Advertisement

Vietnam Journal of Mathematics

, Volume 46, Issue 2, pp 365–379 | Cite as

Subdifferential Stability Analysis for Convex Optimization Problems via Multiplier Sets

  • Duong Thi Viet An
  • Nguyen Dong Yen
Article

Abstract

This paper discusses differential stability of convex programming problems in Hausdorff locally convex topological vector spaces. Among other things, we obtain formulas for computing or estimating the subdifferential and the singular subdifferential of the optimal value function via suitable multiplier sets.

Keywords

Hausdorff locally convex topological vector space Convex programming Optimal value function Subdifferential Multiplier set 

Mathematics Subject Classification (2010)

49J27 49K40 90C25 90C30 90C31 90C46 

Notes

Acknowledgements

The authors would like to thank the two anonymous referees for their very careful readings and valuable suggestions which have helped to greatly improve the presentation.

Funding information

The first author was supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and Thai Nguyen University of Sciences. This research is funded by the National Foundation for Science and Technology Development (Vietnam) under grant number 101.01-2014.37.

References

  1. 1.
    An, D.T.V., Yao, J.-C.: Further results on differential stability of convex optimization problems. J. Optim. Theory Appl. 170, 28–42 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108–128 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aubin, J.-P.: Optima and Equilibria: An Introduction to Nonlinear Analysis, 2nd edn. Springer, Berlin (1998)CrossRefGoogle Scholar
  4. 4.
    Auslender, A.: Differentiable stability in non convex and non differentiable programming. In: Huard, P (ed.) Point-to-set maps and mathematical programming. Mathematical Programming Studies, vol. 10, pp 29–41. Springer, Berlin (1979)Google Scholar
  5. 5.
    Bartl, D.: A short algebraic proof of the Farkas lemma. SIAM J. Optim. 19, 234–239 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Burke, J.: Linear optimization. Class Notes, University of Washington. (https://sites.math.washington.edu/burke/crs/407/notes/)
  8. 8.
    Dien, P.H., Yen, N.D.: On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints. Appl. Math. Optim. 24, 35–54 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. In: Guignard, M (ed.) Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, vol. 19, pp 101–119. Springer, Berlin (1982)Google Scholar
  10. 10.
    Gauvin, J., Dubeau, F.: Some examples and counterexamples for the stability analysis of nonlinear programming problems. In: Fiacco, A. (ed.) Sensitivity, Stability and Parametric Analysis. Mathematical Programming Studies, vol. 21, pp 69–78. Springer, Berlin (1984)Google Scholar
  11. 11.
    Gauvin, J., Tolle, J.W.: Differential stability in nonlinear programming. SIAM J. Control Optim. 15, 294–311 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gollan, B.: On the marginal function in nonlinear programming. Math. Oper. Res. 9, 208–221 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)zbMATHGoogle Scholar
  14. 14.
    Lempio, F., Maurer, H.: Differential stability in infinite-dimensional nonlinear programming. Appl. Math. Optim. 6, 139–152 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  16. 16.
    Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Synthesis Lectures on Mathematics and Statistics, vol. 14. Morgan and Claypool Publishers, Williston (2014)Google Scholar
  17. 17.
    Mordukhovich, B.S., Nam, N.M.: Geometric approach to convex subdifferential calculus. Optimization 66, 839–873 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mordukhovich, B.S., Nam, N.M.: Extremality of convex sets with some applications. Optim. Lett. 11, 1201–1215 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mordukhovich, B.S., Nam, N.M., Rector, R.B., Tran, T.: Variational geometric approach to generalized differential and conjugate calculi in convex analysis. Set-Valued Var. Anal. 25, 731–755 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B 116, 369–396 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Outrata, J.V.: On generalized gradients in optimization problems with set-valued constraints. Math. Oper. Res. 15, 626–639 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Penot, J.-P.: Multipliers and generalized derivatives of performance functions. J. Optim. Theory Appl. 93, 609–618 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. In: Sorensen, D. C., Wets, R. J.-B (eds.) Nondifferential and Variational Techniques in Optimization. Mathematical Programming Studies, vol. 17, pp 28–66. Springer, Berlin (1982)Google Scholar
  25. 25.
    Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsThai Nguyen University of SciencesThai NguyenVietnam
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

Personalised recommendations