Sparse portfolio selection with uncertain probability distribution


Designed as remedies for uncertain parameters and tiny optimal weights in the portfolio selection problem, we consider a class of distributionally robust portfolio optimization models with cardinality constraints. For considering the statistical significance and tractability, we construct two kinds of ambiguity sets based on L1-norm and moment information, respectively. The nominal distribution, as the core of the first ambiguity set, is determined by non-parametric estimation method. To reduce the disturbing error of the second ambiguity set, we apply a shrinkage estimation method to determine the moment information based on historical data. By introducing a binary variable, the proposed sparse portfolio optimization model can be converted equivalently to a tractable mixed-integer 0-1 programming problem, which can be dealt with efficiently by a modified primal-dual Benders’ decomposition method. Through the actual market data, we test the proposed models and show their validity. Furthermore, performances measured by net portfolio return, Sharpe ratio, and cumulative return are superior to the classical portfolio selection models in the back-testing of out-of-sample data.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Given N historical data, \(\boldsymbol {\xi }^{j}\in \mathbb {R}^{n}, j=1,...,N\), \(\boldsymbol {X}^{j}=\boldsymbol {\xi }^{j}-\frac {1}{N}{\sum }^{N}_{j=1}\boldsymbol {\xi }^{j}\).

  2. 2.


  1. 1.

    Markowitz H (1952) J Financ 7(1):77–91.

    Google Scholar 

  2. 2.

    Ogryczak W, Ruszczynski A (1999) From stochastic dominance to mean-risk models: Semideviations as risk measures. Eur J Oper Res 116(1):33–50.

    MATH  Article  Google Scholar 

  3. 3.

    Rockafellar RT, Uryasev S (2000) Optimization of conditional value at risk. J Risk 3(3):21–41.

    Article  Google Scholar 

  4. 4.

    Ledoit O, Wolf M (2003) Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J Empir Financ 10(5):603–621.

    Article  Google Scholar 

  5. 5.

    Benati S, Rizzi R (2007) A mixed integer linear programming formulation of the optimal mean/value-at-risk portfolio problem. Eur J Oper Res 176(1):423–434.

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bjork T, Murgoci A, Zhou X Y (2014) Mean-variance portfolio optimization with state-dependent risk aversion. Math Financ 24(1):1–24.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Kuhn D, Parpas P, Rustem B, Fonseca RJ (2009) Dynamic mean-variance portfolio analysis under model risk. J Comput Financ 12(4):91–115.

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Kim WC, Kim JH, Fabozzi FJ (2015) Shortcomings of mean-variance analysis. In: Robust Equity Portfolio Management + Website. Wiley, pp 22–38

  9. 9.

    Brandimarte P (2002) Numerical methods in finance: A matlab-based introduction. Wiley Interscience

  10. 10.

    Brodie J, Daubechies I, De Mol C, Giannone D, Loris I (2009) Sparse and stable markowitz portfolios. Proc Natl Acad Sci U S A 106(30):12267–12272.

    MATH  Article  Google Scholar 

  11. 11.

    Wang M, Xu F, Wang G (2014) Sparse portfolio rebalancing model based on inverse optimization. Optim Methods Softw 29(2):297–309.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Li J (2015) Sparse and stable portfolio selection with parameter uncertainty. J Bus Econ Stat 33(3):381–392.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Dai Z, Wen F (2018) Some improved sparse and stable portfolio optimization problems. Financ Res Lett 27:46–52.

    Article  Google Scholar 

  14. 14.

    Lai Z, Yang P, Fang L, Wu X (2018) Short-term sparse portfolio optimization based on alternating direction method of multipliers. J Mach Learn Res 19(63):1–28

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Pun CS, Wong HY (2019) A linear programming model for selection of sparse high-dimensional multiperiod portfolios. Eur J Oper Res 273(2):754–771.

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Chang T-J, Meade N, Beasley JE, Sharaiha YM (2000) Heuristics for cardinality constrained portfolio optimisation. Comput Oper Res 27(13):1271–1302.

    MATH  Article  Google Scholar 

  17. 17.

    Di Lorenzo D, Liuzzi G, Rinaldi F, Schoen F, Sciandrone M (2012) A concave optimization-based approach for sparse portfolio selection. Optim Methods Softw 27(6):983–1000.

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Meghwani SS, Thakur M (2017) Multi-objective heuristic algorithms for practical portfolio optimization and rebalancing with transaction cost. Appl Soft Comput 67(6):865–894.

    Google Scholar 

  19. 19.

    Gao J, Li D (2013) Optimal cardinality constrained portfolio selection. Oper Res 61(3):745–761.

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Chen C, Li X, Tolman C, Wang S, Ye Y (2013) Sparse portfolio selection via quasi-norm regularization. arXiv:1312.6350

  21. 21.

    Yen Y (2016) Sparse weighted-norm minimum variance portfolios. Eur Finan Rev 20(3):1259–1287.

    MATH  Article  Google Scholar 

  22. 22.

    Teng Y, Yang L, Yu B, Song X (2017) A penalty palm method for sparse portfolio selection problems. Optim Methods Softw 32(1):126–147.

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Wang Q, Sun H (2017) Sparse Markowitz portfolio selection by using stochastic linear complementarity approach. J Ind Manag Optim 13(2):59–59.

    MathSciNet  Google Scholar 

  24. 24.

    Hardoroudi ND, Keshvari A, Kallio M, Korhonen P (2017) Solving cardinality constrained mean-variance portfolio problems via milp. Ann Oper Res 254(1):47–59.

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Branda M, Bucher M, Červinka M, Schwartz A (2018) Convergence of a scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization. Comput Optim Appl:1–28.

  26. 26.

    Zhao Z, Xu F, Wang M, Zhang C (2019) A sparse enhanced indexation model with norm and its alternating quadratic penalty method. J Oper Res Soc 70(3):433–445.

    Article  Google Scholar 

  27. 27.

    Fastrich B, Paterlini S, Winker P (2014) Cardinality versus q-norm constraints for index tracking. Quant Financ 14(11):2019–2032.

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Xu F, Lu Z, Xu Z (2016) An efficient optimization approach for a cardinality-constrained index tracking problem. Optim Methods Softw 31(2):258–271.

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Mutunge P, Haugland D (2018) Minimizing the tracking error of cardinality constrained portfolios. Comput Oper Res 90:33–41.

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Zhang C, Wang J, Xiu N (2019) Robust and sparse portfolio model for index tracking. J Ind Manag Optim 15(3):1001–1015.

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Zhou Z, Jin Q, Xiao H, Wu Q, Liu W (2018) Estimation of cardinality constrained portfolio efficiency via segmented dea. Omega-Int J Manag Sci 76:28–37.

    Article  Google Scholar 

  32. 32.

    Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press Princeton

  33. 33.

    Bertsimas D, Brown DB, Caramanis C (2010) Theory and applications of robust optimization. SIAM Rev 53(3):464–501.

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Ben-Tal A, Hertog DD, Waegenaere AD, Melenberg B, Rennen G (2013) Robust solutions of optimization problems affected by uncertain probabilities. Manag Sci 59(2):341–57.

    Article  Google Scholar 

  35. 35.

    Laurent El G, Maksim O, Francois O (2003) Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Oper Res 51(4):543–556.

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Björn F, Peter W (2012) Robust portfolio optimization with a hybrid heuristic algorithm. CMS 9:63–88.

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Lotfi S, Zenios SA (2018) Robust VaR and CVaR optimization under joint ambiguity in distributions, means, and covariances. Eur J Oper Res 269(2):556–576.

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Xu F, Wang M, Dai Y, Xu D (2018) A sparse enhanced indexation model with chance and cardinality constraints. J Glob Optim 70(1):5–25.

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Wu D, Wu DD (2019) An enhanced decision support approach for learning and tracking derivative index. Omega-Int J Manag Sci 88:63–76.

    Article  Google Scholar 

  40. 40.

    Konno H, Yamazaki H (1991) Mean-absolute deviation portfolio optimization model and its application to tokyo stock market. Manag Sci 37(5):519–531.

    Article  Google Scholar 

  41. 41.

    Linsmeier TJ, Pearson ND (1996) Risk measurement: An introduction to value at risk. University of Illinois, Urbana-Champaign

  42. 42.

    Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26(7):1443–471.

    Article  Google Scholar 

  43. 43.

    Huang R, Qu S, Gong Z, Goh M, Ji Y (2020) Data-driven two-stage distributionally robust optimization with risk aversion. Appl Soft Comput 87:105978.

    Article  Google Scholar 

  44. 44.

    Fan K (1953) Minimax theorems. Proc Natl Acad Sci U S A 39(1):42–47.

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper Res 58(3):595–612.

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Zymler S, Kuhn D, Rustem B (2013) Distributionally robust joint chance constraints with second-order moment information. Math Program 137(1-2):167–198.

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Luc D, László G (1985) Nonparametric density estimation: The l1 view. Wiley, New York

  48. 48.

    Scott DW (2015) Multivariate density estimation: Theory, practice, and visualization. Wiley

  49. 49.

    Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Am Stat Assoc 58(301):13–30.

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Ledoit O, Wolf M (2004) A well-conditioned estimator for large-dimensional covariance matrices. J Multivar Anal 88(2):365–411.

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Chen Y, Wiesel A, Eldar YC, Hero AO (2010) Shrinkage algorithms for MMSE covariance estimation. IEEE Trans Signal Process 58(10):5016–5029.

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Daniels MJ, Kass RE (2001) Shrinkage estimators for covariance matrices. Biometrics 57 (4):1173–1184.

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Wu WB, Pourahmadi M (2003) Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90(4):831–844.

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Couillet R, Mckay MR (2014) Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators. J Multivar Anal 131:99–120.

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer Math 4(1):238–252.

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Toh KC, Todd MJ, Tütüncü RH (1999) Sdpt3a matlab software package for semidefinite programming, version 1.3. Optim Methods Softw 11(1-4):545–581.

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Labit Y, Peaucelle D, Henrion D (2002) Sedumi interface 1.02: a tool for solving LMI problems with sedumi. IEEE Int Symp Comput Aided Control Syst Des:272–277.

  58. 58.

    Shapiro A (2001) On duality theory of conic linear problems. Kluwer Academic Publishers

  59. 59.

    Löfberg J (2004) Yalmip : a toolbox for modeling and optimization in matlab. Optimization 2004(3):284–289.

    Google Scholar 

  60. 60.

    Jarque CM, Bera AK (1981) Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Econ Lett 7(4):313–318.

    Article  Google Scholar 

  61. 61.

    Miller MB (2019) Quantitative financial risk management. Wiley

  62. 62.

    Demiguel V, Garlappi L, Uppal R (2009) Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy?. Rev Financ Stud 22(5):1915–1953.

    Article  Google Scholar 

  63. 63.

    Fabozzi FJ, Kolm PN, Pachamanova DA, Focardi SM (2007) Robust portfolio optimization. J Portf Manag 33(3):40–48.

    Article  Google Scholar 

  64. 64.

    Sehgal R, Mehra A (2020) Robust portfolio optimization with second order stochastic dominance constraints. Comput Ind Eng 144(6):106396.

    Article  Google Scholar 

  65. 65.

    Cesarone F, Scozzari A, Tardella F (2020) An optimization-diversification approach to portfolio selection. J Glob Optim 76(2):1–21

    MathSciNet  MATH  Article  Google Scholar 

Download references


This research has been supported by National Natural Science Foundation of China (71571055) and partially supported by the NSF of Anhui Educational Committe (KJ2020A0712). Meanwhile, this research was also supported by Initial Scientific Research Fund of Chuzhou University (2020qd39).

Author information



Corresponding author

Correspondence to Shaojian Qu.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huang, R., Qu, S., Yang, X. et al. Sparse portfolio selection with uncertain probability distribution. Appl Intell (2021).

Download citation


  • Data-driven approach
  • Robust optimization
  • Sparse portfolio
  • Distributional uncertainty
  • Modified generalized Benders’ decomposition