A General Framework for Nonconvex Sparse Mean-CVaR Portfolio Optimization Via ADMM

This paper presents a general framework for addressing sparse portfolio optimization problems using the mean-CVaR (Conditional Value-at-Risk) model and regularization techniques. The framework incorporates a non-negative constraint to prevent the portfolio from being too heavily weighted in certain assets. We propose a specific ADMM (alternating directional multiplier method) for solving the model and provide a subsequential convergence analysis for theoretical integrity. To demonstrate the effectiveness of our framework, we consider the & ell;1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} and SCAD (smoothly clipped absolute deviation) penalties as notable instances within our unified framework. Additionally, we introduce a novel synthesis of the CVaR-based model with & ell;1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document}/& ell;2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document} regularization. We explore the subproblems of ADMM associated with CVaR and the presented regularization functions, employing the gradient descent method to solve the subproblem related to CVaR and the proximal operator to evaluate the subproblems with respect to penalty functions. Finally, we evaluate the proposed framework through a series of parametric and out-of-sample experiments, which shows that the proposed framework can achieve favorable out-of-sample performance. We also compare the performance of the proposed nonconvex penalties with that of convex ones, highlighting the advantages of nonconvex penalties such as improved sparsity and better risk control.