Skip to main content
SpringerLink
Log in

Directionally generalized differentiation for multifunctions and applications to set-valued programming problems

  • RAOTA-2016
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The aim of this work is twofold. First, we establish sum rules for the directionally coderivatives of multifunctions and intersection rules for the directionally limiting normal cones. Then, we apply the provided formulas to derive directionally necessary conditions for a set-valued optimization problem with equilibrium constraints.

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.

Institutional subscriptions

Similar content being viewed by others

References

  • Aubin, J. P., & Frankowska, H. (1990). Set-valued analysis. Basel: Birkhuser.

    Google Scholar 

  • Bao, T. Q., & Mordukhovich, B. S. (2007). Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints. Applications of Mathematics, 52(6), 453–472.

  • Bao, T. Q., Gupta, P., & Mordukhovich, B. S. (2007). Necessary conditions in multiobjective optimization with equilibrium constraints. Journal of Optimization Theory and Applications, 135(2), 179–203.

    Article  Google Scholar 

  • Chuong, T. D., & Kim, D. S. (2015). Nondifferentiable minimax programming problems with applications. Annals of Operations Research. doi:10.1007/s10479-015-1843-3.

  • Chuong, T. D., & Kim, D. S. (2016a). A class of nonsmooth fractional multiobjective optimization problems. Annals of Operations Research. doi:10.1007/s10479-016-2130-7.

  • Chuong, T. D., & Kim, D. S. (2016b). Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications. Annals of Operations Research. doi:10.1007/s10479-016-2337-7.

  • Ehrgott, M. (2005). Multicriteria optimization. Berlin: Springer.

    Google Scholar 

  • Gfrerer, H. (2013). On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical problems. Set-Valued and Variational Analysis, 21, 151–176.

    Article  Google Scholar 

  • Gfrerer, H. (2013). On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical problems. SIAM Journal of Optimization, 23(1), 632–665.

    Article  Google Scholar 

  • Ginchev, I., & Mordukhovich, B. S. (2011). On directionally dependent subdifferentials. Comptes Rendus De l’Academie Bulgare Des Sciences, 64(4), 497–508.

    Google Scholar 

  • Ginchev, I., & Mordukhovich, B. S. (2012). Directional subdifferentials and optimality conditions. Positivity, 16, 707–737.

    Article  Google Scholar 

  • Jahn, J. (2011). Vector optimization: Theory, applications and extensions. Berlin: Springer-Verlag.

    Book  Google Scholar 

  • Long, P., Wang, B., & Yang, X. (2016). Calculus of directional subdifferentials and coderivatives in Banach spaces. Positivity,. doi:10.1007/s1117-016-0417-1.

    Google Scholar 

  • Mordukhovich, B. S. (1980). Metric approximations and necessary optimality conditions for general classes of extremal problems. Soviet Mathematics Doklady, 22, 526–530.

    Google Scholar 

  • Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory. Berlin: Springer-Verlag.

    Google Scholar 

  • Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation II: Applications. Berlin: Springer-Verlag.

    Google Scholar 

  • Mordukhovich, B. S. (2009). Multiobjective optimization problems with equilibrium constraints. Mathematical Programming, 117, 331–354.

    Article  Google Scholar 

  • Mordukhovich, B. S., Nam, N. M., & Nhi, N. T. Y. (2014). Partial second-order subdifferentials in variational analysis and optimization. Numerical Functional Analysis and Optimization, 35(7–9), 1113–1151.

    Article  Google Scholar 

  • Yu, G. L., & Liu, S. Y. (2009). Globally proper saddle point in ic-cone-convexlike set-valued optimization problems. Acta Mathematica Sinica (English Series), 25(11), 1921–1928.

    Article  Google Scholar 

  • Zheng, X. Y., & Ng, K. F. (2006). The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM Journal on Optimization, 17(14), 1154–1175.

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions, which greatly improved the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics & Informatics Teacher Education, Dongthap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Dong Thap Province, Vietnam

    Vo Duc Thinh

  2. Department of Mathematics & Applications, Saigon University, 273 An Duong Vuong Street, Ward 3, District 5, Ho Chi Minh City, Vietnam

    Thai Doan Chuong

Authors
  1. Vo Duc Thinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thai Doan Chuong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thai Doan Chuong.

Additional information

This work was supported by a grant from Dongthap University (CS2015.01.33).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thinh, V.D., Chuong, T.D. Directionally generalized differentiation for multifunctions and applications to set-valued programming problems. Ann Oper Res 269, 727–751 (2018). https://doi.org/10.1007/s10479-017-2400-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-017-2400-z

Keywords

Search

Navigation