Sparse portfolio optimization via [formula omitted] over [formula omitted] regularization

Abstract

Sparse portfolio optimization, which significantly boosts the out-of-sample performance of traditional mean–variance methods, is widely studied in the fields of optimization and financial economics. In this paper, we explore the ℓ1/ℓ2 fractional regularization constructed by the ratio of the ℓ1 and ℓ2 norms on the mean–variance model to promote sparse portfolio selection. We present an ℓ1/ℓ2 regularized sparse portfolio optimization model and provide financial insights regarding short positions and estimation errors. Then, we develop an efficient alternating direction method of multipliers (ADMM) method to solve it numerically. Due to the nonconvexity and noncoercivity of the ℓ1/ℓ2 term, we give the convergence analysis for the proposed ADMM based on the nonconvex optimization framework. Furthermore, we discuss an extension of the model to incorporate a more general ℓ1/ℓq regularization, where q>1. Moreover, we conduct numerical experiments on four stock datasets to demonstrate the effectiveness and superiority of the proposed model in promoting sparse portfolios while achieving the desired level of expected return.