Variable Selection and Regularization via Arbitrary Rectangle-range Generalized Elastic Net

Abstract

We introduce the arbitrary rectangle-range generalized elastic net penalty method, abbreviated to ARGEN, for performing constrained variable selection and regularization in high-dimensional sparse linear models. As a natural extension of the nonnegative elastic net penalty method, ARGEN is proved to have variable selection consistency and estimation consistency under some conditions. The asymptotic behavior in distribution of the ARGEN estimators have been studied. We also propose an algorithm called MU-QP-RR-W-l1 to efficiently solve ARGEN. By conducting simulation study we show that ARGEN outperforms the elastic net in a number of settings. Finally an application of S&P 500 index tracking with constraints on the stock allocations is performed to provide general guidance for adapting ARGEN to solve real-world problems.

Boaren: Improving Regularization in Linear Regression With an Application to Index Tracking

“…which is a particular case of the Arbitrary Rectangle-range Generalized Elastic Net (ARGEN) studied in Ding et al (2021). As a result, the AREN problem, Equation (2.2), can be solved numerically using the so-called "multiplicative updates for solving quadratic programming with rectangle-range l 1 regularization" algorithm.…”

“…AREN is a special case of the more general ARGEN model studied in Ding et al (2021) and is ideal in this context because it is broadly applicable, and its important properties (tractability, estimation consistency, and variable selection consistency) follow from the more general ARGEN, allowing these results to be described without lengthy proofs. Progress in this field of research has been accelerating along with the influence of data science and the availability of extensive and inexpensive computing resources.…”

“…Section 3 is devoted to an application of two-step AREN and bootstrapped two-step AREN to the practical problem of S&P 500 index-tracking. The interested reader may refer to Ding et al (2021) for simulations that compare the performance of a variety of similar methods of this type.…”