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A constrained optimization reformulation and a feasible descent direction method for \(L_{1/2}\) regularization

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Abstract

In this paper, we first propose a constrained optimization reformulation to the \(L_{1/2}\) regularization problem. The constrained problem is to minimize a smooth function subject to some quadratic constraints and nonnegative constraints. A good property of the constrained problem is that at any feasible point, the set of all feasible directions coincides with the set of all linearized feasible directions. Consequently, the KKT point always exists. Moreover, we will show that the KKT points are the same as the stationary points of the \(L_{1/2}\) regularization problem. Based on the constrained optimization reformulation, we propose a feasible descent direction method called feasible steepest descent method for solving the unconstrained \(L_{1/2}\) regularization problem. It is an extension of the steepest descent method for solving smooth unconstrained optimization problem. The feasible steepest descent direction has an explicit expression and the method is easy to implement. Under very mild conditions, we show that the proposed method is globally convergent. We apply the proposed method to solve some practical problems arising from compressed sensing. The results show its efficiency.

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Acknowledgments

The authors would like to thank two anonymous referees for their valuable suggestions and comments. Supported by the NSF of China Grant 11371154, 11071087, 11201197 and 11126147.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Dong-Hui Li & Xiong-ji Zhang

  2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

    Lei Wu & Zhe Sun

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  1. Dong-Hui Li
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  2. Lei Wu
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  3. Zhe Sun
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  4. Xiong-ji Zhang
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Correspondence to Lei Wu.

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Li, DH., Wu, L., Sun, Z. et al. A constrained optimization reformulation and a feasible descent direction method for \(L_{1/2}\) regularization. Comput Optim Appl 59, 263–284 (2014). https://doi.org/10.1007/s10589-014-9683-7

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