Object of the `penalty`

class to handle the AO penalty (Ulbricht, 2010).

1 |

`lambda` |
two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter |

`...` |
further arguments. |

The basic idea of the AO penalty is
to use a linear combination of *L_1*-norm and the bridge penalty with *γ > 1* where the amount of
the bridge penalty part is driven by empirical
correlation. So, consider the penalty

*
P_{\tilde{λ}}^{ao}(\boldsymbol{β}) = ∑_{i = 2}^p ∑_{j< i} p_{\tilde{λ},ij}
(\boldsymbol{β}), \quad \tilde{λ} = (λ, γ)
*

where

*
p_{\tilde{λ},ij} = λ[(1 - |\varrho_{ij}|) (|β_i| + |β_j|) + |\varrho_{ij}|(|β_i|^γ + |β_j|^γ)],
*

and *\varrho_{ij}* denotes the value of the (empirical) correlation of the i-th and j-th regressor. Since we are going to
approximate an octagonal polytope in two dimensions, we will refer to this penalty as *approximated octagon*
(AO) penalty. Note that *P_{\tilde{λ}}^{ao}(\boldsymbol{β})* leads to a dominating lasso term if the regressors are uncorrelated and to a
dominating bridge term if they are nearly perfectly correlated.

The penalty can be rearranged as

*
P_{\tilde{λ}}^{ao}(\boldsymbol{β}) = ∑_{i=1}^p p_{\tilde{λ},i}^{ao}(β_i),
*

where

*
p_{\tilde{λ},i}^{ao}(β_i) = λ ≤ft\{|β_i|∑_{j \neq i} (1 - |\varrho_{ij}|) + |β_i|^γ ∑_{j \neq i} |\varrho_{ij}|\right\}.
*

It uses two tuning parameters *\tilde{λ} = (λ, γ)*, where *λ* controls the penalty amount and *γ*
manages the approximation of the pairwise *L_∞*-norm.

An object of the class `penalty`

. This is a list with elements

`penalty` |
character: the penalty name. |

`lambda` |
double: the (nonnegative) regularization parameter. |

`getpenmat` |
function: computes the diagonal penalty matrix. |

Jan Ulbricht

Ulbricht, Jan (2010) *Variable Selection in Generalized Linear Models*. Ph.D. Thesis. LMU Munich.

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