Provides a general interface for risk and error measures generated through the expectation risk quadrangle. More...

#include <ROL_ExpectationQuad.hpp>

Inheritance diagram for ROL::ExpectationQuad< Real >:

Public Member Functions
virtual	~ExpectationQuad (void)

	ExpectationQuad (void)

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

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

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

Detailed Description

template<class Real>
class ROL::ExpectationQuad< Real >

Provides a general interface for risk and error measures generated through the expectation risk quadrangle.

Provides a general interface for risk measures generated through the expectation risk quadrangle.

Provides a general interface for deviation measures generated through the expectation risk quadrangle.

The expectation risk quadrangle is a specialization of the general risk quadrangle that provides a rigorous connection between risk-averse optimization and statistical estimation. The risk quadrangle provides fundamental relationships between measures of risk, regret, error and deviation. An expectation risk quadrangle is defined through scalar regret and error functions. The scalar regret function, \(v:\mathbb{R}\to(-\infty,\infty]\), must be proper, closed, convex and satisfy \(v(0)=0\) and \(v(x) > x\) for all \(x\neq 0\). Similarly, the scalar error function, \(e:\mathbb{R}\to[0,\infty]\), must be proper, closed, convex and satisfy \(e(0)=0\) and \(e(x) > 0\) for all \(x\neq 0\). \(v\) and \(e\) are obtained from one another through the relations

\[ v(x) = e(x) + x \quad\text{and}\quad e(x) = v(x) - x. \]

Given \(v\) (or equivalently \(e\)), the associated risk measure is

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

In general, \(\mathcal{R}\) is convex and translation equivariant. Moreover, \(\mathcal{R}\) is monotonic if \(v\) is increasing and \(\mathcal{R}\) is positive homogeneous if \(v\) is. ROL implements this by augmenting the optimization vector \(x_0\) with the parameter \(t\), then minimizes jointly for \((x_0,t)\).

Definition at line 53 of file ROL_ExpectationQuad.hpp.