Provides the interface to implement risk measures. More...

#include <ROL_RiskMeasure.hpp>

Public Member Functions
virtual	~RiskMeasure ()

	RiskMeasure (void)

void	setRiskVectorInfo (const int comp, const int index)

int	getComponent (void) const

int	getIndex (void) const

virtual void	reset (ROL::Ptr< Vector< Real > > &x0, const Vector< Real > &x)
	Reset internal risk measure storage. Called for value and gradient computation. More...

virtual void	reset (ROL::Ptr< Vector< Real > > &x0, const Vector< Real > &x, ROL::Ptr< Vector< Real > > &v0, const Vector< Real > &v)
	Reset internal risk measure storage. Called for Hessian-times-a-vector computation. More...

virtual void	update (const Real val, const Real weight)
	Update internal risk measure storage for value computation. More...

virtual void	update (const Real val, const Vector< Real > &g, const Real weight)
	Update internal risk measure storage for gradient computation. More...

virtual void	update (const Real val, const Vector< Real > &g, const Real gv, const Vector< Real > &hv, const Real weight)
	Update internal risk measure storage for Hessian-time-a-vector computation. More...

virtual Real	getValue (SampleGenerator< Real > &sampler)
	Return risk measure value. More...

virtual void	getGradient (Vector< Real > &g, SampleGenerator< Real > &sampler)
	Return risk measure (sub)gradient. More...

virtual void	getHessVec (Vector< Real > &hv, SampleGenerator< Real > &sampler)
	Return risk measure Hessian-times-a-vector. More...

Protected Attributes
Real	val_

Real	gv_

ROL::Ptr< Vector< Real > >	g_

ROL::Ptr< Vector< Real > >	hv_

ROL::Ptr< Vector< Real > >	dualVector_

bool	firstReset_

int	comp_

int	index_

Detailed Description

template<class Real>
class ROL::RiskMeasure< Real >

Provides the interface to implement risk measures.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a complete space. Here, \(\Omega\) is the set of outcomes, \(\mathcal{F}\subseteq 2^\Omega\) is a \(\sigma\)-algebra of events and \(\mathbb{P}:\mathcal{F}\to[0,1]\) is a probability measure. Moreover, let \(\mathcal{X}\) be a class of random variables. A risk measure is an extended real-valued functional that associates numerical values to random variables, i.e., \(\mathcal{R}:\mathcal{X}\to\mathbb{R}\cup\{+\infty\}\). In most cases, \(\mathcal{X} = L^p(\Omega,\mathcal{F},\mathbb{P})\) for some \(1\le p\le \infty\).

There are a number of classifications for risk measures. One important class are the coherent risk measures. \(\mathcal{R}\) is coherent if it satisfies the following four axioms:

Convexity: \(\mathcal{R}(\lambda X + (1-\lambda)X') \le \lambda \mathcal{R}(X) + (1-\lambda)\mathcal{R}(X')\) for all \(0 \le \lambda \le 1\);
Translation Equivariance: \(\mathcal{R}(X+C) = \mathcal{R}(X) + C\) for all constants \(C\);
Monotonicity: If \(X \le X'\) \(\mathbb{P}\) almost everywhere then \(\mathcal{R}(X) \le \mathcal{R}(X')\);
Positive Homogeneity: \(\mathcal{R}(tX) = t \mathcal{R}(X)\) for all \(t \ge 0\).

Another useful characterization is law invariance. \(\mathcal{R}\) is law invariant if \(\mathcal{R}(X) = \mathcal{R}(X')\) whenever \(\mathbb{P}(X\le t) = \mathbb{P}(X'\le t)\) for all \(t\in\mathbb{R}\). Law invariant risk measures are only functions of the distribution of the input random variable.

ROL's risk measure base class is written in a way to exploit parallel sampling. General risk measures may depend on global information such as the expected value of a random variable, \(\mathbb{E}[X]\). Thus, ROL::RiskMeasure contains functions to update intermediate information and to compute desired quantities such as risk values, gradients and Hessians applied to vectors.

Definition at line 62 of file ROL_RiskMeasure.hpp.

Constructor & Destructor Documentation

template<class Real >

virtual ROL::RiskMeasure< Real >::~RiskMeasure ( )

inlinevirtual

Definition at line 75 of file ROL_RiskMeasure.hpp.

template<class Real >

ROL::RiskMeasure< Real >::RiskMeasure ( void )

inline

Definition at line 77 of file ROL_RiskMeasure.hpp.

Member Function Documentation

template<class Real >

void ROL::RiskMeasure< Real >::setRiskVectorInfo	(	const int	comp,
		const int	index
	)

inline

Definition at line 80 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::comp_, and ROL::RiskMeasure< Real >::index_.

template<class Real >

int ROL::RiskMeasure< Real >::getComponent ( void ) const

inline

Definition at line 85 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::comp_.

template<class Real >

int ROL::RiskMeasure< Real >::getIndex ( void ) const

inline

Definition at line 89 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::index_.

template<class Real >

virtual void ROL::RiskMeasure< Real >::reset	(	ROL::Ptr< Vector< Real > > &	x0,
		const Vector< Real > &	x
	)

inlinevirtual

Reset internal risk measure storage. Called for value and gradient computation.

Parameters

[out]	x0	is a user-provided optimization vector
[in]	x	is a (potentially) augmented risk vector

On input, \(x\) carries \(x_0\) and any statistics (scalars) associated with the risk measure.

Definition at line 102 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::dualVector_, ROL::RiskMeasure< Real >::firstReset_, ROL::RiskMeasure< Real >::g_, ROL::RiskMeasure< Real >::gv_, ROL::RiskMeasure< Real >::hv_, ROL::RiskMeasure< Real >::val_, and zero.

Referenced by ROL::RiskMeasure< Real >::reset().

template<class Real >

virtual void ROL::RiskMeasure< Real >::reset	(	ROL::Ptr< Vector< Real > > &	x0,
		const Vector< Real > &	x,
		ROL::Ptr< Vector< Real > > &	v0,
		const Vector< Real > &	v
	)

inlinevirtual

Reset internal risk measure storage. Called for Hessian-times-a-vector computation.

  @param[out]  x0 is a user-provided optimization vector
  @param[in]   x  is a (potentially) augmented risk vector
  @param[out]  v0 is a user-provided direction vector
  @param[in]   v  is a (potentially) augmented risk vector

  On input, \form#46 carries \form#447 and any statistics (scalars)
  associated with the risk measure.  Similarly, \form#63 carries

\(v_0\) and any statistics (scalars) associated with the risk measure.

Definition at line 131 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::reset().

template<class Real >

virtual void ROL::RiskMeasure< Real >::update	(	const Real	val,
		const Real	weight
	)

inlinevirtual

Update internal risk measure storage for value computation.

Parameters

[in]	val	is the value of the random variable objective function at the current sample point
[in]	weight	is the weight associated with the current sample point

Definition at line 146 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::val_.

template<class Real >

virtual void ROL::RiskMeasure< Real >::update	(	const Real	val,
		const Vector< Real > &	g,
		const Real	weight
	)

inlinevirtual

Update internal risk measure storage for gradient computation.

Parameters

[in]	val	is the value of the random variable objective function at the current sample point
[in]	g	is the gradient of the random variable objective function at the current sample point
[in]	weight	is the weight associated with the current sample point

Definition at line 159 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::g_.

template<class Real >

virtual void ROL::RiskMeasure< Real >::update	(	const Real	val,
		const Vector< Real > &	g,
		const Real	gv,
		const Vector< Real > &	hv,
		const Real	weight
	)

inlinevirtual

Update internal risk measure storage for Hessian-time-a-vector computation.

Parameters

[in]	val	is the value of the random variable objective function at the current sample point
[in]	g	is the gradient of the random variable objective function at the current sample point
[in]	gv	is the gradient of the random variable objective function at the current sample point applied to the vector v0
[in]	hv	is the Hessian of the random variable objective function at the current sample point applied to the vector v0
[in]	weight	is the weight associated with the current sample point

Definition at line 178 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::hv_.

template<class Real >

virtual Real ROL::RiskMeasure< Real >::getValue ( SampleGenerator< Real > & sampler )

inlinevirtual

Return risk measure value.

Parameters

[in] sampler is the ROL::SampleGenerator used to sample the objective function

Upon return, getValue returns \(\mathcal{R}(f(x_0))\) where \(f(x_0)\) denotes the random variable objective function evaluated at \(x_0\).

Definition at line 191 of file ROL_RiskMeasure.hpp.

References ROL::SampleGenerator< Real >::sumAll(), and ROL::RiskMeasure< Real >::val_.

template<class Real >

virtual void ROL::RiskMeasure< Real >::getGradient	(	Vector< Real > &	g,
		SampleGenerator< Real > &	sampler
	)

inlinevirtual

Return risk measure (sub)gradient.

Parameters

[out]	g	is the (sub)gradient of the risk measure
[in]	sampler	is the ROL::SampleGenerator used to sample the objective function

Upon return, getGradient returns \(\theta\in\partial\mathcal{R}(f(x_0))\) where \(f(x_0)\) denotes the random variable objective function evaluated at \(x_0\) and \(\partial\mathcal{R}(X)\) denotes the subdifferential of \(\mathcal{R}\) at \(X\).

Definition at line 208 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::dualVector_, ROL::RiskMeasure< Real >::g_, and ROL::SampleGenerator< Real >::sumAll().

template<class Real >

virtual void ROL::RiskMeasure< Real >::getHessVec	(	Vector< Real > &	hv,
		SampleGenerator< Real > &	sampler
	)

inlinevirtual

Return risk measure Hessian-times-a-vector.

Parameters

[out]	hv	is the Hessian-times-a-vector of the risk measure
[in]	sampler	is the ROL::SampleGenerator used to sample the objective function

Upon return, getHessVec returns \(\nabla^2 \mathcal{R}(f(x_0))v_0\) (if available) where \(f(x_0)\) denotes the random variable objective function evaluated at \(x_0\).

Definition at line 224 of file ROL_RiskMeasure.hpp.

References ROL::RiskMeasure< Real >::dualVector_, ROL::RiskMeasure< Real >::hv_, and ROL::SampleGenerator< Real >::sumAll().