Provides an interface for spectral risk measures. More...

#include <ROL_SpectralRisk.hpp>

Inheritance diagram for ROL::SpectralRisk< Real >:

Public Member Functions
	SpectralRisk (void)

	SpectralRisk (const ROL::Ptr< Distribution< Real > > &dist, const int nQuad, const ROL::Ptr< PlusFunction< Real > > &pf)

	SpectralRisk (ROL::ParameterList &parlist)

	SpectralRisk (const std::vector< Real > &pts, const std::vector< Real > &wts, const ROL::Ptr< PlusFunction< Real > > &pf)

Real	computeStatistic (const std::vector< Real > &xstat)

void	setStorage (const Ptr< SampledScalar< Real >> &value_storage, const Ptr< SampledVector< Real >> &gradient_storage)

void	setHessVecStorage (const Ptr< SampledScalar< Real >> &gradvec_storage, const Ptr< SampledVector< Real >> &hessvec_storage)

void	setSample (const std::vector< Real > &point, const Real weight)

void	resetStorage (bool flag=true)
	Reset internal storage. More...

void	initialize (const Vector< Real > &x)
	Initialize temporary variables. More...

Real	computeStatistic (const Ptr< std::vector< Real >> &xstat) const

void	updateValue (Objective< Real > &obj, const Vector< Real > &x, const std::vector< Real > &xstat, Real &tol)
	Update internal storage for value computation. More...

void	updateGradient (Objective< Real > &obj, const Vector< Real > &x, const std::vector< Real > &xstat, Real &tol)
	Update internal risk measure storage for gradient computation. More...

void	updateHessVec (Objective< Real > &obj, const Vector< Real > &v, const std::vector< Real > &vstat, const Vector< Real > &x, const std::vector< Real > &xstat, Real &tol)
	Update internal risk measure storage for Hessian-time-a-vector computation. More...

Real	getValue (const Vector< Real > &x, const std::vector< Real > &xstat, SampleGenerator< Real > &sampler)
	Return risk measure value. More...

void	getGradient (Vector< Real > &g, std::vector< Real > &gstat, const Vector< Real > &x, const std::vector< Real > &xstat, SampleGenerator< Real > &sampler)
	Return risk measure (sub)gradient. More...

void	getHessVec (Vector< Real > &hv, std::vector< Real > &hvstat, const Vector< Real > &v, const std::vector< Real > &vstat, const Vector< Real > &x, const std::vector< Real > &xstat, SampleGenerator< Real > &sampler)
	Return risk measure Hessian-times-a-vector. More...

Public Member Functions inherited from ROL::RandVarFunctional< Real >
virtual	~RandVarFunctional ()

	RandVarFunctional (void)

	weight_ (0)

void	useStorage (bool storage)

void	useHessVecStorage (bool storage)

virtual Real	computeStatistic (const Ptr< const std::vector< Real >> &xstat) const
	Compute statistic. More...

Protected Member Functions
void	buildMixedQuantile (const std::vector< Real > &pts, const std::vector< Real > &wts, const ROL::Ptr< PlusFunction< Real > > &pf)

void	buildQuadFromDist (std::vector< Real > &pts, std::vector< Real > &wts, const int nQuad, const ROL::Ptr< Distribution< Real > > &dist) const

void	printQuad (const std::vector< Real > &pts, const std::vector< Real > &wts, const bool print=false) const

Protected Member Functions inherited from ROL::RandVarFunctional< Real >
Real	computeValue (Objective< Real > &obj, const Vector< Real > &x, Real &tol)

void	computeGradient (Vector< Real > &g, Objective< Real > &obj, const Vector< Real > &x, Real &tol)

Real	computeGradVec (Vector< Real > &g, Objective< Real > &obj, const Vector< Real > &v, const Vector< Real > &x, Real &tol)

void	computeHessVec (Vector< Real > &hv, Objective< Real > &obj, const Vector< Real > &v, const Vector< Real > &x, Real &tol)

Private Member Functions
void	checkInputs (ROL::Ptr< Distribution< Real > > &dist=ROL::nullPtr) const

Private Attributes
ROL::Ptr< MixedCVaR< Real > >	mqq_

ROL::Ptr< PlusFunction< Real > >	plusFunction_

std::vector< Real >	wts_

std::vector< Real >	pts_

Additional Inherited Members
Protected Attributes inherited from ROL::RandVarFunctional< Real >
Real	val_

Real	gv_

Ptr< Vector< Real > >	g_

Ptr< Vector< Real > >	hv_

Ptr< Vector< Real > >	dualVector_

bool	firstReset_

std::vector< Real >	point_

Real	weight_

Detailed Description

template<class Real>
class ROL::SpectralRisk< Real >

Provides an interface for spectral risk measures.

Kusuoka's representation for law-invariant risk measures is

\[ \mathcal{R}(X) = \sup_{\mu\in\mathfrak{M}} \int_0^1 \mathrm{CVaR}_{\alpha}(X)\,\mathrm{d}\mu(\alpha) \]

where the conditional value-at-risk (CVaR) with confidence level \(0\le \alpha < 1\) is

\[ \mathrm{CVaR}_\alpha(X) = \inf_{t\in\mathbb{R}} \left\{ t + \frac{1}{1-\alpha} \mathbb{E}\left[(X-t)_+\right] \right\}, \quad (x)_+ = \max\{0,x\}, \]

and \(\mathfrak{M}\) is a subset of distributions on the interval \([0,1)\). By spectral risk measures, we refer to the case where the set \(\mathfrak{M}\) is a singleton. If the distribution \(\mu\in\mathfrak{M}\) is discrete, then the corresponding risk measure is a mixed quantile quadrangle risk measure.

If the distribution of \(X\) is continuous, then \(\mathrm{CVaR}_{\alpha}(X)\) is the conditional expectation of \(X\) exceeding the \(\alpha\)-quantile of \(X\) and the optimal \(t\) is the \(\alpha\)-quantile. Additionally, \(\mathcal{R}\) is a law-invariant coherent risk measure.

ROL implements \(\mathcal{R}\) by approximating the integral with Gauss-Chebyshev quadrature of the first kind. The corresponding quadrature points and weights are then used to construct a ROL::MixedCVaR risk measure. When using derivative-based optimization, the user can provide a smooth approximation of \((\cdot)_+\) using the ROL::PlusFunction class.

Definition at line 89 of file ROL_SpectralRisk.hpp.

Constructor & Destructor Documentation

template<class Real >

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

inline

Definition at line 166 of file ROL_SpectralRisk.hpp.

template<class Real >

ROL::SpectralRisk< Real >::SpectralRisk	(	const ROL::Ptr< Distribution< Real > > &	dist,
		const int	nQuad,
		const ROL::Ptr< PlusFunction< Real > > &	pf
	)

inline

Definition at line 168 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::buildMixedQuantile(), ROL::SpectralRisk< Real >::buildQuadFromDist(), and ROL::SpectralRisk< Real >::checkInputs().

template<class Real >

ROL::SpectralRisk< Real >::SpectralRisk ( ROL::ParameterList & parlist )

inline

Definition at line 181 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::buildMixedQuantile(), ROL::SpectralRisk< Real >::buildQuadFromDist(), ROL::SpectralRisk< Real >::checkInputs(), and ROL::SpectralRisk< Real >::printQuad().

template<class Real >

ROL::SpectralRisk< Real >::SpectralRisk	(	const std::vector< Real > &	pts,
		const std::vector< Real > &	wts,
		const ROL::Ptr< PlusFunction< Real > > &	pf
	)

inline

Definition at line 202 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::buildMixedQuantile(), and ROL::SpectralRisk< Real >::checkInputs().

Member Function Documentation

template<class Real >

void ROL::SpectralRisk< Real >::checkInputs ( ROL::Ptr< Distribution< Real > > & dist = ROL::nullPtr ) const

inlineprivate

Definition at line 97 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::plusFunction_.

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

template<class Real >

void ROL::SpectralRisk< Real >::buildMixedQuantile	(	const std::vector< Real > &	pts,
		const std::vector< Real > &	wts,
		const ROL::Ptr< PlusFunction< Real > > &	pf
	)

inlineprotected

Definition at line 111 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_, ROL::SpectralRisk< Real >::plusFunction_, ROL::SpectralRisk< Real >::pts_, and ROL::SpectralRisk< Real >::wts_.

Referenced by ROL::SecondOrderCVaR< Real >::initializeQuad(), ROL::ChebyshevSpectral< Real >::initializeQuad(), and ROL::SpectralRisk< Real >::SpectralRisk().

template<class Real >

void ROL::SpectralRisk< Real >::buildQuadFromDist	(	std::vector< Real > &	pts,
		std::vector< Real > &	wts,
		const int	nQuad,
		const ROL::Ptr< Distribution< Real > > &	dist
	)		const

inlineprotected

Definition at line 119 of file ROL_SpectralRisk.hpp.

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

template<class Real >

void ROL::SpectralRisk< Real >::printQuad	(	const std::vector< Real > &	pts,
		const std::vector< Real > &	wts,
		const bool	print = `false`
	)		const

inlineprotected

Definition at line 145 of file ROL_SpectralRisk.hpp.

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

template<class Real >

Real ROL::SpectralRisk< Real >::computeStatistic ( const std::vector< Real > & xstat )

inline

Definition at line 210 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::wts_.

template<class Real >

void ROL::SpectralRisk< Real >::setStorage	(	const Ptr< SampledScalar< Real >> &	value_storage,
		const Ptr< SampledVector< Real >> &	gradient_storage
	)

inlinevirtual

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 219 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_, and ROL::RandVarFunctional< Real >::setStorage().

template<class Real >

void ROL::SpectralRisk< Real >::setHessVecStorage	(	const Ptr< SampledScalar< Real >> &	gradvec_storage,
		const Ptr< SampledVector< Real >> &	hessvec_storage
	)

inlinevirtual

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 225 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_, and ROL::RandVarFunctional< Real >::setHessVecStorage().

template<class Real >

void ROL::SpectralRisk< Real >::setSample	(	const std::vector< Real > &	point,
		const Real	weight
	)

inlinevirtual

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 231 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_, and ROL::RandVarFunctional< Real >::setSample().

template<class Real >

void ROL::SpectralRisk< Real >::resetStorage ( bool flag = true )

inlinevirtual

Reset internal storage.

Parameters

[in] x is a vector used for initializing storage

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 236 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_, and ROL::RandVarFunctional< Real >::resetStorage().

template<class Real >

void ROL::SpectralRisk< Real >::initialize ( const Vector< Real > & x )

inlinevirtual

Initialize temporary variables.

Parameters

[in] x is a vector used for initializing storage

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 241 of file ROL_SpectralRisk.hpp.

References ROL::RandVarFunctional< Real >::initialize(), and ROL::SpectralRisk< Real >::mqq_.

template<class Real >

Real ROL::SpectralRisk< Real >::computeStatistic ( const Ptr< std::vector< Real >> & xstat ) const

inline

Definition at line 246 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.

template<class Real >

void ROL::SpectralRisk< Real >::updateValue	(	Objective< Real > &	obj,
		const Vector< Real > &	x,
		const std::vector< Real > &	xstat,
		Real &	tol
	)

inlinevirtual

Update internal 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

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 250 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.

template<class Real >

void ROL::SpectralRisk< Real >::updateGradient	(	Objective< Real > &	obj,
		const Vector< Real > &	x,
		const std::vector< Real > &	xstat,
		Real &	tol
	)

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

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 257 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.

template<class Real >

void ROL::SpectralRisk< Real >::updateHessVec	(	Objective< Real > &	obj,
		const Vector< Real > &	v,
		const std::vector< Real > &	vstat,
		const Vector< Real > &	x,
		const std::vector< Real > &	xstat,
		Real &	tol
	)

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

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 264 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.

template<class Real >

Real ROL::SpectralRisk< Real >::getValue	(	const Vector< Real > &	x,
		const std::vector< Real > &	xstat,
		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 \(x_0\).

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 273 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.

template<class Real >

void ROL::SpectralRisk< Real >::getGradient	(	Vector< Real > &	g,
		std::vector< Real > &	gstat,
		const Vector< Real > &	x,
		const std::vector< Real > &	xstat,
		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 \(X\).

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 279 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.

template<class Real >

void ROL::SpectralRisk< Real >::getHessVec	(	Vector< Real > &	hv,
		std::vector< Real > &	hvstat,
		const Vector< Real > &	v,
		const std::vector< Real > &	vstat,
		const Vector< Real > &	x,
		const std::vector< Real > &	xstat,
		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 \(x_0\).

Reimplemented from ROL::RandVarFunctional< Real >.

Definition at line 287 of file ROL_SpectralRisk.hpp.

References ROL::SpectralRisk< Real >::mqq_.