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Minimal Zero Norm Solutions of Linear Complementarity Problems

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Abstract

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.

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Acknowledgments

This research was supported by the National Basic Research Program of China (2010CB732501), the National Natural Science Foundation of China (71271021,11001011), the Fundamental Research Funds for the Central Universities of China (2013JBZ005) and the Scientific Research Fund of Hebei Provincial Education Department(QN20132030). The authors thank the editor and the anonymous referees for their valuable comments for improving the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, China

    Meijuan Shang, Chao Zhang & Naihua Xiu

  2. Department of Mathematics, Shijiazhuang University, Shijiazhuang, China

    Meijuan Shang

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  1. Meijuan Shang
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  2. Chao Zhang
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  3. Naihua Xiu
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Correspondence to Chao Zhang.

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Communicated by Lionel Thibault.

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Shang, M., Zhang, C. & Xiu, N. Minimal Zero Norm Solutions of Linear Complementarity Problems. J Optim Theory Appl 163, 795–814 (2014). https://doi.org/10.1007/s10957-014-0549-z

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  • DOI: https://doi.org/10.1007/s10957-014-0549-z

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