Cardinality-constrained maximal predictability portfolios with an $ \ell_2 $ regularization

The maximal predictability portfolio (MPP) is a portfolio optimization framework that explicitly incorporates return predictability into the objective function. While MPP has been shown to achieve favorable empirical performance, it is prone to overfitting when using rich sets of candidate assets and factors. To address this issue, we study a cardinality-constrained MPP formulation with an $ \ell_2 $ regularization term and investigate how regularization reshapes the structure and out-of-sample behavior of MPP. The resulting formulation leads to a challenging optimization problem involving binary variables and nonconvex constraints. We show that, through an appropriate bilevel reformulation, the problem can be handled within the cutting-plane algorithm (CPA) framework. In particular, by exploiting a globally optimal solution of the primal lower-level problem, we derive an equivalent convex formulation that preserves the finite termination and $ \varepsilon $-optimality guarantees of CPA. We also derive a subgradient expression tailored to the MPP structure, which enables the construction of valid cutting planes despite the nonconvexity of the original formulation. For practical large-scale computation, we further incorporate normalized linearization as a heuristic accelerator for the lower-level problem, which improves tractability. However, the exact finite termination and $ \varepsilon $-optimality guarantees of CPA no longer directly apply. Numerical experiments demonstrate that the proposed approach, combined with normalized linearization, can efficiently compute high-quality solutions for large-scale instances where exact methods become computationally impractical. The results further illustrate that moderate $ \ell_2 $ regularization improves out-of-sample predictability, whereas excessive regularization diminishes the predictive structure of MPP. Overall, our findings highlight the role of $ \ell_2 $ regularization in balancing predictability, stability, and practical feasibility within the MPP framework.