  In this paper, we propose $\ell_p$-norm regularized models to seek
near-optimal sparse portfolios. These sparse solutions reduce the complexity of
portfolio implementation and management. Theoretical results are established to
guarantee the sparsity of the second-order KKT points of the $\ell_p$-norm
regularized models. More interestingly, we present a theory that relates
sparsity of the KKT points with Projected correlation and Projected Sharpe
ratio. We also design an interior point algorithm to obtain an approximate
second-order KKT solution of the $\ell_p$-norm models in polynomial time with a
fixed error tolerance, and then test our $\ell_p$-norm modes on S&P 500
(2008-2012) data and international market data.\ The computational results
illustrate that the $\ell_p$-norm regularized models can generate portfolios of
any desired sparsity with portfolio variance and portfolio return comparable to
those of the unregularized Markowitz model with cardinality constraint. Our
analysis of a combined model lead us to conclude that sparsity is not directly
related to overfitting at all. Instead, we find that sparsity moderates
overfitting only indirectly. A combined $\ell_1$-$\ell_p$ model shows that the
proper choose of leverage, which is the amount of additional buying-power
generated by selling short can mitigate overfitting; A combined
$\ell_2$-$\ell_p$ model is able to produce extremely high performing portfolios
that exceeded the 1/N strategy and all $\ell_1$ and $\ell_2$ regularized
portfolios.
