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Abstract and Applied Analysis
Volume 2013, Article ID 250943, 8 pages
Research Article

Properties and Iterative Methods for the -Lasso

1Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 24 September 2013; Accepted 27 November 2013

Academic Editor: Chi-Keung Ng

Copyright © 2013 Maryam A. Alghamdi 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.


We introduce the -lasso which generalizes the well-known lasso of Tibshirani (1996) with a closed convex subset of a Euclidean m-space for some integer . This set can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the lasso. Solutions of the -lasso depend on a tuning parameter . In this paper, we obtain basic properties of the solutions as a function of . Because of ill posedness, we also apply regularization to the -lasso. In addition, we discuss iterative methods for solving the -lasso which include the proximal-gradient algorithm and the projection-gradient algorithm.