$$l_1$$-Regularization for multi-period portfolio selection

Split Bregman iteration for multi-period mean variance portfolio optimization

“…This improves the control on the investment and reduces holding cost. Moreover, we have observed that it penalizes negative components, thus resulting in a penalty on shorting [21] , [22] . This turns out to be useful when short positions are not allowed, since one can obtain positive weights by properly tuning τ 1 .…”

“…Moreover, it seems that the estimation errors for variances and covariances are reduced in this case [18] , [19] . Finally, it has been observed that the application of l 1 regularization has the effect of penalizing short positions [20] , [21] , forbidden in several markets. The structure of Bregman iteration has been exploited both in the static and the dynamic case, to develop procedures which adaptively fix the value of the regularization parameter; this value is chosen as one that provides solutions with certain financial properties, while preserving fidelity to data [21] , [22] .…”

Sparse Approximations with Interior Point Methods

“…A common strategy to estimate Markowitz model parameters is to use historical data as predictive of the future behavior of asset returns. Different regularization techniques have been proposed to deal with ill-conditioning due to asset correlation; in the last years the ℓ 1 -regularization has been used to promote sparsity in the solution [21].…”

“…Another useful interpretation of the ℓ 1 norm is related to the amount of short positions, which indicate an investment strategy where an investor is selling borrowed stocks in the open market, expecting that the market will drop, in order to realize a profit. A suitable tuning of the regularization parameter permits a short controlling in both the single-and the multi-period case [18,21]. However, in the multi-period case, the sparsity in the solution does not guarantee the control of the transaction costs, especially if the pattern of the active positions completely changes across periods.…”

A subspace-accelerated split Bregman method for sparse data recovery with joint l1-type regularizers

“…Different regularization techniques have been suggested with the aim of improving the problem conditioning. In the last years, ℓ 1 regularization techniques have been considered to obtain sparse solutions in both the single and multi-period cases, with the aim of reducing costs [7,13,15]. Another useful interpretation of the ℓ 1 regularization is related to the amount of shorting in the portfolio.…”