes.

Reimplemented from itk::SingleValuedNonLinearOptimizer.

virtual void itk::LightObject::PrintTrailer	(	std::ostream &	os,
		Indent	indent
	)			const `[protected, virtual, inherited]`

Methods invoked by Print() to print information about the object including superclasses. Typically not called by the user (use Print() instead) but used in the hierarchical print process to combine the output of several classes.

virtual void itk::Object::Register ( ) const [virtual, inherited]

Increase the reference count (mark as used by another object).

Reimplemented from itk::LightObject.

void itk::Object::RemoveAllObservers ( ) [inherited]

Remove all observers .

void itk::Object::RemoveObserver ( unsigned long tag ) [inherited]

Remove the observer with this tag value.

void itk::RegularStepGradientDescentBaseOptimizer::ResumeOptimization ( void ) [inherited]

Resume previously stopped optimization with current parameters.

See also:: StopOptimization

virtual void itk::SingleValuedNonLinearOptimizer::SetCostFunction ( CostFunctionType * costFunction ) [virtual, inherited]

Set the cost function.

virtual void itk::Optimizer::SetCurrentPosition ( const ParametersType & param ) [protected, virtual, inherited]

Set the current position.

void itk::Object::SetDebug ( bool debugFlag ) const [inherited]

Set the value of the debug flag. A non-zero value turns debugging on.

static void itk::Object::SetGlobalWarningDisplay ( bool flag ) [static, inherited]

This is a global flag that controls whether any debug, warning or error messages are displayed.

Referenced by itk::Object::GlobalWarningDisplayOff(), and itk::Object::GlobalWarningDisplayOn().

virtual void itk::RegularStepGradientDescentBaseOptimizer::SetGradientMagnitudeTolerance ( double _arg ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual void itk::Optimizer::SetInitialPosition ( const ParametersType & param ) [virtual, inherited]

Set the position to initialize the optimization.

virtual void itk::RegularStepGradientDescentBaseOptimizer::SetMaximize ( bool _arg ) [virtual, inherited]

Specify whether to minimize or maximize the cost function.

virtual void itk::RegularStepGradientDescentBaseOptimizer::SetMaximumStepLength ( double _arg ) [virtual, inherited]

Set/Get parameters to control the optimization process.

void itk::Object::SetMetaDataDictionary ( const MetaDataDictionary & rhs ) [inherited]

Returns:: Set the MetaDataDictionary

void itk::RegularStepGradientDescentBaseOptimizer::SetMinimize ( bool v ) [inline, inherited]

Specify whether to minimize or maximize the cost function.

Definition at line 64 of file itkRegularStepGradientDescentBaseOptimizer.h.

virtual void itk::RegularStepGradientDescentBaseOptimizer::SetMinimumStepLength ( double _arg ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual void itk::RegularStepGradientDescentBaseOptimizer::SetNumberOfIterations ( unsigned long _arg ) [virtual, inherited]

Set/Get parameters to control the optimization process.

virtual void itk::Object::SetReferenceCount ( int ) [virtual, inherited]

Sets the reference count (use with care)

Reimplemented from itk::LightObject.

virtual void itk::RegularStepGradientDescentBaseOptimizer::SetRelaxationFactor ( double _arg ) [virtual, inherited]

Set/Get parameters to control the optimization process.

void itk::Optimizer::SetScales ( const ScalesType & scales ) [inherited]

Set current parameters scaling.

void itk::RegularStepGradientDescentBaseOptimizer::StartOptimization ( void ) [virtual, inherited]

Start optimization.

Reimplemented from itk::Optimizer.

virtual void itk::RegularStepGradientDescentBaseOptimizer::StepAlongGradient	(	double	,
		const DerivativeType &
	)			`[inline, protected, virtual, inherited]`

Advance one step along the corrected gradient taking into account the steplength represented by factor. This method is invoked by AdvanceOneStep. It is expected to be overrided by optimization methods in non-vector spaces

See also:: AdvanceOneStep

Definition at line 119 of file itkRegularStepGradientDescentBaseOptimizer.h.

virtual void itk::RegularStepGradientDescentOptimizer::StepAlongGradient	(	double	factor,
		const DerivativeType &	transformedGradient
	)			`[protected, virtual]`

Advance one step along the corrected gradient taking into account the steplength represented by factor. This method is invoked by AdvanceOneStep. It is expected to be overrided by optimization methods in non-vector spaces