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

doi:10.1007/s40305-024-00551-2

Abstract

Abstract: 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 themodel and provide a subsequential convergence analysis for theoretical integrity. To demonstrate the effectiveness of our framework, we consider the $\ell$₁ 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/$\ell$₂ 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.

Key words: Portfolio optimization, Mean-CVaR, Sparse regularization, Alternating direction method of multipliers

CLC Number:

Ke-Xin Sun, Zhong-Ming Wu, Neng Wan. A General Framework for Nonconvex Sparse Mean-CVaR Portfolio Optimization Via ADMM[J]. Journal of the Operations Research Society of China, 2024, 12(4): 1022-1047.

TrendMD

References

[1] Alexander, G.J., Baptista, A.M.: Economic implications of using a mean-VaR model for portfolio selection: a comparison with mean-variance analysis. J. Econ. Dyn. Control. 26(7-8), 1159-1193(2002)
[2] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203- 228(1999)
[3] Bodnar, T., Lindholm, M., Niklasson, V., Thorsén, E.: Bayesian portfolio selection using VaR and CVaR. Appl. Math. Comput. 427, 127120(2022)
[4] Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1-2), 459-494(2014)
[5] Brodie, J., Daubechies, I., De Mol, C., Giannone, D., Loris, I.: Sparse and stable Markowitz portfolios. Proc. Natl. Acad. Sci. 106(30), 12267-12272(2009)
[6] Corsaro, S., De Simone, V., Marino, Z.: Split Bregman iteration for multi-period mean variance portfolio optimization. Appl. Math. Comput. 392, 125715(2009)
[7] Corsaro, S., Simone, V.D., Marino, Z.: Fused lasso approach in portfolio selection. Ann. Oper. Res. 299(1), 47-59(2021)
[8] Cui, A., Peng, J., Zhang, C., Li, H., Wen, M.: Sparse portfolio selection via non-convex fraction function (2021). arXiv:1801.09171
[9] DeMiguel, V., Garlappi, L., Nogales, F.J., Uppal, R.: A generalized approach to portfolio optimization: improving performance by constraining portfolio norms. Manage. Sci. 55(5), 798-812(2009)
[10] Derumigny, A.: Improved bounds for Square-Root Lasso and Square-Root Slope. Electr. J. Stat. 12(1), 741-766(2018)
[11] Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348-1360(2001)
[12] Goel, A., Sharma, A., Mehra, A.: Index tracking and enhanced indexing using mixed conditional value-at-risk. J. Comput. Appl. Math. 335, 361-380(2018)
[13] Guo, K., Han, D., Wang, D.Z., Wu, T.: Convergence of ADMM for multi-block nonconvex separable optimization models. Front. Math. China. 12, 1139-1162(2017)
[14] Guo, K., Han, D., Wu, T.: Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints. Int. J. Comput. Math. 94(8), 1653-1669(2017)
[15] Han, D.: A survey on some recent developments of alternating direction method of multipliers. J. Op. Res. Soc. China. 10, 1-52(2022)
[16] Jia, Z., Gao, X., Cai, X., Han, D.: Local linear convergence of the alternating direction method of multipliers for nonconvex separable optimization problems. J. Optim. Theory Appl. 188, 1-25(2021)
[17] Kim, Y., Choi, H., Oh, H.-S.: Smoothly clipped absolute deviation on high dimensions. J. Am. Stat. Assoc. 103(484), 1665-1673(2008)
[18] Lai, Z.-R., Yang, P.-Y., Fang, L., Wu, X.: Short-term sparse portfolio optimization based on alternating direction method of multipliers. J. Mach. Learn. Res. 19(1), 2547-2574(2018)
[19] Lee, S.-I., Ganapathi, V., Koller, D.: Efficient structure learning of Markov networks using L₁- regularization. Adv. Neural Inf. Process. Syst. (2006). https://doi.org/10.7551/mitpress/7503.003.0107
[20] Li, B., Teo, K.L.: Portfolio optimization in real financial markets with both uncertainty and randomness. Appl. Math. Model. 100, 125-137(2021)
[21] Liu, J., Chen, Z., Lisser, A., Xu, Z.: Closed-form optimal portfolios of distributionally robust meanCVaR problems with unknown mean and variance. Appl. Math. Optim. 79(3), 671-693(2019)
[22] Lwin, K.T., Qu, R., MacCarthy, B.L.: Mean-VaR portfolio optimization: a nonparametric approach. Eur. J. Oper. Res. 260(2), 751-76(2017)
[23] Markowitz, H.M.: Portfolio selection. J. Finance. 7(1), 71-91(1952)
[24] Munos, R.: Performance bounds in $\ell$_p-norm for approximate value iteration. SIAM J. Control. Optim. 46(2), 541-561(2007)
[25] Pflug, G.C.: Some remarks on the value-at-risk and the conditional value-at-risk, pp. 272-281. Probabilistic constrained optimization. Springer, Berlin (2000)
[26] Ramirez-Pico, C., Moreno, E.: Generalized adaptive partition-based method for two-stage stochastic linear programs with fixed recourse. Math. Program. 196(1-2), 755-774(2022)
[27] Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21-42(2000)
[28] Roy, A.D.: Safety first and the holding of assets. Econom.: J. Econom. Soc. 20(3), 431-449(1952)
[29] Simon, N., Friedman, J., Hastie, T., Tibshirani, R.: A sparse-group lasso. J. Comput. Graph. Stat. 22(2), 231-245(2013)
[30] Tao, M.: Minimization of L₁ over L₂ for sparse signal recovery with convergence guarantee. SIAM J. Sci. Comput. 44(2), A770-A797(2022)
[31] Tao, M., Zhang, X.-P.: A unified study on $\ell$₁ over $\ell$₂ minimization (2021). arXiv:2108.01269
[32] Wang, C., Tao, M., Nagy, J.G., Lou, Y.: Limited-angle CT reconstruction via the $\ell$₁/$\ell$₂ minimization. SIAM J. Imag. Sci. 14(2), 749-777(2021)
[33] Wang, H., Zhang, W., He, Y., Cao, W.: 0-norm based short-term sparse portfo- lio optimization algorithm based on alternating direction method of multipliers (2023). Available at SSRN: https://ssrn.com/abstract=4115395
[34] Wu, Y., Liu, Y.: Variable selection in quantile regression. Statistica Sinica, pp. 801-817(2009)
[35] Wu, Z., Li, M.: General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems. Comput. Optim. Appl. 73, 129-158(2019)
[36] Wu, Z., Sun, K., Ge, Z., Zeng, T.: Sparse portfolio optimization via a novel fractional regularization (2023). Available at SSRN: https://ssrn.com/abstract=4666990
[37] Xie, H., Huang, J.: SCAD-penalized regression in high-dimensional partially linear models. Ann. Stat. 37(2), 673-696(2009)
[38] Xu, Q., Zhou, Y., Jiang, C., Yu, K., Niu, X.: A large CVaR-based portfolio selection model with weight constraints. Econ. Model. 59, 436-447(2016)
[39] Yang, L., Pong, T.K., Chen, X.: Alternating direction method of multipliers for a class of nonconvex and non-smooth problems with applications to back ground/foreground extraction. SIAM J. Imag. Sci. 10(1), 74-110(2017)
[40] Zhang, C.-H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894-942(2010)
[41] Zhao, H., Kong, L., Qi, H.-D.: Optimal portfolio selections via $\ell$_1,2-norm regularization. Comput. Optim. Appl. 80(3), 853-881(2021)
[42] Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B Stat Methodol. 67(2), 301-320(2005)