Split Bregman iteration for multi-period mean variance portfolio optimization

This paper investigates the problem of defining an optimal long-term investment strategy, where the investor can exit the investment before maturity without severe loss. Our setting is a multi-period one, where the aim is to make a plan for allocating all of wealth among the n assets within a time horizon of m periods. In addition, the investor can rebalance the portfolio at the beginning of each period. We develop a model in Markowitz context, based on a fused lasso approach. According to it, both wealth and its variation across periods are penalized using the l1 norm, so to produce sparse portfolios, with limited number of transactions. The model leads to a non-smooth constrained optimization problem, where the inequality constraints are aimed to guarantee at least a minimum level of expected wealth at each date. We solve it by using split Bregman method, that has proved to be efficient in the solution of this type of problems. Due to the additive structure of the objective function, the alternating split Bregman at each iteration yields to easier subproblems to be solved, which either admit closed form solutions or can be solved very quickly. Numerical results on data sets generated using real-world price values show the effectiveness of the proposed model.