Skip to main content
SpringerLink
Log in

Parametric Variational System with a Smooth-Boundary Constraint Set

  • Chapter
  • First Online:
Variational Analysis and Generalized Differentiation in Optimization and Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 47))

Abstract

Solution stability of parametric variational systems with smooth-boundary constraint sets is investigated. Sufficient conditions for the lower semicontinuity, Lipschitz-like property, and local metric regularity in Robinson’s sense of the solution map are obtained by using a calculus rule for the normal second-order subdifferential from B.S. Mordukhovich (Variational Analysis and Generalized Differentiation, Vol.I: Basic Theory, Vol.II: Applications, Springer, Berlin, 2006) and the implicit function theorems for multifunctions from G.M. Lee, N.N. Tam and N.D. Yen (J Math Anal Appl 338:11–22, 2008).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F. J. Aragón Artacho and B. S. Mordukhovich, Metric regularity and Lipschitzian stability of parametric variational systems, Nonlinear Anal. 72 (2010), 1149–1170.

    Article  MATH  MathSciNet  Google Scholar 

  2. J.-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87–111.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. H. Chieu, J.-C. Yao and N. D. Yen, Relationships between the Robinson robust stability and the Aubin continuity property of implicit multifunctions, Nonlinear Anal. 72 (2010), 3594–3601.

    Article  MATH  MathSciNet  Google Scholar 

  4. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

    MATH  Google Scholar 

  5. A. L. Dontchev and R. T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim. 6 (1996), 1087–1105.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Henrion, B. S. Mordukhovich and N. M. Nam, Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities, SIAM J. Optim. 20 (2010), 2199–2227.

    Article  MathSciNet  Google Scholar 

  7. R. Henrion, J. Outrata and T. Surowiec, On the co-derivative of normal cone mappings to inequality systems, Nonlinear Anal. 71 (2009), 1213–1226.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Henrion and W. Römisch, On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling, Appl. Math. 52 (2007), 473–494.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. M. Lee, N. N. Tam and N. D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008), 11–22.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. M. Lee, N. N. Tam and N. D. Yen, Stability of linear-quadratic minimization over Euclidean balls, Institute of Mathematics, Vietnamese Academy of Science and Technology, E-Preprint No.2007/10/02. (Submitted).

    Google Scholar 

  11. S. Lucidi, L. Palagi and M. Roma, On some properties of quadratic programs with a convex quadratic constraint, SIAM J. Optim. 8 (1998), 105–122.

    Article  MATH  MathSciNet  Google Scholar 

  12. B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol.I: Basic Theory, Vol.II: Applications, Springer, Berlin, 2006.

    Google Scholar 

  13. B. S. Mordukhovich, J. V. Outrata, On second-order subdifferentials and their applications, SIAM J. Optim. 12 (2001), 139–169.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Am. Math. Soc. 348 (1996), 1235–1280.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. R. Phelps, Convex Functions, Monotone Operators and Differentiablity, 2nd Edition, Spinger, Berlin, 1993.

    Google Scholar 

  16. S. M. Robinson, Stability theory for systems of inequalities, II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497–513.

    Google Scholar 

  17. S. M. Robinson, Generalized equations and their solutions, Part I: Basic theory, Math. Program. Study 10 (1979), 128–141.

    MATH  Google Scholar 

  18. J.-C. Yao and N. D. Yen, Coderivative calculation related to a parametric affine variational inequality. Part 1: Basic calculations. Acta Math. Vietnam. 34 (2009), 155–170.

    Google Scholar 

  19. J.-C. Yao and N. D. Yen, Coderivative calculation related to a parametric affine variational inequality. Part 2: Applications. Pacific J. Optim. 3 (2009), 493–506.

    Google Scholar 

  20. N. D. Yen, Hölder continuity of solutions to a parametric variational inequality, Appl. Math. Optim. 31 (1995), 245–255.

    Article  MATH  MathSciNet  Google Scholar 

  21. N. D. Yen and J.-C. Yao, Pointbased sufficient conditions for metric regularity of implicit multifunctions, Nonlinear Anal. 70 (2009), 2806–2815.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Sun Yat-Sen University, Kaohsiung, Taiwan and the National Foundation for Science & Technology Development, Vietnam. The authors are indebted to Dr. N.Q. Huy, Mr. N.H. Chieu and Mr. T.D. Chuong for an useful discussion on Asplund spaces. We thank the referee for helpful comments.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan

    J.-C. Yao

  2. Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    N. D. Yen

Authors
  1. J.-C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. D. Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.-C. Yao .

Editor information

Editors and Affiliations

  1. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  2. , Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan R.O.C.

    Jen-Chih Yao

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Yao, JC., Yen, N.D. (2010). Parametric Variational System with a Smooth-Boundary Constraint Set. In: Burachik, R., Yao, JC. (eds) Variational Analysis and Generalized Differentiation in Optimization and Control. Springer Optimization and Its Applications, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0437-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation