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On a Reformulation of Mathematical Programs with Cardinality Constraints

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Advances in Global Optimization

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 95))

Abstract

Mathematical programs with cardinality constraints are optimization problems with an additional constraint which requires the solution to be sparse in the sense that the number of nonzero elements, i.e. the cardinality, is bounded by a given constant. Such programs can be reformulated as a mixed-integer ones in which the sparsity is modeled with the use of complementarity-type constraints. It is shown that the standard relaxation of the integrality leads to a nonlinear optimization program of the striking property that its solutions (global minimizers) are the same as the solutions of the original program with cardinality constraints. Since the number of local minimizers of the relaxed program is typically larger than the number of local minimizers of the cardinality-constrained problem, the relationship between the local minimizers is also discussed in detail. Furthermore, we show under which assumptions the standard KKT conditions are necessary optimality conditions for the relaxed program. The main result obtained for such conditions is significantly different from the existing optimality conditions that are known for the somewhat related class of mathematical programs with complementarity constraints.

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References

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

    MATH  MathSciNet  Google Scholar 

  2. Gao, J., Li, D.: A polynomial case of the cardinality-constrained quadratic optimization problem. J. Glob. Optim. 56, 1441–1455 (2013)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. 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 

  5. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. 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 

  8. Streichert, F., Ulmer, H., Zell, A.: Evolutionary algorithms and the cardinality constrained portfolio optimization problem. In: Operations Research Proceedings, vol. 2003, pp. 253–260. Springer, Berlin (2004)

    Google Scholar 

  9. Tropp, J.A., Wright, S.J.: Computational methods for sparse solution of linear inverse problems. Proc. IEEE 98, 948–958 (2010)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Mitchell, J., Pang, J.-S., Waechter, A., Bai, L., Feng, M., Shen, X.: Complementarity formulations for L 0 norm optimization problems. In: Presentation at the 11-th Workshop on Advances in Continuous Optimization, Florence, 26–28 June 2013

    Google Scholar 

  12. Bazaraa, M.S., Shetty, C.M.: Foundations of Optimization. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  13. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Linköping University, 58183, Linköping, Sweden

    Oleg Burdakov

  2. Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

    Christian Kanzow & Alexandra Schwartz

Authors
  1. Oleg Burdakov
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  2. Christian Kanzow
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  3. Alexandra Schwartz
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Corresponding author

Correspondence to Oleg Burdakov .

Editor information

Editors and Affiliations

  1. School of Science, Information Technology and Engineering, Federation University Australia, Ballarat, Victoria, Australia

    David Gao

  2. School Science, Inform., Tech. & Engg., Federation University Australia, Ballarat, Victoria, Australia

    Ning Ruan

  3. Dept. of Mathematical Sciences, Tsinghua University, Beijing, China

    Wenxun Xing

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Burdakov, O., Kanzow, C., Schwartz, A. (2015). On a Reformulation of Mathematical Programs with Cardinality Constraints. In: Gao, D., Ruan, N., Xing, W. (eds) Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-08377-3_1

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