An efficient ADMM for multi-period sparse behavioral portfolio optimization based on cumulative prospect theory

Qingyang Wang; Kunpeng Zhu; Yanjing Guo; Liu Yang; Zhongming Wu; Qingyang Wang; Kunpeng Zhu; Yanjing Guo; Liu Yang; Zhongming Wu

doi:10.3934/jimo.2026064

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 4: 1726-1757. doi: 10.3934/jimo.2026064

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Research article

An efficient ADMM for multi-period sparse behavioral portfolio optimization based on cumulative prospect theory

1.
School of Management and Engineering, Nanjing University, Nanjing 210093, China
2.
School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received: 15 December 2025 Revised: 08 February 2026 Accepted: 24 February 2026 Published: 11 March 2026
91G10, 65K05, 90C06

Traditional portfolio optimization approaches typically presume a completely rational market, overlooking investors' behavioral inclinations and the complexity of sparsity in high-dimensional data. To tackle these concerns, this study presents a novel multi-period sparse behavioral portfolio optimization model. In a multi-period context, the model incorporates a utility function based on cumulative prospect theory, effectively capturing the irrational behavioral characteristics of investors. By leveraging $ \ell_1 $-norm regularization, it attains portfolio sparsity in each period and minimizes turnover across periods. Next, we proposed a hybrid algorithm that integrates the alternating direction method of multipliers with the pooling-adjacent-violators algorithm to efficiently solve the newly formulated model. Furthermore, the framework incorporates environmental, social, and governance factors to evaluate their influence on investors' behavioral portfolios. Numerical experiments and empirical analyses demonstrated that the proposed method can efficiently solve the model, and the new model is capable of reducing risk and transaction costs.
- portfolio optimization,
- multi-period,
- $ \ell_1 $-norm regularization,
- cumulative prospect theory,
- alternating direction method of multipliers
Citation: Qingyang Wang, Kunpeng Zhu, Yanjing Guo, Liu Yang, Zhongming Wu. An efficient ADMM for multi-period sparse behavioral portfolio optimization based on cumulative prospect theory[J]. Journal of Industrial and Management Optimization, 2026, 22(4): 1726-1757. doi: 10.3934/jimo.2026064

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Abstract

Traditional portfolio optimization approaches typically presume a completely rational market, overlooking investors' behavioral inclinations and the complexity of sparsity in high-dimensional data. To tackle these concerns, this study presents a novel multi-period sparse behavioral portfolio optimization model. In a multi-period context, the model incorporates a utility function based on cumulative prospect theory, effectively capturing the irrational behavioral characteristics of investors. By leveraging $ \ell_1 $-norm regularization, it attains portfolio sparsity in each period and minimizes turnover across periods. Next, we proposed a hybrid algorithm that integrates the alternating direction method of multipliers with the pooling-adjacent-violators algorithm to efficiently solve the newly formulated model. Furthermore, the framework incorporates environmental, social, and governance factors to evaluate their influence on investors' behavioral portfolios. Numerical experiments and empirical analyses demonstrated that the proposed method can efficiently solve the model, and the new model is capable of reducing risk and transaction costs.

References

[1]	A. Gunjan, S. Bhattacharyya, A brief review of portfolio optimization techniques, Artif. Intell. Rev., 56 (2023), 3847–3886. http://doi.org/10.1007/s10462-022-10273-7 doi: 10.1007/s10462-022-10273-7
[2]	G. Curtis, Modern portfolio theory and behavioural finance, J. Wealth Manag., 7 (2004), 16–22. http://doi.org/10.3905/jwm.2004.434562 doi: 10.3905/jwm.2004.434562
[3]	V. N. C. Mushinada, Are individual investors irrational or adaptive to market dynamics? J. Behav. Exp. Finance, 25 (2020), 100243. http://doi.org/10.1016/j.jbef.2019.100243
[4]	Y. Chen, B. Li, An uncertainty theory based tri-objective behavioral portfolio selection model with loss aversion and reference level using a modified evolutionary root system growth algorithm, J. Comput. Appl. Math., 446 (2024), 115859. http://doi.org/10.1016/j.cam.2024.115859 doi: 10.1016/j.cam.2024.115859
[5]	X. Li, B. Li, T. Jin, P. Zheng, Uncertain random portfolio optimization with non-dominated sorting genetic algorithm-II and optimal solution criterion, Artif. Intell. Rev., 56 (2023), 8511–8546. http://doi.org/10.1007/s10462-022-10388-x doi: 10.1007/s10462-022-10388-x
[6]	X. Li, A. S. Uysal, J. M. Mulvey, Multi-period portfolio optimization using model predictive control with mean-variance and risk parity frameworks, Eur. J. Oper. Res., 299 (2022), 1158–1176. http://doi.org/10.1016/j.ejor.2021.10.002 doi: 10.1016/j.ejor.2021.10.002
[7]	D. Li, W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Math. Finance, 10 (2000), 387–406. http://doi.org/10.1111/1467-9965.00100 doi: 10.1111/1467-9965.00100
[8]	X. Cui, J. Gao, X. Li, D. Li, Optimal multi-period mean–variance policy under no-shorting constraint, Eur. J. Oper. Res., 234 (2014), 459–468. http://doi.org/10.1016/j.ejor.2013.02.040 doi: 10.1016/j.ejor.2013.02.040
[9]	Z. Chen, G. Consigli, J. Liu, G. Li, T. Fu, Q. Hu, Multi-period risk measures and optimal investment policies, In: Optimal financial decision making under uncertainty, 2016. http://doi.org/10.1007/978-3-319-41613-7_1
[10]	M. A. Moghadam, S. B. Ebrahimi, D. Rahmani, A constrained multi-period robust portfolio model with behavioral factors and an interval semi-absolute deviation, J. Comput. Appl. Math., 374 (2020), 112742. http://doi.org/10.1016/j.cam.2020.112742 doi: 10.1016/j.cam.2020.112742
[11]	Y. Dai, Z. Qin, Multi-period uncertain portfolio optimization model with minimum transaction lots and dynamic risk preference, Appl. Soft Comput., 109 (2021), 107519. http://doi.org/10.1016/j.asoc.2021.107519 doi: 10.1016/j.asoc.2021.107519
[12]	R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B, 58 (1996), 267–288. http://doi.org/10.1111/j.2517-6161.1996.tb02080.x doi: 10.1111/j.2517-6161.1996.tb02080.x
[13]	R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, K. Knight, Sparsity and smoothness via the fused LASSO, J. R. Stat. Soc. Ser. B, 67 (2005), 91–108. http://doi.org/10.1111/j.1467-9868.2005.00490.x doi: 10.1111/j.1467-9868.2005.00490.x
[14]	S. Corsaro, V. De Simone, Z. Marino, Fused lasso approach in portfolio selection, Ann. Oper. Res., 299 (2021), 47–59. http://doi.org/10.1007/s10479-019-03289-w doi: 10.1007/s10479-019-03289-w
[15]	F. Cesarone, J. Moretti, F. Tardella, Optimally chosen small portfolios are better than large ones, Econ. Bull., 36 (2016), 1876–1891.
[16]	M. Rezaei, H. Nezamabadi-Pour, A taxonomy of literature reviews and experimental study of deep reinforcement learning in portfolio management, Artif. Intell. Rev., 58 (2025), 94. http://doi.org/10.1007/s10462-024-11066-w doi: 10.1007/s10462-024-11066-w
[17]	P. N. Kolm, R. Tümentüncü, F. J. Fabozzi, 60 years of portfolio optimization: Practical challenges and current trends, Eur. J. Oper. Res., 234 (2014), 356–371. http://doi.org/10.1016/j.ejor.2013.10.060 doi: 10.1016/j.ejor.2013.10.060
[18]	Z. Yang, Y. Zhao, Hybrid SGD algorithms to solve stochastic composite optimization problems with application in sparse portfolio selection problems, J. Comput. Appl. Math., 436 (2024), 115425. http://doi.org/10.1016/j.cam.2023.115425 doi: 10.1016/j.cam.2023.115425
[19]	Y. Wang, L. Kong, H. Qi, An efficient Lagrange–Newton algorithm for long-only cardinality constrained portfolio selection on real data sets, J. Comput. Appl. Math., 461 (2025), 116453. http://doi.org/10.1016/j.cam.2024.116453 doi: 10.1016/j.cam.2024.116453
[20]	W. Guo, W. G. Zhang, X. Chen, Portfolio selection models considering fuzzy preference relations of decision makers, IEEE Trans. Syst. Man Cybern. Syst., 54 (2024), 2254–2265. http://doi.org/10.1109/TSMC.2023.3342038 doi: 10.1109/TSMC.2023.3342038
[21]	D. Kahneman, Prospect theory: An analysis of decisions under risk, Econometrica, 47 (1979), 278. http://doi.org/10.2307/1914185 doi: 10.2307/1914185
[22]	A. Tversky, D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, J. Risk Uncertain., 5 (1992), 297–323. http://doi.org/10.1007/BF00122574 doi: 10.1007/BF00122574
[23]	X. D. He, X. Y. Zhou, Portfolio choice under cumulative prospect theory: An analytical treatment, Manag. Sci., 57 (2011), 315–331. http://doi.org/10.1287/mnsc.1100.1269 doi: 10.1287/mnsc.1100.1269
[24]	X. Chen, X. Liu, Q. Zhu, Z. Wang, DEA cross-efficiency models with prospect theory and distance entropy: An empirical study on high-tech industries, Expert Syst. Appl., 244 (2024), 122941. http://doi.org/10.1016/j.eswa.2023.122941 doi: 10.1016/j.eswa.2023.122941
[25]	Y. Shi, X. Cui, J. Yao, D. Li, Dynamic trading with reference point adaptation and loss aversion, Oper. Res., 63 (2015), 789–806. http://doi.org/10.1287/opre.2015.1399 doi: 10.1287/opre.2015.1399
[26]	Y. E. Rodríguez, J. M. Gómez, J. Contreras, Diversified behavioral portfolio as an alternative to modern portfolio theory, North Am. J. Econ. Finance, 58 (2021), 101508. http://doi.org/10.1016/j.najef.2021.101508 doi: 10.1016/j.najef.2021.101508
[27]	S. van Bilsen, R. J. A. Laeven, Dynamic consumption and portfolio choice under prospect theory, Insur. Math. Econ., 91 (2020), 224–237. http://doi.org/10.1016/j.insmatheco.2020.02.004 doi: 10.1016/j.insmatheco.2020.02.004
[28]	E. Luxenberg, P. Schiele, S. Boyd, Portfolio Optimization with Cumulative Prospect Theory Utility via Convex Optimization, Comput. Econ., 64 (2024), 3027–3047. https://doi.org/10.1007/s10614-024-10556-x doi: 10.1007/s10614-024-10556-x
[29]	Y. Shi, X. Cui, D. Li, Discrete-time behavioural portfolio selection under cumulative prospect theory, J. Econ. Dyn. Control, 61 (2015), 283–302. http://doi.org/10.1016/j.jedc.2015.10.002 doi: 10.1016/j.jedc.2015.10.002
[30]	S. Srivastava, A. Aggarwal, A. Mehra, Portfolio selection by cumulative prospect theory and its comparison with mean-variance model, Granul. Comput., 7 (2022), 903–916. http://doi.org/10.1007/s41066-021-00302-1 doi: 10.1007/s41066-021-00302-1
[31]	T. Hens, J. Mayer, Cumulative prospect theory and mean variance analysis: A rigorous comparison, Swiss Finance Institute Research Paper, 2014. http://doi.org/10.2139/ssrn.2417191
[32]	C. Fulga, Portfolio optimization under loss aversion, Eur. J. Oper. Res., 251 (2016), 310–322. http://doi.org/10.1016/j.ejor.2015.11.038 doi: 10.1016/j.ejor.2015.11.038
[33]	J. Bi, H. Jin, Q. Meng, Behavioural mean-variance portfolio selection, Eur. J. Oper. Res., 271 (2018), 644–663. http://doi.org/10.1016/j.ejor.2018.05.065 doi: 10.1016/j.ejor.2018.05.065
[34]	M. Kaucic, F. Piccotto, G. Sbaiz, G. Valentinuz and others, Optimal portfolio with sustainable attitudes under cumulative prospect theory, J. Appl. Finance Bank., 13 (2023), 65–86. http://doi.org/10.47260/jafb/1344
[35]	J. Wei, Y. Yang, M. Jiang, J. Liu, Dynamic multi-period sparse portfolio selection model with asymmetric investors' sentiments, Expert Syst. Appl., 177 (2021), 114945. http://doi.org/10.1016/j.eswa.2021.114945 doi: 10.1016/j.eswa.2021.114945
[36]	Z. Wu, L. Yang, V. De Simone, Behavioral portfolio optimization via cumulative prospect theory with a symmetric alternating direction method of multipliers, Comput. Optim. Appl., 93 (2025), 303–334. http://doi.org/10.1007/s10589-025-00721-9 doi: 10.1007/s10589-025-00721-9
[37]	K. Blay, A. Gosh, S. Kusiak, H. Markowitz, N. Savoulides, Q. Zheng, Multiperiod portfolio selection: A practical simulation-based framework, J. Invest. Manag., 18 (2020), 94–129.
[38]	G. N. Raj, Adaptive and Regime-Aware RL for Portfolio Optimization, arXiv: 2509.14385, 2025. http://doi.org/10.48550/arXiv.2509.14385
[39]	S. Corsaro, V. De Simone, Z. Marino, Split Bregman iteration for multi-period mean variance portfolio optimization, Appl. Math. Comput., 392 (2021), 125715. http://doi.org/10.1016/j.amc.2020.125715 doi: 10.1016/j.amc.2020.125715
[40]	S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), 1–122.
[41]	K. Guo, D. Han, T. Wu, Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints, Int. J. Comput. Math., 94 (2017), 1653–1669. http://doi.org/10.1080/00207160.2016.1227432 doi: 10.1080/00207160.2016.1227432
[42]	D. Han, A survey on some recent developments of alternating direction method of multipliers, J. Oper. Res. Soc. China, 10 (2022), 1–52. http://doi.org/10.1007/s40305-021-00368-3 doi: 10.1007/s40305-021-00368-3
[43]	M. Ayer, H. D. Brunk, G. M. Ewing, W. T. Reid, E. Silverman, An empirical distribution function for sampling with incomplete information, Ann. Math. Stat., 26 (1955), 641–647. http://doi.org/10.1214/aoms/1177728423 doi: 10.1214/aoms/1177728423
[44]	H. D. Brunk, R. E. Barlow, D. J. Bartholomew, J. M. Bremner, Statistical inference under order restrictions the theory and application of isotonic regression, Int. Stat. Rev., 41 (1972), 195.
[45]	R. K. Ahuja, J. B. Orlin, A fast scaling algorithm for minimizing separable convex functions subject to chain constraints, Oper. Res., 49 (2001), 784–789. http://doi.org/10.1287/opre.49.5.784.10601 doi: 10.1287/opre.49.5.784.10601
[46]	Z. Cui, C. Lee, L. Zhu, Y. Zhu, Non-convex isotonic regression via the Myersonian approach, Stat. Probab. Lett., 179 (2021), 109210. http://doi.org/10.1016/j.spl.2021.109210 doi: 10.1016/j.spl.2021.109210
[47]	Z. Yu, X. Chen, X. Li, A dynamic programming approach for generalized nearly isotonic optimization, Math. Program. Comput., 15 (2023), 195–225. http://doi.org/10.1007/s12532-022-00229-x doi: 10.1007/s12532-022-00229-x
[48]	X. Cui, R. Jiang, Y. Shi, R. Xiao, Y. Yan, Decision making under cumulative prospect theory: An alternating direction method of multipliers, INFORMS J. Comput., 37 (2024), 856–873. http://doi.org/10.1287/ijoc.2023.0243 doi: 10.1287/ijoc.2023.0243
[49]	M. J. Best, N. Chakravarti, V. A. Ubhaya, Minimizing separable convex functions subject to simple chain constraints, SIAM J. Optim., 10 (2000), 658–672. http://doi.org/10.1137/S1052623497314970 doi: 10.1137/S1052623497314970
[50]	R. T. Rockafellar, R. J. B. Wets, Variational Analysis, In: Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1998. http://doi.org/10.1007/978-3-642-02431-3
[51]	F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, PA, 1990. http://doi.org/10.1137/1.9781611971309
[52]	Z. Wu, K. Sun, Z. Ge, Z. Allen-Zhao, T. Zeng, Sparse portfolio optimization via $\ell_1$ over $\ell_2$ regularization, Eur. J. Oper. Res., 319 (2024), 820–833. http://doi.org/10.1016/j.ejor.2024.07.017 doi: 10.1016/j.ejor.2024.07.017
[53]	H. Attouch, J. Bolte, B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods, Math. Program., 137 (2013), 91–129. http://doi.org/10.1007/s10107-011-0484-9 doi: 10.1007/s10107-011-0484-9
[54]	J. Bolte, S. Sabach, M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), 459–494. http://doi.org/10.1007/s10107-013-0701-9 doi: 10.1007/s10107-013-0701-9
[55]	Z. Wu, M. Li, D. Wang, D. Han, A symmetric alternating direction method of multipliers for separable nonconvex minimization problems, Asia-Pac. J. Oper. Res., 34 (2017), 1750030. http://doi.org/10.1142/S0217595917500300 doi: 10.1142/S0217595917500300
[56]	S. Utz, M. Wimmer, R. Steuer, Tri-criterion modeling for constructing more-sustainable mutual funds, Eur. J. Oper. Res., 246 (2015), 331–338. http://doi.org/10.1016/j.ejor.2015.04.035 doi: 10.1016/j.ejor.2015.04.035
[57]	L. H. Pedersen, S. Fitzgibbons, L. Pomorski, Responsible investing: The ESG-efficient frontier, J. Financ. Econ., 142 (2021), 572–597. http://doi.org/10.1016/j.jfineco.2020.11.001 doi: 10.1016/j.jfineco.2020.11.001
[58]	Z. Wu, L. Yang, Y. Fei, X. Wang, Regularization methods for sparse ESG-valued multi-period portfolio optimization with return prediction using machine learning, Expert Syst. Appl., 232 (2023), 120850. http://doi.org/10.1016/j.eswa.2023.120850 doi: 10.1016/j.eswa.2023.120850
[59]	D. Lauria, W. Brent Lindquist, S. Mittnik, Svetlozar T. Rachev, ESG-valued portfolio optimization and dynamic asset pricing, arXiv: 2206.02854, 2022. http://doi.org/10.48550/arXiv.2206.02854

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