Skip to main content
Log in

Apart Sets and Functions: an Application to the Stability of Penalized Optimization Problems

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

Abstract

The notion of apart sets and functions, a technical concept introduced by Luc and Penot, is studied in the framework of star-shaped sets and functions. As an application, we provide a necessary and sufficient condition ensuring that a penalized optimization problem is stable with respect to perturbations of both the penalty and the objective functions, in the case when the perturbations are small in the sense of the Attouch–Wets topology.

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

Access this article

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.: Variational Convergence for Functions and Operators. Appl. Math. Series, Pitman, London (1984)

  2. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academics Publishers, Dordrecht (1993)

    Book  MATH  Google Scholar 

  3. Beer, G., Lucchetti, R.: Convex optimization and the epi-distance topology. Trans. Am. Math. Soc. 327, 795–813 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beer, G., Théra, M.A.: Attouch–wets convergence and a differential operator for convex functions. Proc. Am. Math. Soc. 122, 851–858 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borwein, J., Lewis, A.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Birkhäuser, New York (2006)

    Book  MATH  Google Scholar 

  6. Cominetti, R., Dussault, J.-P.: Stable exponential-penalty algorithm with superlinear convergence. J. Optim. Theory Appl. 83, 285–309 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dontchev, A., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berline (1993)

    Google Scholar 

  9. Dussault, J.-P.: Numerical stability and efficiency of penalty algorithms. SIAM J. Numer. Anal. 32, 296–317 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eberhard, A., Wenczel, R.: Epi-distance convergence of parametrized sums of convex functions in non-reflexive spaces. J. Convex Anal. 7, 47–71 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Ernst, E., Théra, M.A.: Minimizing irregular convex functions: Ulam stability for approximate minima. Set-valued Var Anal. 18, 447–466 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)

    MATH  Google Scholar 

  13. Forsgren, A., Gill, P.E., Wright, M.H.: Interior methods for nonlinear optimization. SIAM Rev. 44, 525–597 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gilbert, J.-C., Gonzaga, V., Karas, E.: Examples of ill-behaved central paths in convex optimization. Math. Program. 103, 63–94 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ioffe, A., Lucchetti, R.E.: Generic well-posedness in minimization problems. Abstr. Appl. Anal. 2005, 343–360 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luc, D.T., Penot, J.-P.: Convergence of asymptotic directions. Trans. Am. Math. Soc. 353, 4095–4121 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Motzkin, T.S.: New Techniques for Linear Inequalities and Optimisation. Project SCOOP Symposium on Linear Inequalities and Programming, Planning Research Division, U.S. Air Force, Washington D.C. (1951)

    Google Scholar 

  18. Penot, J.-P., Zalinescu, C.: Bounded (Hausdorff) convergence: basic facts and applications. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications, pp 827–854. Springer, Boston (2002)

  19. Penot, J.-P.: A metric approach to asymptotic analysis. Bull. Sci. Math. 127, 815–833 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to warmly thank the anonymous referee. His careful reading of the paper allowed us to correct a significant number of typos and errors and largely contributed to the final form of the proofs of several important results of the article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Emil Ernst.

Additional information

Dedicated to Michel Théra in honor of his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ernst, E. Apart Sets and Functions: an Application to the Stability of Penalized Optimization Problems. Vietnam J. Math. 46, 343–358 (2018). https://doi.org/10.1007/s10013-018-0288-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-018-0288-9

Keywords

Mathematics Subject Classification (2010)

Navigation