Abstract
In this paper, we aim to find sparse solutions of co-coercive nonlinear complementarity problems (NCPs). Mathematically, the underlying model is NP-hard in general. Thus an $\ell_{1}$ regularized projection minimization model is proposed for relaxation and an extragradient thresholding algorithm (ETA) is then designed for this regularized model. Furthermore, we analyze the convergence of this algorithm and show any cluster point of the sequence generated by ETA is a solution of NCP. Numerical results demonstrate that the ETA can effectively solve the $\ell_{1}$ regularized model and output very sparse solutions of co-coercive NCPs with high quality.
Highlights
The nonlinear complementarity problem, denoted by the nonlinear complementarity problems (NCPs)(F), is to find a vector x ∈ Rn such that x ≥, F(x) ≥, x F(x) =, where F is a mapping from Rn into itself
In Section, we propose an extragradient thresholding algorithm (ETA) for ( ) and analyze the convergence of this algorithm
3 Algorithm and convergence we suggest the extragradient thresholding algorithm (ETA) to solve regularization projection minimization problem ( ) and give the convergence analysis of ETA
Summary
The set of solutions to this problem is denoted by SOL(F). Numerical methods for solving NCPs, such as filter method, continuation method, nonsmooth Newton’s method, smoothing Newton methods, Levenberg-Marquardt method, projection method, descent method, interior-point method have been extensively investigated in the literature. It seems that there is a vacant study of sparse solutions for NCPs. in real applications, it is very necessary to investigate the sparse solution of the NCPs. in real applications, it is very necessary to investigate the sparse solution of the NCPs For example this is so in bimatrix games [ ] and portfolio selections [ ].
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