ROL
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ROL::MoreauYosidaCVaR< Real > Class Template Reference
Risk Measures

Provides an interface for a smooth approximation of the conditional value-at-risk. More...

#include <ROL_MoreauYosidaCVaR.hpp>

+ Inheritance diagram for ROL::MoreauYosidaCVaR< Real >:

Public Member Functions

 MoreauYosidaCVaR (Real prob, Real eps)
 Constructor. More...
 
 MoreauYosidaCVaR (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

Real prob_
 
Real eps_
 
Real omp_
 
Real ub_
 

Detailed Description

template<class Real>
class ROL::MoreauYosidaCVaR< Real >

Provides an interface for a smooth approximation of the conditional value-at-risk.

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.

The conditional value-at-risk is in general not smooth due to \((\cdot)_+\). One approach to smoothing the conditional-value-at-risk is to regularize its biconjugate form. That is, since \(\mathcal{R}\) is coherent, we have that

\[ \mathcal{R}(X) = \sup_{\vartheta\in\mathfrak{A}}\mathbb{E}[\vartheta X] \]

where \(\mathfrak{A}\) is the effective domain of the conjugate of \(\mathcal{R}\), i.e.,

\[ \mathfrak{A} = \mathrm{dom}\,\mathcal{R}^* = \{\vartheta\in\mathcal{X}^*\,:\, \mathcal{R}^*(\vartheta) < \infty\} \]

where \(\mathcal{R}^*\) denotes the Legendre-Fenchel transformation of \(\mathcal{R}\). This risk measure implements the penalized conditional value-at-risk

\[ \mathcal{R}(X) = \sup_{\vartheta\in\mathfrak{A}} \left\{\mathbb{E}[\vartheta X] - \frac{\gamma}{2}\mathbb{E}[(\vartheta-1)^2]\right\} \]

for \(\gamma > 0\). This is implemented using the expectation risk quadrangle interface. Thus, we represent \(\mathcal{R}\) as

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

for an appropriately defined scalar regret function \(v\). ROL implements this by augmenting the optimization vector \(x_0\) with the parameter \(t\), then minimizes jointly for \((x_0,t)\).

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.

The conditional value-at-risk is in general not smooth due to \((\cdot)_+\). One approach to smoothing the conditional-value-at-risk is to regularize its biconjugate form. That is, since \(\mathcal{R}\) is coherent, we have that

\[ \mathcal{R}(X) = \sup_{\vartheta\in\mathfrak{A}}\mathbb{E}[\vartheta X] \]

where \(\mathfrak{A}\) is the effective domain of the conjugate of \(\mathcal{R}\), i.e.,

\[ \mathfrak{A} = \mathrm{dom}\,\mathcal{R}^* = \{\vartheta\in\mathcal{X}^*\,:\, \mathcal{R}^*(\vartheta) < \infty\} \]

where \(\mathcal{R}^*\) denotes the Legendre-Fenchel transformation of \(\mathcal{R}\). This risk measure implements the penalized conditional value-at-risk

\[ \mathcal{R}(X) = \sup_{\vartheta\in\mathfrak{A}} \left\{\mathbb{E}[\vartheta X] - \frac{\gamma}{2}\mathbb{E}[\vartheta^2]\right\} \]

for \(\gamma > 0\). This is implemented using the expectation risk quadrangle interface. Thus, we represent \(\mathcal{R}\) as

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

for an appropriately defined scalar regret function \(v\). ROL implements this by augmenting the optimization vector \(x_0\) with the parameter \(t\), then minimizes jointly for \((x_0,t)\).

Definition at line 71 of file ROL_MoreauYosidaCVaR.hpp.

Constructor & Destructor Documentation

template<class Real >
ROL::MoreauYosidaCVaR< Real >::MoreauYosidaCVaR ( Real  prob,
Real  eps 
)
inline

Constructor.

Parameters
[in]probis the confidence level
[in]epsis the regularization parameter

Definition at line 119 of file ROL_MoreauYosidaCVaR.hpp.

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

template<class Real >
ROL::MoreauYosidaCVaR< Real >::MoreauYosidaCVaR ( ROL::ParameterList &  parlist)
inline

Constructor.

Parameters
[in]parlistis a parameter list specifying inputs

parlist should contain sublists "SOL"->"Risk Measure"->"Moreau-Yosida CVaR" and within the "Moreau-Yosida CVaR" sublist should have the following parameters

  • "Confidence Level" (between 0 and 1)
  • "Smoothing Parameter" (must be positive)

Definition at line 134 of file ROL_MoreauYosidaCVaR.hpp.

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

Member Function Documentation

template<class Real >
void ROL::MoreauYosidaCVaR< Real >::parseParameterList ( ROL::ParameterList &  parlist)
inlineprivate

Definition at line 80 of file ROL_MoreauYosidaCVaR.hpp.

References ROL::MoreauYosidaCVaR< Real >::eps_, and ROL::MoreauYosidaCVaR< Real >::prob_.

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

template<class Real >
void ROL::MoreauYosidaCVaR< Real >::checkInputs ( void  ) const
inlineprivate

Definition at line 99 of file ROL_MoreauYosidaCVaR.hpp.

References ROL::MoreauYosidaCVaR< Real >::eps_, ROL::MoreauYosidaCVaR< Real >::prob_, and zero.

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

template<class Real >
void ROL::MoreauYosidaCVaR< Real >::setParameters ( void  )
inlineprivate

Definition at line 107 of file ROL_MoreauYosidaCVaR.hpp.

References ROL::MoreauYosidaCVaR< Real >::eps_, ROL::MoreauYosidaCVaR< Real >::omp_, ROL::MoreauYosidaCVaR< Real >::prob_, and ROL::MoreauYosidaCVaR< Real >::ub_.

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

template<class Real >
Real ROL::MoreauYosidaCVaR< Real >::error ( Real  x,
int  deriv = 0 
)
inlinevirtual

Evaluate the scalar error function at x.

Parameters
[in]xis the scalar input
[in]derivis the derivative order

This function returns \(e(x)\) or a derivative of \(e(x)\).

Reimplemented from ROL::ExpectationQuad< Real >.

Definition at line 141 of file ROL_MoreauYosidaCVaR.hpp.

References ROL::MoreauYosidaCVaR< Real >::regret(), and zero.

template<class Real >
Real ROL::MoreauYosidaCVaR< Real >::regret ( Real  x,
int  deriv = 0 
)
inlinevirtual

Evaluate the scalar regret function at x.

Parameters
[in]xis the scalar input
[in]derivis the derivative order

This function returns \(v(x)\) or a derivative of \(v(x)\).

Implements ROL::ExpectationQuad< Real >.

Definition at line 147 of file ROL_MoreauYosidaCVaR.hpp.

References ROL::MoreauYosidaCVaR< Real >::eps_, ROL::MoreauYosidaCVaR< Real >::omp_, ROL::MoreauYosidaCVaR< Real >::ub_, and zero.

Referenced by ROL::MoreauYosidaCVaR< Real >::check(), and ROL::MoreauYosidaCVaR< Real >::error().

template<class Real >
void ROL::MoreauYosidaCVaR< Real >::check ( void  )
inlinevirtual

Run default derivative tests for the scalar regret function.

Reimplemented from ROL::ExpectationQuad< Real >.

Definition at line 163 of file ROL_MoreauYosidaCVaR.hpp.

References ROL::ExpectationQuad< Real >::check(), ROL::MoreauYosidaCVaR< Real >::eps_, ROL::MoreauYosidaCVaR< Real >::regret(), and zero.

Member Data Documentation

template<class Real >
Real ROL::MoreauYosidaCVaR< Real >::prob_
private

Definition at line 74 of file ROL_MoreauYosidaCVaR.hpp.

Referenced by ROL::MoreauYosidaCVaR< Real >::checkInputs(), ROL::MoreauYosidaCVaR< Real >::parseParameterList(), and ROL::MoreauYosidaCVaR< Real >::setParameters().

template<class Real >
Real ROL::MoreauYosidaCVaR< Real >::eps_
private

Definition at line 75 of file ROL_MoreauYosidaCVaR.hpp.

Referenced by ROL::MoreauYosidaCVaR< Real >::check(), ROL::MoreauYosidaCVaR< Real >::checkInputs(), ROL::MoreauYosidaCVaR< Real >::parseParameterList(), ROL::MoreauYosidaCVaR< Real >::regret(), and ROL::MoreauYosidaCVaR< Real >::setParameters().

template<class Real >
Real ROL::MoreauYosidaCVaR< Real >::omp_
private

Definition at line 77 of file ROL_MoreauYosidaCVaR.hpp.

Referenced by ROL::MoreauYosidaCVaR< Real >::regret(), and ROL::MoreauYosidaCVaR< Real >::setParameters().

template<class Real >
Real ROL::MoreauYosidaCVaR< Real >::ub_
private

Definition at line 78 of file ROL_MoreauYosidaCVaR.hpp.

Referenced by ROL::MoreauYosidaCVaR< Real >::regret(), and ROL::MoreauYosidaCVaR< Real >::setParameters().

The documentation for this class was generated from the following file:
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