ao: Approximated Octagon Penalty

Description Usage Arguments Details Value Author(s) References See Also


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


ao (lambda = NULL, ...)



two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter λ. This must be a nonnegative real number. The second component indicates the exponent γ of the bridge penalty term. See details below. It must hold that γ > 1.


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)


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),


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


character: the penalty name.


double: the (nonnegative) regularization parameter.


function: computes the diagonal penalty matrix.


Jan Ulbricht


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

See Also

penalty, genet

Search within the lqa package
Search all R packages, documentation and source code

Questions? Problems? Suggestions? or email at

Please suggest features or report bugs with the GitHub issue tracker.

All documentation is copyright its authors; we didn't write any of that.