Skip to main content
SpringerLink
Log in

Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper we consider cardinality-constrained convex programs that minimize a convex function subject to a cardinality constraint and other linear constraints. This class of problems has found many applications, including portfolio selection, subset selection and compressed sensing. We propose a successive convex approximation method for this class of problems in which the cardinality function is first approximated by a piecewise linear DC function (difference of two convex functions) and a sequence of convex subproblems is then constructed by successively linearizing the concave terms of the DC function. Under some mild assumptions, we establish that any accumulation point of the sequence generated by the method is a KKT point of the DC approximation problem. We show that the basic algorithm can be refined by adding strengthening cuts in the subproblems. Finally, we report some preliminary computational results on cardinality-constrained portfolio selection problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Arthanari, T.S., Dodge, Y.: Mathematical Programming in Statistics. Wiley, New York (1993)

    MATH  Google Scholar 

  2. Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43, 1–22 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Blog, B., van der Hoek, G., Rinnooy Kan, A.H.G., Timmer, G.T.: The optimal selection of small portfolios. Manag. Sci. 29, 792–798 (1983)

    Article  MATH  Google Scholar 

  5. Bonami, P., Lejeune, M.A.: An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Oper. Res. 57, 650–670 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51, 34–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Candes, E., Wakin, M., Boyd, S.: Enhancing sparsity by reweighted ℓ 1 minimization. J. Fourier Anal. Appl. 14(5), 877–905 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chang, T.J., Meade, N., Beasley, J.E., Sharaiha, Y.M.: Heuristics for cardinality constrained portfolio optimisation. Comput. Oper. Res. 27, 1271–1302 (2000)

    Article  MATH  Google Scholar 

  9. Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of unconstrained l 2-l p minimization. Math. Program. (2011). arXiv:1105.0638

  10. Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of ℓ 2-ℓ p minimization. SIAM J. Sci. Comput. 32, 2832–2852 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, X., Zhou, W.: Convergence of reweighted l 1 minimization algorithms and unique solution of truncated l p minimization. Tech. rep, Hong Kong Polytechnic University (2010). Available at http://www.polyu.edu.hk/ama/staff/xjchen/IRL1-4-8.pdf

  12. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  13. Cui, X.T., Zheng, X.J., Zhu, S.S., Sun, X.L.: Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems. J. Glob. Optim. (2012). 10.1007/s10898-012-9845-z

    Google Scholar 

  14. Fazel, M., Hindi, H., Boyd, S.: Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices. In: Proceedings of American Control Conference, pp. 2156–2162 (2003)

    Google Scholar 

  15. Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106, 225–236 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Frangioni, A., Gentile, C.: SDP diagonalizations and perspective cuts for a class of nonseparable MIQP. Oper. Res. Lett. 35, 181–185 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Ge, D., Jiang, X., Ye, Y.: A note on complexity of ℓ p minimization. Math. Program. 129, 285–299 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hong, L., Yang, Y., Zhang, L.: Sequential convex approximations to joint chance constrained programs: a Monte Carlo approach. Oper. Res. 59, 617–630 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. IBM ILOG CPLEX: IBM ILOG CPLEX 12. 3 User’s Manual for CPLEX (2011)

  21. Jacob, N.L.: A limited-diversification portfolio selection model for the small investor. J. Finance 29, 847–856 (1974)

    Article  Google Scholar 

  22. Jobst, N., Horniman, M., Lucas, C., Mitra, G.: Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quant. Finance 1, 489–501 (2001)

    Article  MathSciNet  Google Scholar 

  23. Li, D., Sun, X.L., Wang, J.: Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection. Math. Finance 16, 83–101 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lu, Z., Zhang, Y.: Penalty decomposition methods for rank minimization. Tech. rep, Department of Mathematics, Simon Fraser University (2012). Available at http://www.optimization-online.org/DB_HTML/2010/08/2718.html

  25. Lu, Z., Zhang, Y.: Sparse approximation via penalty decomposition methods. Tech. rep, Department of Mathematics, Simon Fraser University (2012). Available at http://www.optimization-online.org/DB_HTML/2010/08/2719.html

  26. Mangasarian, O.: Minimum-support solutions of polyhedral concave programs. Optimization 45, 149–162 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Maringer, D., Kellerer, H.: Optimization of cardinality constrained portfolios with a hybrid local search algorithm. OR Spektrum 25, 481–495 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Miller, A.J.: Subset Selection in Regression, 2nd edn. Chapman and Hall, London (2002)

    Book  MATH  Google Scholar 

  29. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  30. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (2004)

    Google Scholar 

  31. Shaw, D.X., Liu, S., Kopman, L.: Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optim. Methods Softw. 23, 411–420 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sriperumbudur, B., Torres, D., Lanckriet, G.: A majorization-minimization approach to the sparse generalized eigenvalue problem. Mach. Learn. 85, 3–39 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Vielma, J.P., Ahmed, S., Nemhauser, G.L.: A lifted linear programming branch-and-bound algorithm for mixed integer conic quadratic programs. INFORMS J. Comput. 20, 438–450 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Weston, J., Elisseeff, A., Schölkopf, B., Tipping, M.: Use of the zero norm with linear models and kernel methods. J. Mach. Learn. Res. 3, 1439–1461 (2003)

    MATH  MathSciNet  Google Scholar 

  35. Woodside-Oriakhi, M., Lucas, C., Beasley, J.E.: Heuristic algorithms for the cardinality constrained efficient frontier. Eur. J. Oper. Res. 213, 538–550 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zheng, X.J., Sun, X.L., Li, D.: Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach. Tech. report (2011). Available at http://www.optimization-online.org/DB_HTML/2010/11/2797.html

Download references

Author information

Authors and Affiliations

  1. School of Economics and Management, Tongji University, Shanghai, 200092, P.R. China

    Xiaojin Zheng

  2. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, P.R. China

    Xiaoling Sun

  3. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Duan Li

  4. Department of Decision Sciences and Risk Management Institute, National University of Singapore, Singapore, 117245, Singapore

    Jie Sun

Authors
  1. Xiaojin Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoling Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Duan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jie Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoling Sun.

Additional information

Dedicated to Professor Masao Fukushima in celebration of his 65th birthday.

This research was supported by the NSFC grants 10971034, 11101092 and the Joint NSFC/RGC grant 71061160506, and by Hong Kong RGC Grants CUHK414610 and 2050524.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, X., Sun, X., Li, D. et al. Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach. Comput Optim Appl 59, 379–397 (2014). https://doi.org/10.1007/s10589-013-9582-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-013-9582-3

Keywords

Search

Navigation