  Recently, $L_1$ regularization have been attracted extensive attention and
successfully applied in mean-variance portfolio selection for promoting
out-of-sample properties and decreasing transaction costs. However, $L_1$
regularization approach is ineffective in promoting sparsity and selecting
regularization parameter on index tracking with the budget and no-short selling
constraints, since the 1-norm of the asset weights will have a constant value
of one. Our recent research on $L_{1/2}$ regularization has found that the half
thresholding algorithm with optimal regularization parameter setting strategy
is the fast solver of $L_{1/2}$ regularization, which can provide the more
sparse solution. In this paper we apply $L_{1/2}$ regularization method to
stock index tracking and establish a new sparse index tracking model. A hybrid
half thresholding algorithm is proposed for solving the model. Empirical tests
of model and algorithm are carried out on the eight data sets from OR-library.
The optimal tracking portfolio obtained from the new model and algorithm has
lower out-of-sample prediction error and consistency both in-sample and
out-of-sample. Moreover, since the automatic regularization parameters are
selected for the fixed number of optimal portfolio, our algorithm is a fast
solver, especially for the large scale problem.
