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Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential

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Abstract

In this paper, we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential. We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency, which used by the concepts of supremum and infimum of a set. We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map. Finally, we define the sub-optimal path related to the dual problem and show that every cluster point of this sub-optimal path is a primal optimal solution of the object optimization problem. In addition, we consider a generalized vector variational inequality as an application of abstract subdifferential.

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References

  1. Altangerel, L., Bot, R.I., Wanka, G. Conjugate duality in vector optimization and some applications to the vector variational inequality. J. Math. Anal. Appl. 329: 1010–1035 (2007)

    Article  MathSciNet  Google Scholar 

  2. Burachik, R.S., Iusem, A.N., Melo, J.G. Duality and exact penalization for general augmented Lagrangians. J. Optim. Theory Appl., 147: 125–140 (2010)

    Article  MathSciNet  Google Scholar 

  3. Burachik, R.S., Rubinov, A.M. Abstract convexity and augmented Lagrangians. SIAM J. Optim., 18: 413–436 (2007)

    Article  MathSciNet  Google Scholar 

  4. Huy, N.Q., Kim, D.S. Duality in vector optimization via augmented Lagrangian. J. Math. Anal. Appl., 386: 473–486 (2012)

    Article  MathSciNet  Google Scholar 

  5. Huang, X.X., Yang, X.Q. Duality and exact Penalization for Vector Optimization via Augmented Lagrangian. J. Optim. Theory Appl., 3: 615–640 (2001)

    Article  MathSciNet  Google Scholar 

  6. Huang, X.X., Yang, X.Q. A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res., 28: 533–552 (2003)

    Article  MathSciNet  Google Scholar 

  7. Khan, A.A., Zalinescu, C., Tammer, C. Set-valued optimization, IEEE, 1900

  8. Li, S.J., Chen, C.R., Wu, S.Y. Conjugate dual problems in constrained set-valued optimization and applications. Euro. J. Oper. Res., 196: 21–32 (2009)

    Article  MathSciNet  Google Scholar 

  9. Li, T.Y., Xu, Y.H. The strictly efficient subgradient of set-valued optimization. Bull. Austral. Math. Soc., 75: 361–371 (2007)

    Article  MathSciNet  Google Scholar 

  10. Nguyen, L.H.A. Duality and its applications to optimality conditions with nonsolid cones. Bull. Malays. Math. Sci. Soc., DOI: https://doi.org/10.1007/s40840-016-0375-6

  11. Patriche, M. Minimax theorems for set-valued maps without continuity assumptions. Optimization, 65:957–976 (2016)

    Article  MathSciNet  Google Scholar 

  12. Rockafellar, R.T., Wets, R.J-B. Variational Analysis. Springer, Berlin, 2009

    MATH  Google Scholar 

  13. Rubinov, A.M., Huang, X.X., Yang, X.Q. The zero duality gap property and lower semicontinuity of the perturbation function. Math. Oper. Res., 27: 775–791 (2002)

    Article  MathSciNet  Google Scholar 

  14. Tanino, T. On supremum of a set in a multidimensional space. J. Math. Anal. Appl., 130: 386–397 (1988)

    Article  MathSciNet  Google Scholar 

  15. Tanino, T. Conjugate duality in vector optimization. J. Math. Anal. Appl., 167: 84–97 (1992)

    Article  MathSciNet  Google Scholar 

  16. Wang, C.Y., Yang, X.Q., Yang, X.M. Unified nonlinear Lagrangian approach to duality and optimal paths. J. Optim. Theory Appl., 135: 85–100 (2007)

    Article  MathSciNet  Google Scholar 

  17. Zhou, Y.Y., Yang, X.Q. Duality and penalization in optimization via an augmented Lagrangian function with applications. J. Optim. Theory Appl., 140: 171–188 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Xi’an Polytechnic University, Xi’an, 710048, China

    Yan-fei Chai

  2. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    San-yang Liu

  3. College of Transportation Engineering, Chang’an University, Xi’an, 710064, China

    Si-qi Wang

Authors
  1. Yan-fei Chai
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  2. San-yang Liu
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  3. Si-qi Wang
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Corresponding author

Correspondence to Yan-fei Chai.

Additional information

This work was supported by National Science Foundation of China (No.11401487), the Education Department of Shaanxi Province (No.17JK0330), the Fundamental Research Funds for the Central Universities (No.300102341101) and State Key Laboratory of Rail Transit Engineering Informatization (No.211934210083).

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Chai, Yf., Liu, Sy. & Wang, Sq. Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential. Acta Math. Appl. Sin. Engl. Ser. 38, 337–351 (2022). https://doi.org/10.1007/s10255-022-1079-3

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  • DOI: https://doi.org/10.1007/s10255-022-1079-3

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