Provides an interface for a convex combination of the expected value and the conditional value-at-risk using the expectation risk quadrangle. More...

#include <ROL_QuantileQuadrangle.hpp>

Inheritance diagram for ROL::QuantileQuadrangle< Real >:

Public Member Functions
	QuantileQuadrangle (Real prob, Real eps, ROL::Ptr< PlusFunction< Real > > &pf)
	Constructor. More...

	QuantileQuadrangle (Real prob, Real lam, Real eps, ROL::Ptr< PlusFunction< Real > > &pf)
	Constructor. More...

	QuantileQuadrangle (ROL::ParameterList &parlist)
	Constructor. More...

Real	error (Real x, int deriv=0)
	Evaluate the scalar error function at x. More...

Real	regret (Real x, int deriv=0)
	Evaluate the scalar regret function at x. More...

void	check (void)
	Run default derivative tests for the scalar regret function. More...

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

	ExpectationQuad (void)

Private Member Functions
void	parseParameterList (ROL::ParameterList &parlist)

void	checkInputs (void) const

void	setParameters (void)

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

Real	prob_

Real	lam_

Real	eps_

Real	alpha_

Real	beta_

Detailed Description

template<class Real>
class ROL::QuantileQuadrangle< Real >

Provides an interface for a convex combination of the expected value and the conditional value-at-risk using the expectation risk quadrangle.

The conditional value-at-risk (also called the average value-at-risk or the expected shortfall) with confidence level \(0\le \beta < 1\) is

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

where \((x)_+ = \max\{0,x\}\). If the distribution of \(X\) is continuous, then \(\mathcal{R}\) is the conditional expectation of \(X\) exceeding the \(\beta\)-quantile of \(X\) and the optimal \(t\) is the \(\beta\)-quantile. Additionally, \(\mathcal{R}\) is a law-invariant coherent risk measure.

This class defines a convex combination of expected value and the conditional value-at-risk using the expectation risk quadrangle. In this case, the scalar regret function is

\[ v(x) = \alpha (x)_+ - \lambda (-x)_+ \]

for \(\alpha > 1\) and \(0 \le \lambda < 1\). The associated confidence level for the conditional value-at-risk is

\[ \beta = \frac{\alpha-1}{\alpha-\lambda}. \]

This convex combination of expected value and the conditional value-at-risk is then realized as

\[ \mathcal{R}(X) = \inf_{t\in\mathbb{R}}\left\{ t + \mathbb{E}[v(X-t)] \right\}. \]

ROL implements this by augmenting the optimization vector \(x_0\) with the parameter \(t\), then minimizes jointly for \((x_0,t)\).

When using derivative-based optimization, the user can provide a smooth approximation of \((\cdot)_+\) using the ROL::PlusFunction class.

Definition at line 63 of file ROL_QuantileQuadrangle.hpp.

Constructor & Destructor Documentation

template<class Real >

ROL::QuantileQuadrangle< Real >::QuantileQuadrangle	(	Real	prob,
		Real	eps,
		ROL::Ptr< PlusFunction< Real > > &	pf
	)

inline

Constructor.

Parameters

[in]	prob	is the confidence level
[in]	eps	is the smoothing parameter for the plus function approximation
[in]	pf	is the plus function or an approximation

Definition at line 123 of file ROL_QuantileQuadrangle.hpp.

References ROL::QuantileQuadrangle< Real >::checkInputs(), and ROL::QuantileQuadrangle< Real >::setParameters().

template<class Real >

ROL::QuantileQuadrangle< Real >::QuantileQuadrangle	(	Real	prob,
		Real	lam,
		Real	eps,
		ROL::Ptr< PlusFunction< Real > > &	pf
	)

inline

Constructor.

Parameters

[in]	prob	is the confidence level
[in]	lam	is the convex combination parameter (coeff=0 corresponds to the expected value whereas coeff=1 corresponds to the conditional value-at-risk)
[in]	eps	is the smoothing parameter for the plus function approximation
[in]	pf	is the plus function or an approximation