Multi-period mean–variance portfolio optimization with capital injections

In this study, we explore the portfolio optimization problem where the investors initially allocate portions of their capital across a large asset pool, followed by gradual capital injections over the subsequent periods. We introduce a multi-period mean–variance model with capital injections to develop a sparse long-term investment strategy within this framework. This model adopts the fused Lasso technique, integrating two ${\ell }_1$ penalty terms designed to lower both holding and trading costs. We utilize a two-block alternating direction method of multipliers algorithm to solve this complex, non-smooth optimization problem involving multiple variables. A thorough analysis of the convergence of the algorithm is provided. In addition, we empirically validate the efficacy of our model using two real datasets, demonstrating its practical applicability and effectiveness in real-world scenarios.