Provides an interface for the Chebyshev-Spectral risk measure. More...

#include <ROL_ChebyshevSpectral.hpp>

Inheritance diagram for ROL::ChebyshevSpectral< Real >:

Public Member Functions
	ChebyshevSpectral (ROL::ParameterList &parlist)
	Constructor. More...

	ChebyshevSpectral (const Real lower, const Real upper, const int nQuad, const int wType, const ROL::Ptr< PlusFunction< Real > > &pf)
	Constructor. More...

Public Member Functions inherited from ROL::SpectralRisk< Real >
	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...

Private Member Functions
void	checkInputs (void) const

void	initializeQuad (void)

Private Attributes
ROL::Ptr< PlusFunction< Real > >	plusFunction_

Real	lower_

Real	upper_

int	nQuad_

int	wType_

std::vector< Real >	wts_

std::vector< Real >	pts_

Additional Inherited Members
Protected Member Functions inherited from ROL::SpectralRisk< Real >
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)

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::ChebyshevSpectral< Real >

Provides an interface for the Chebyshev-Spectral risk measure.

The Chebyshev-Spectral risk measure is defined as

\[ \mathcal{R}(X) = \int_{\alpha_0}^{\alpha_1} w(\alpha) \mathrm{CVaR}_{\alpha}(X) \,\mathrm{d}\alpha \]

where \(0\le \alpha_0 < \alpha_1 < 1\) and the conditional value-at-risk (CVaR) with confidence level \(0\le \alpha < 1\) is

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

There are three choices of weight functions \(w\): (i) the first weight function generates the Chebyshev polynomials of the first kind and has the specific form

\[ w(x) = \frac{1}{\sqrt{(x-\alpha_0)(\alpha_1-x)}}; \]

(ii) the second weight function generates the Chebyshev polynomials of the second kind and has the specific form

\[ w(x) = \sqrt{(x-\alpha_0)(\alpha_1-x)}; \]

and (iii) the third weight function is related again to the Chebyshev polynomials of the first kind and has the specific form

\[ w(x) = \sqrt{\frac{x-\alpha_0}{\alpha_1-x}}. \]

As defined, \(\mathcal{R}\) is a law-invariant coherent risk measure.

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

Definition at line 97 of file ROL_ChebyshevSpectral.hpp.

Constructor & Destructor Documentation

template<class Real >

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

inline

Constructor.

Parameters

[in] parlist is a parameter list specifying inputs

parlist should contain sublists "SOL"->"Risk Measure"->"Chebyshev-Spectral" and the "Chebyshev-Spectral" sublist should have the following parameters

"Lower Bound" (between 0 and 1)
"Upper Bound" (between 0 and 1, greater than "Lower Bound")
"Weight Type" (either 1, 2, or 3)
"Number of Quadrature Points"
A sublist for plus function information.

Definition at line 158 of file ROL_ChebyshevSpectral.hpp.

References ROL::ChebyshevSpectral< Real >::checkInputs(), ROL::ChebyshevSpectral< Real >::initializeQuad(), ROL::ChebyshevSpectral< Real >::lower_, ROL::ChebyshevSpectral< Real >::nQuad_, ROL::ChebyshevSpectral< Real >::plusFunction_, ROL::ChebyshevSpectral< Real >::upper_, and ROL::ChebyshevSpectral< Real >::wType_.

template<class Real >

ROL::ChebyshevSpectral< Real >::ChebyshevSpectral	(	const Real	lower,
		const Real	upper,
		const int	nQuad,
		const int	wType,
		const ROL::Ptr< PlusFunction< Real > > &	pf
	)

inline

Constructor.

Parameters

[in]	lower	is the lower confidence level (between 0 and 1)
[in]	upper	is the upper confidence level (between 0 and 1, greater than lower)
[in]	nQuad	is the number of quadrature points
[in]	wType	is the weight type (either 1, 2, or 3)
[in]	pf	is the plus function or an approximation