Skip to main content
SpringerLink
Log in

Generic Existence of Solutions and Generic Well-Posedness of Optimization Problems

  • Conference paper
  • First Online:
Book cover Computational and Analytical Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

  • 1871 Accesses

Abstract

We exhibit a large class of topological spaces in which the generic attainability of the infimum by the bounded continuous perturbations of a lower semicontinuous function implies generic well-posedness of the perturbed optimization problems. The class consists of spaces which admit a winning strategy for one of the players in a certain topological game and contains, in particular, all metrizable spaces and all spaces that are homeomorphic to a Borel subset of a compact space.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

References

  1. Arhangel’skii, A.V.: On a class containing all metric and all locally bicompact spaces. Dokl. Akad. Nauk. SSSR 151, 751–754 (1963)

    MathSciNet  Google Scholar 

  2. Arhangel’skii, A.V., Čoban, M.M., Kenderov, P.S.: Topological games and continuity of group operations. Topology Appl. 157, 2542–2552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and differentiability of convex functions. Trans. Amer. Math. Soc. 303, 517–527 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Čoban, M.M., Kenderov, P.S.: Dense Gâteaux differentiability of the sup-norm in C ∗(T) and the topological properties of T. Compt. Rend. Acad. Bulg. Sci. 38, 1603–1604 (1985)

    Google Scholar 

  5. Čoban, M.M., Kenderov P.S., Revalski, J.P.: Generic well-posedness of optimization problems in topological spaces. Mathematika 36, 301–324 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Čoban, M.M., Kenderov P.S., Revalski, J.P.: Variational principles and topological games. Topol. Appl., 159(17), 3550–3562 (2012)

    Article  Google Scholar 

  7. Deville, R., Godefroy G., Zizler, V.: A smooth variational principle with applications to Hamilton–Jacobi equations in infinite dimensions. J. Funct. Anal. 111, 197–212 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deville, R., Godefroy G., Zizler, V.: Smoothness and renormings in Banach spaces. Pitman monographs and Surveys in Pure and Applied Mathematics. Longman Scientific and Technical, Harlow (1993)

    MATH  Google Scholar 

  9. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jayne, J.E., Rogers, C.A.: Borel selectors for upper semi-continuous set-valued maps. Acta Math. 155, 41–79 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kenderov, P.S., Kortezov, I.S., Moors, W.B.: Topological games and topological groups. Topol. Appl. 109, 157–165 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kenderov, P.S., Kortezov, I.S., Moors, W.B.: Continuity points of quasi-continuous mappings. Topol. Appl. 109, 321–346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kenderov, P.S., Moors, W.B.: Game characterization of fragmentability of topological spaces. In: Mathematics and Education in Mathematics, Proceedings of the 25-th Spring conference of the Union of Bulgarian Mathematicians, April 1996, pp. 8–18, Kazanlak, Bulgaria (1996)

    Google Scholar 

  14. Kenderov, P.S., Moors, W.B.: Fragmentability of Banach spaces. Compt. Rend. Acad. Bulg. Sci. 49(2), 9–12 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Kenderov, P.S., Moors, W.B.: Fragmentability and sigma-fragmentability of Banach spaces. J. London Math. Soc. 60, 203–223 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kenderov, P.S., Revalski, J.P.: The Banach-Mazur game and generic existence of solutions to optimization problems. Proc. Amer. Math. Soc. 118, 911–917 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kenderov, P.S., Revalski, J.P.: Dense existence of solutions of perturbed optimization problems and topological games. Compt. Rend. Acad. Bulg. Sci. 63(7), 937–942 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Kenderov, P.S., Revalski, J.P.: Variational principles in non metrizable spaces. TOP. DOI: 10.1007/s11750-011-0243-3 (2011)

    Google Scholar 

  19. Michael, E.: A note on completely metrizable spaces. Proc. Amer. Math. Soc. 96, 513–522 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oxtoby, J.C.: The Banach-Mazur game and Banach Category Theorem. In: Contributions to the Theory of Games, Vol. III, Annals of Mathamatics Studies, PrincetonUniversity Press, Princeton 39, 159–163 (1957)

    MathSciNet  MATH  Google Scholar 

  21. Ribarska, N.K.: Internal characterization of fragmentable spaces. Mathematika 34, 243–257 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stegall, C.: Optimization of functions on certain subsets of Banach space. Math. Ann. 236, 171–176 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank an anonymous referee for the helpful suggestions to improve the presentation of the results. Both authors have been partially supported by the Bulgarian National Fund for Scientific Research, under grant DO02-360/2008.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, block 8, 1113, Sofia, Bulgaria

    P. S. Kenderov & J. P. Revalski

Authors
  1. P. S. Kenderov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. P. Revalski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. P. Revalski .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

Additional information

Dedicated to Jonathan Borwein on the occasion of his 60th birthday

Communicated By Jon D. Vanderwerff.

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this paper

Cite this paper

Kenderov, P.S., Revalski, J.P. (2013). Generic Existence of Solutions and Generic Well-Posedness of Optimization Problems. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_20

Download citation

Publish with us

Policies and ethics

Search

Navigation