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A Comparative Study of Sequential Optimality Conditions for Mathematical Programs with Cardinality Constraints

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Abstract

We propose a comparative study of sequential optimality conditions for mathematical programs with cardinality constraints. Besides analyzing some of the classical approximate conditions for nonlinear programming, such as AKKT, CAKKT and PAKKT, we also propose an approximate weak stationarity (\({ AW}\)-stationarity) concept designed to deal with this class of problems and we prove that it is a legitimate optimality condition independently of any constraint qualification. We point out that, despite the computational appeal of the sequential optimality conditions, in this work we are not concerned with algorithmic consequences. Our aim is purely to discuss theoretical aspects of such conditions for MPCaC problems.

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References

  1. Andreani, R., Fazzio, N.S., Schuverdt, M.L., Secchin, L.D.: A sequential optimality condition related to the quasinormality constraint qualification and its algorithmic consequences. SIAM J. Optim. 29, 743–766 (2019)

    Article  MathSciNet  Google Scholar 

  2. Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60, 627–641 (2011)

    Article  MathSciNet  Google Scholar 

  3. Andreani, R., Haeser, G., Secchin, L.D., Silva, P.J.S.: New sequential optimality conditions for mathematical problems with complementarity constraints and algorithmic consequences. SIAM J. Optim. 29(4), 3201–3230 (2019)

    Article  MathSciNet  Google Scholar 

  4. Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: Strict constraint qualifications and sequential optimality conditions for constrained optimization. Math. Oper. Res. 43(3), 693–717 (2018)

    Article  MathSciNet  Google Scholar 

  5. Andreani, R., Martínez, J.M., Svaiter, B.F.: A new sequential optimality condition for constrained optimization and algorithmic consequences. SIAM J. Optim. 6, 3533–3554 (2010)

    Article  MathSciNet  Google Scholar 

  6. Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM, Philadelphia (2014)

    Book  Google Scholar 

  7. 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(2), 503–530 (2018)

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. Burdakov, O., Kanzow, C., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type conditions and a regularization method. SIAM J. Optim. 26(1), 397–425 (2016)

    Article  MathSciNet  Google Scholar 

  10. Červinka, 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 

  11. Helou, E.S., Santos, S.A., Simões, L.E.A.: Analysis of a new sequential optimality condition applied to mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 185, 433–447 (2020)

    Article  MathSciNet  Google Scholar 

  12. Helou, E.S., Santos, S.A., Simões, L.E.A.: A new sequential optimality condition for constrained nonsmooth optimization. SIAM J. Optim. 30(2), 1610–1637 (2020)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. 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 

  15. Krulikovski, E.H.M., Ribeiro, A.A., Sachine, M.: On the weak stationarity conditions for mathematical programs with cardinality constraints: a unified approach. Appl. Math. Optim. 84, 3451–3473 (2021)

    Article  MathSciNet  Google Scholar 

  16. Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118, 117–133 (2003)

    Article  MathSciNet  Google Scholar 

  17. Pang, L., Xue, M., Xu, N: A New Sequential Optimality Condition of Cardinality-Constrained Optimization Problems and Application. arXiv:2110.01220v1 (2021)

  18. Ramos, A.: Mathematical programs with equilibrium constraints: a sequential optimality condition, new constraint qualifications and algorithmic consequences. Optim. Methods Softw. 36(1), 45–81 (2021)

    Article  MathSciNet  Google Scholar 

  19. Ribeiro, A.A., Sachine, M., Santos, S.A.: On the approximate solutions of augmented subproblems within sequential methods for nonlinear programming. Comp. Appl. Math. 37, 6601–6618 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was partially supported by CNPq (Grant 309437/2016-4).

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Authors and Affiliations

  1. Federal University of Paraná, Curitiba, Brazil

    Ademir A. Ribeiro, Mael Sachine & Evelin H. M. Krulikovski

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  1. Ademir A. Ribeiro
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  2. Mael Sachine
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  3. Evelin H. M. Krulikovski
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Correspondence to Ademir A. Ribeiro.

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Communicated by Wei Bian.

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Ribeiro, A.A., Sachine, M. & Krulikovski, E.H.M. A Comparative Study of Sequential Optimality Conditions for Mathematical Programs with Cardinality Constraints. J Optim Theory Appl 192, 1067–1083 (2022). https://doi.org/10.1007/s10957-022-02007-0

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  • DOI: https://doi.org/10.1007/s10957-022-02007-0

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