Robust and Sparse Portfolio: Optimization Models and Algorithms

Abstract

The robust and sparse portfolio selection problem is one of the most-popular and -frequently studied problems in the optimization and financial literature. By considering the uncertainty of the parameters, the goal is to construct a sparse portfolio with low volatility and decent returns, subject to other investment constraints. In this paper, we propose a new portfolio selection model, which considers the perturbation in the asset return matrix and the parameter uncertainty in the expected asset return. We define three types of stationary points of the penalty problem: the Karush–Kuhn–Tucker point, the strong Karush–Kuhn–Tucker point, and the partial minimizer. We analyze the relationship between these stationary points and the local/global minimizer of the penalty model under mild conditions. We design a penalty alternating-direction method to obtain the solutions. Compared with several existing portfolio models on seven real-world datasets, extensive numerical experiments demonstrate the robustness and effectiveness of our model in generating lower volatility.

Keywords

• portfolio optimization
• robustness
• sparsity
• uncertainty set
• penalty-alternating-direction method