Skip to main content
SpringerLink
Log in

On Epsilon-Stability in Optimization

  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study stability of optimization problems in the separated locally convex topological vector space setting. We use the concepts of 𝜖-stability, dual 𝜖-stability, and 𝜖-duality gap, and establish geometric characterizations of these notions by an epigraphical analysis approach. Under a constraint qualification involving quasi relative interiors, we obtain some criteria for 𝜖-stability and 𝜖-duality gap, which are shown to be useful for obtaining relevant stability and duality theorems in infinite dimensional optimization. We apply our approach to study cone constrained problems, Fenchel duality, conjugate duality, and the subdifferentiability of functions associated with epigraphical type sets. Several duality and stability results of recent publications can be deduced from and sometimes improved by a geometric characterization of 𝜖-stability in a unifying way.

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

  1. Attouch, H., Brezis, H.: Duality for the sum of convex functions in general Banach spaces. In: Barroso, J.A. (ed.) Aspect of Mathematics and Its Applications. North Holland Mathematical Library, vol. 34, pp. 125–133. Elsevier, North-Holland (1986)

  2. Attouch, H., Wets, R.J.-B., et al.: Epigraphical analysis. In: Attouch, H. (ed.) Analyse Non Lineaire, pp 73–100. Gauthier-Villars, Paris (1989)

  3. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. Revised edition. Translated from the Romanian. Editura Academiei, Bucharest. Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn (1978)

    Google Scholar 

  4. Boncea, H.-V., Grad, S.-M.: Characterizations of 𝜖-duality gap statements for composed optimization problems. Nonlinear Anal. 92, 96–107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borwein, J.M., Burachik, R.S., Yao, L.: Conditions for zero duality gap in convex programming. J. Nonlinear Convex Anal. 15, 167–190 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  MATH  Google Scholar 

  7. Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)

    MATH  Google Scholar 

  8. Boţ, R.I., Csetnek, E.R.: Regualarity conditions via generalized interiority notions in convex optimizations: New achievements and their relation to some classical statements. Optimization 61, 35–65 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boţ, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boţ, R. I., Csetnek, E.R., Wanka, G.: Sequential optimality conditions in convex programming via perturbation approach. J. Convex Anal. 15, 149–164 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Boţ, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boţ, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal. 64, 2787–2804 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cammaroto, F., Di Bella, B.: Separation theorem based on the quasirelative interior and application to duality theory. J. Optim. Theory Appl. 125, 223–229 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Champion, T.: Duality gap in convex programming. Math. Program. 99, 487–498 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Daniele, P., Giuffrè, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dinh, N., López, M.A., Volle, M.: Functional inequalities in the absence of convexity and lower semicontinuity with applications to optimization. SIAM J. Optim. 20, 2540–2559 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dinh, N., Goberna, M.A., López, M.A., Volle, M.: Convex inequalities without constraint qualification nor closedness condition, and their applications in optimization. Set-Valued Var. Anal. 18, 423–445 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Translated from the French. Studies in Mathematics and Its Applications, vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York (1976)

  19. Ernst, E., Volle, M.: Zero duality gap and attainment with possibly non-convex data. J. Convex Anal. 23, 615–629 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Fenchel, W.: A remark on convex sets and polarity. Comm. Sem. Math. Univ. Lund, Tome supplémentaire, pp. 82–89 (1952)

  21. Flores-Bazán, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Flores-Bazán, F., Flores-Bazán, F., Vera, C.: A complete characterization of strong duality in nonconvex optimization with a single constraint. J. Glob. Optim. 53, 185–201 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giannessi, F., Mastroeni, G.: Separation of sets and Wolf duality. J. Glob. Optim. 42, 401–412 (2008)

    Article  MATH  Google Scholar 

  24. Grad, A.: Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints. J. Math. Anal. Appl. 361, 86–95 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grad, S.-M.: Closedness type regularity conditions in convex optimization and beyond. Front. Appl. Math. Stat. (2016). https://doi.org/https://doi.org/10.3389/fams.2016.00014

  26. Grad, S.-M.: Vector Optimization and Monotone Operators via Convex Duality: Recent Advances. Springer, Cham (2014)

    MATH  Google Scholar 

  27. Goberna, M.A., Ridolfi, A.B., Vera de Serio, V.N.: Stability of the duality gap in linear optimization. Set-Valued Var. Anal. 25, 617–636 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hiriart-Urruty, J.-B., López, M.A., Volle, M.: The 𝜖-strategy in variational analysis: illustration with the closed convexification of a function. Rev. Mat. Iberoam 27, 449–474 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24, 1727–1754 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jeyakumar, V., Wolkovicz, H.: Zero duality gap in infinite-dimensional programming. J. Optim. Theory Appl. 67, 87–108 (1990)

    Article  MathSciNet  Google Scholar 

  31. Jeyakumar, V., Li, G.Y.: Stable zero duality gaps in convex programming: complete dual characterisations with applications to semidefinite programs. J. Math. Anal. Appl. 360, 156–167 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Joly, J.-L., Laurent, P.-J.: Stability and duality in convex minimization problems. M.R.C. Techn. Report 1090, Univ. of Wisc. (1970); RIRO, pp. 3–42 (1971)

  33. Moussaoui, M., Volle, M.: Quasicontinuity and united functions in convex duality theory. Commun. Appl. Nonlinear Anal. 4, 73–89 (1997)

    MathSciNet  MATH  Google Scholar 

  34. Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelpia (1974)

    Book  MATH  Google Scholar 

  35. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  36. Vicente-Pérez, J., Volle, M.: Even convexity, subdifferentiability, and Γ-regularization in general topological vector spaces. J. Math. Anal. Appl. 429, 956–968 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zălinescu, C.: A note on d-stability of convex programs and limiting Lagrangians. Math. Program. 53, 267–277 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  39. Zălinescu, C.: On three open problems related to quasi relative interior. J. Convex Anal. 22, 641–645 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Zălinescu, C.: On the use of the quasi-relative interior in optimization. Optimization 64, 1795–1823 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the two anonymous referees for their careful reading and useful comments on the first draft of this submission.

Author information

Authors and Affiliations

  1. Avignon University, LMA EA 2151, Avignon, France

    Dinh The Luc & Michel Volle

  2. Center for Informatics and Computing, VAST, Hanoi, Vietnam

    Dinh The Luc

Authors
  1. Dinh The Luc
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Volle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Volle.

Additional information

Dedicated to Michel Théra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luc, D.T., Volle, M. On Epsilon-Stability in Optimization. Vietnam J. Math. 46, 149–167 (2018). https://doi.org/10.1007/s10013-017-0265-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-017-0265-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation