Provides an interface for the risk measure associated with the super quantile quadrangle. More...

#include <ROL_SecondOrderCVaR.hpp>

Inheritance diagram for ROL::SecondOrderCVaR< Real >:

Public Member Functions
	SecondOrderCVaR (ROL::ParameterList &parlist)

	SecondOrderCVaR (const Real alpha, const int nQuad, const ROL::Ptr< PlusFunction< Real > > &pf, const bool useGauss=true)

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	alpha_

int	nQuad_

bool	useGauss_

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::SecondOrderCVaR< Real >

Provides an interface for the risk measure associated with the super quantile quadrangle.

The risk measure associated with the super quantile quadrangle is defined as

\[ \mathcal{R}(X) = \frac{1}{1-\beta}\int_\beta^1\mathrm{CVaR}_{\alpha}(X) \,\mathrm{d}\alpha \]

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

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

where \((x)_+ = \max\{0,x\}\). 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-Legendre or Fejer 2 quadrature. 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 85 of file ROL_SecondOrderCVaR.hpp.

Constructor & Destructor Documentation

template<class Real >

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

inline

Definition at line 126 of file ROL_SecondOrderCVaR.hpp.

References ROL::SecondOrderCVaR< Real >::alpha_, ROL::SecondOrderCVaR< Real >::checkInputs(), ROL::SecondOrderCVaR< Real >::initializeQuad(), ROL::SecondOrderCVaR< Real >::nQuad_, ROL::SecondOrderCVaR< Real >::plusFunction_, and ROL::SecondOrderCVaR< Real >::useGauss_.

template<class Real >

ROL::SecondOrderCVaR< Real >::SecondOrderCVaR	(	const Real	alpha,
		const int	nQuad,
		const ROL::Ptr< PlusFunction< Real > > &	pf,
		const bool	useGauss = `true`
	)