# *l* _{1}-*l* _{2} regularization of split feasibility problems

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## Abstract

Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP); see for example Censor and Elfving (Numer. Algorithms **8**, 221–239 1994). Each function is typically either a data fidelity term or a regularization term enforcing some properties on the solution; see for example Chaux et al. (SIAM J. Imag. Sci. **2**, 730–762 2009) and references therein. In this paper, we are interested in split feasibility problems which can be seen as a general form of *Q*-Lasso introduced in Alghamdi et al. (2013) that extended the well-known Lasso of Tibshirani (J. R. Stat. Soc. Ser. B **58**, 267–288 1996). *Q* is a closed convex subset of a Euclidean *m*-space, for some integer *m* ≥ 1, that 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. Inspired by recent works by Lou and Yan (2016), Xu (IEEE Trans. Neural Netw. Learn. Syst. **23**, 1013–1027 2012), we are interested in a nonconvex regularization of SFP and propose three split algorithms for solving this general case. The first one is based on the DC (difference of convex) algorithm (DCA) introduced by Pham Dinh Tao, the second one is nothing else than the celebrate forward-backward algorithm, and the third one uses a method introduced by Mine and Fukushima. It is worth mentioning that the SFP model a number of applied problems arising from signal/image processing and specially optimization problems for intensity-modulated radiation therapy (IMRT) treatment planning; see for example Censor et al. (Phys. Med. Biol. **51**, 2353–2365, 2006).

## Keywords

Q-Lasso Split feasibility Soft-thresholding Regularization DCA algorithm Forward-backward iterations Mine-Fukushima algorithm Douglas-Rachford algorithm## Mathematics Subject Classification (2010)

Primary 49J53, 65K10 Secondary 49M37, 90C25## Preview

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## Notes

### Acknowledgments

We wish to thank the anonymous referee for the thorough analysis and review; all the comments and suggestions helped tremendously in improving the quality of this paper and made it suitable for publication. The first author wish to thank his team “Image & Modèle” and the Computer Science and System Laboratory (L.I.S) of Aix-Marseille University. The second author work is supported by the EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669.

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