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Minimal Zero Norm Solutions of Linear Complementarity Problems

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  • Meijuan Shang

    (Beijing Jiaotong University
    Shijiazhuang University)

  • Chao Zhang

    (Beijing Jiaotong University)

  • Naihua Xiu

    (Beijing Jiaotong University)

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.

Suggested Citation

  • Meijuan Shang & Chao Zhang & Naihua Xiu, 2014. "Minimal Zero Norm Solutions of Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 795-814, December.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:3:d:10.1007_s10957-014-0549-z
    DOI: 10.1007/s10957-014-0549-z
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    References listed on IDEAS

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    1. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    2. B. Blog & G. van der Hoek & A. H. G. Rinnooy Kan & G. T. Timmer, 1983. "The Optimal Selection of Small Portfolios," Management Science, INFORMS, vol. 29(7), pages 792-798, July.
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    Cited by:

    1. Li, Xiangli & Guo, Xiao, 2015. "Spectral residual methods with two new non-monotone line searches for large-scale nonlinear systems of equations," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 59-69.

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