Journal of Applied Mathematics and Decision Sciences
Volume 2007 (2007), Article ID 24053, 13 pages
Research Article

On Solving Lq-Penalized Regressions

Tracy Zhou Wu,1,2 Yingyi Chu,3 and Yan Yu2

1JPMorgan Chase Bank, 1111 Polaris Pkwy, Columbus 43240, OH, USA
2Department of Quantitative Analysis and Operations Management, University of Cincinnati, P.O. Box 210130, Cincinnati 45221, OH, USA
3ABN AMRO Bank, 250 Bishopsgate, London EC2M 4AA, UK

Received 1 November 2006; Accepted 18 July 2007

Academic Editor: Fernando Beltran

Copyright © 2007 Tracy Zhou Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Lq-penalized regression arises in multidimensional statistical modelling where all or part of the regression coefficients are penalized to achieve both accuracy and parsimony of statistical models. There is often substantial computational difficulty except for the quadratic penalty case. The difficulty is partly due to the nonsmoothness of the objective function inherited from the use of the absolute value. We propose a new solution method for the general Lq-penalized regression problem based on space transformation and thus efficient optimization algorithms. The new method has immediate applications in statistics, notably in penalized spline smoothing problems. In particular, the LASSO problem is shown to be polynomial time solvable. Numerical studies show promise of our approach.