Skip to main content
SpringerLink
Log in

Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications

  • Multiple Objective Optimization
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

We establish verifiable conditions for the feasible set of a nonsmooth semi-infinite multiobjective optimization problem to have the normal regularity (that is, the coincidence of the Fréchet normal cone and the limiting normal one) at a given point. In this way, both the Fréchet normal cone and the limiting normal one to the considered set are then computed via active constraint multipliers and limiting subdifferentials of the involved constraints. In order to achieve such goals, two classes of nonsmooth functions are introduced and exploited. Finally, the obtained results are applied to provide necessary optimality conditions for semi-infinite multiobjective optimization problems.

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

  • Bonnans, J. F., & Shapiro, A. (2000). Perturbation analysis of optimization problems. New York: Springer.

    Book  Google Scholar 

  • Canovas, M. J., Lopez, M. A., Mordukhovich, B. S., & Parra, J. (2009). Variational analysis in semi-infinite and infinite programming. I. Stability of linear inequality systems of feasible solutions. SIAM Journal on Optimization, 20, 1504–1526.

    Article  Google Scholar 

  • Chuong, T. D. (2013). Derivatives of the efficient point multifunction in parametric vector optimization problems. Journal of Optimization Theory and Applications, 156, 247–265.

    Article  Google Scholar 

  • Chuong, T. D. (2016). Nondifferentiable fractional semi-infinite multiobjective optimization problems. Operations Research Letters, 44(2), 260–266.

    Article  Google Scholar 

  • Chuong, T. D., Huy, N. Q., & Yao, J.-C. (2009). Subdifferentials of marginal functions in semi-infinite programming. SIAM Journal on Optimization, 20, 1462–1477.

    Article  Google Scholar 

  • Chuong, T. D., & Kim, D. S. (2014a). Nonsmooth semi-infinite multiobjective optimization problems. Journal of Optimization Theory and Applications, 160, 748–762.

  • Chuong, T. D., & Kim, D. S. (2014b). Optimality conditions and duality in nonsmooth multiobjective optimization problems. Annals of Operations Research, 217, 117–136.

  • 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. (2016). Holder-like property and metric regularity of a positive-order for implicit multifunctions. Mathematics of Operations Research, 41(2), 596–611.

    Article  Google Scholar 

  • Chuong, T. D., & Yao, J.-C. (2010). Generalized Clarke epiderivatives of parametric vector optimization problems. Journal of Optimization Theory and Applications, 146, 77–94.

    Article  Google Scholar 

  • Clarke, F. H. (1983). Optimization and nonsmooth analysis. New York: Wiley.

    Google Scholar 

  • Dinh, N., Goberna, M. A., López, M. A., & Son, T. Q. (2007). New Farkas-type constraint qualifications in convex infinite programming. ESAIM: Control, Optimisation and Calculus of Variations, 13, 580–597.

    Article  Google Scholar 

  • Dinh, N., Mordukhovich, B. S., & Nghia, T. T. A. (2009). Qualification and optimality conditions for DC programs with infinite constraints. Acta Mathematica Vietnamica, 34, 123–153.

    Google Scholar 

  • Dinh, N., Mordukhovich, B. S., & Nghia, T. T. A. (2010). Subifferentials of value functions and optimality conditions for some classes of DC and bilevel infinite and semi-infinite programs. Mathematical Programming, 123, 101–138.

    Article  Google Scholar 

  • Goberna, M. A., & López, M. A. (1998). Linear semi-infinite optimization. Chichester: Wiley.

    Google Scholar 

  • Goberna, M. A., & López, M. A. (2014). Post-optimal analysis in linear semi-infinite optimization. New York: Springer.

    Book  Google Scholar 

  • Huy, N. Q., Giang, N. D., & Yao, J.-C. (2012). Subdifferential of optimal value functions in nonlinear infinite programming. Applied Mathematics and Optimization, 65, 91–109.

    Article  Google Scholar 

  • Li, C., Ng, K. F., & Pong, T. K. (2007). The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces. SIAM Journal on Optimization, 18(2), 643–665.

    Article  Google Scholar 

  • Li, C., Ng, K. F., & Pong, T. K. (2008). Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM Journal on Optimization, 19(1), 163–187.

    Article  Google Scholar 

  • Luc, D. T. (1989). Theory of vector optimization. Berlin: Springer.

    Book  Google Scholar 

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

    Google Scholar 

  • Mordukhovich, B. S., Nam, N. M., & Yen, N. D. (2009). Subgradients of marginal functions in parametric mathematical programming. Mathematical Programming, 116(1–2), 369–396.

    Article  Google Scholar 

  • Mordukhovich, B. S., & Nghia, T. T. A. (2013a). Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs. Mathematical Programming, 139, 271–300.

  • Mordukhovich, B. S., & Nghia, T. T. A. (2013b). Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data. SIAM Journal on Optimization, 23, 406–431.

  • Mordukhovich, B. S., & Nghia, T. T. A. (2014). Nonsmooth cone-constrained optimization with applications to semi-infinite programming. Mathematics of Operations Research, 39, 301–324.

    Article  Google Scholar 

  • Ngai, H. V., Luc, D. T., & Thera, M. (2000). Approximate convex functions. Journal of Nonlinear and Convex Analysis, 1(2), 155–176.

    Google Scholar 

  • Seidman, T. I. (2010). Normal cones to infinite intersections. Nonlinear Analysis, 72(11), 3911–3917.

    Article  Google Scholar 

  • Zheng, X. Y., & Ng, K. F. (2012). Subsmooth semi-infinite and infinite optimization problems. Mathematical Programming Series A, 134, 365–393.

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank an anonymous referee for valuable comments and suggestions which greatly improved the representation of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Thai Doan Chuong

  2. Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

    Do Sang Kim

Authors
  1. Thai Doan Chuong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Do Sang Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Do Sang Kim.

Additional information

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2A10008908) and by the UNSW Vice-Chancellor’s Postdoctoral Research Fellowship (RG134608/SIR50).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chuong, T.D., Kim, D.S. Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications. Ann Oper Res 267, 81–99 (2018). https://doi.org/10.1007/s10479-016-2337-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-016-2337-7

Keywords

Mathematics Subject Classification

Search

Navigation