A Modified ADMM Approach for the Sparse Large-Dimensional Vector Estimation with Application to Portfolio Management

Abstract This article proposes a novel methodology for estimating a sparse large-dimensional vector, specifically the product of the precision matrix of asset returns and a vector of ones. This estimation is highly effective for constructing an optimal portfolio strategy within a substantial investment pool. We propose a well-defined objective function with two l1-norm penalties: one for the large-dimensional vector being estimated and another for changes in portfolio weights with each update. This dual-penalty approach aims to achieve sparse portfolio weights while effectively minimizing transaction costs. To solve the resulting constrained convex minimization problem, we have developed an efficient algorithm based on the Alternating Direction Method of Multipliers. The advantages of our proposed approach are validated through both theoretical and empirical analyses. Comprehensive empirical studies demonstrate that our method surpasses plug-in global minimum variance portfolio strategies, yielding lower out-of-sample volatility, comparable Sharpe ratios, and reduced turnover of portfolio weights.