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Relaxed method for optimization problems with cardinality constraints

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Abstract

In this paper, we review optimality conditions and constraint qualifications for the optimization problems with cardinality constraints (OPCC). OPCC is a class of optimization problems with important applications. In this paper, we provide a relaxed method for OPCC. We show that the Mangasarian-Fromovitz constraint qualification or constant positive linear dependence constraint qualification holds for the relaxed problem under some mild conditions. We provide that the local solution of the relaxed problem converges to the M-stationarity of OPCC under appropriate conditions. Furthermore, we obtain that the inexact stationary points of relaxed problem converges to the M-stationarity of OPCC under very weaker conditions. Numerical experiments show the effectiveness of the proposed method.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Branda, M., Bucher, M., Červinka, M., Schwartz, A.: Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization. Comput. Optim. Appl. 70, 503–530 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bucher, M., Schwartz, A.: Second-order optimality conditions and improved convergence results for regularization methods for cardinality-constrained optimization problems. J. Optim. Theory Appl. 178, 383–410 (2018)

    Article  MathSciNet  Google Scholar 

  5. Burdakov, O.P., Kanzow, C., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type conditions and a regularization method. SIAM J. Optim. 26, 397–425 (2016)

    Article  MathSciNet  Google Scholar 

  6. Candès, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008)

    Article  ADS  Google Scholar 

  7. Cervinka, M., Kanzow, C., Schwartz, A.: Constraint qualifications and optimality conditions for optimization problems with cardinality constraints. Math. Program. 160, 353–377 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Kadrani, A., Dussault, J.-P., Benchakroun, A.: A new regularization scheme for mathematical programs with complimentarity constraints. SIAM J. Optim. 20, 78–103 (2009)

    Article  MathSciNet  Google Scholar 

  11. Kanzow, C., Raharja, A.B., Schwartz, A.: An augmented lagrangian method for cardinality-constrained optimization problems. J. Optim. Theory Appl. 189, 793–813 (2021)

    Article  MathSciNet  Google Scholar 

  12. Kanzow, C., Raharja, A.B., Schwartz, A.: Sequential optimality conditions for cardinality-constrained optimization problems with applications. Comput. Optim. Appl. 80, 185–211 (2021)

    Article  MathSciNet  Google Scholar 

  13. Kanzow, C., Schwartz, A.: The price of inexactness: convergence properties of relaxation methods for mathematical programs with complementarity constraints revisited. Math. Oper. Res. 40, 253–275 (2015)

    Article  MathSciNet  Google Scholar 

  14. Miller, A.: Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, Boca Raton (2002)

    Book  Google Scholar 

  15. Murray, W., Shek, H.: A local relaxation method for the cardinality-constrained portfolio optimization problem. Comput. Optim. Appl. 53, 681–709 (2012)

    Article  MathSciNet  Google Scholar 

  16. Ruiz-Torrubiano, R., García-Moratilla, S., Suárez, A.: Optimization problems with cardinality constraints. In: Tenne, Y., Goh, C.-K. (eds.) Comput. Intell. Optim., pp. 105–130. Springer, Berlin (2010)

    Google Scholar 

  17. Sun, X., Zheng, X., Li, D.: Recent advances in mathematical programming with semi-continuous variables and cardinality constraint. J. Oper. Res. Soc. China 1, 55–77 (2013)

    Article  Google Scholar 

  18. Zheng, X., Sun, X., Li, D., Sun, J.: Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach. Comput. Optim. Appl. 59, 379–397 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Yan-Chao Liang

  2. School of Management, Shanghai University, Shanghai, 200444, China

    Gui-Hua Lin

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  1. Yan-Chao Liang
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  2. Gui-Hua Lin
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Correspondence to Yan-Chao Liang.

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The first author’s work was supported in part by NSFC Grant #11801152, #12071133. The second author’s work was supported in part by NSFC Grant #12071280. The authors are grateful to the anonymous referees for their helpful suggestions and comments.

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Liang, YC., Lin, GH. Relaxed method for optimization problems with cardinality constraints. J Glob Optim 88, 359–375 (2024). https://doi.org/10.1007/s10898-023-01317-5

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  • DOI: https://doi.org/10.1007/s10898-023-01317-5

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