Skip to main content

Advertisement

Springer Nature Link
Log in

Semi-continuous quadratic optimization: existence conditions and duality scheme

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this work, we study the class of problems called semi-continuous optimization, which contains constrained minimization (maximization) problems with lower (upper) semi-continuous objective functions. We show some existence conditions for solutions based on asymptotic techniques, as well as a duality scheme based on the Fenchel–Moreau conjugation specifically applied to semi-continuous problems. Promising results are obtained, when we apply this scheme to minimize quadratic functions (whose Hessians can be symmetric indefinite) over nonempty, closed and convex polyhedral sets.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003)

    MATH  Google Scholar 

  2. Rockafellar, R.T.: Convex Analysis, Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  3. Adler, S., Goeleven, D., Théra, M.: Recession mappings and noncoercive variational inequalities. Nonlinear Anal. Theory Methods Appl. 26, 1573–1603 (1996)

    Article  Google Scholar 

  4. Auslender, A.: Noncoercive optimization problems. Math. Oper. Res. 21, 769–782 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baiocchi, C., Buttazo, G., Gastaldi, F., Tomarelli, F.: General existence theorem for unilateral problems in continuum mechanics. Arch. Ration. Mech. Anal. 100, 149–189 (1988)

    Article  MATH  Google Scholar 

  6. Iusem, A.N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. Theory Methods Appl. 52, 621–635 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic programming and affine variational inequalities: a qualitative study. In: Pardalos, P.M. (ed.) Nonconvex Optimization and its Applications, vol. 78. Springer, New York (2005)

    Google Scholar 

  8. Bertsekas, D., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  9. Crouzeix, J.P., Ocana, E., Sosa, W.: Análisis Convexo, Monografias del IMCA 33 (in Spanish) (2003)

  10. Martínez-Legaz, J.E.: Generalized convex duality and its economic applications. In: Handbook of Generalized Convexity and Generalized Monotonicity, vol. 76, Nonconvex Optimization Application, pp. 237–292. Springer, New York (2005)

  11. Svaiter, B.F.: A new duality theory for mathematical programming. Optimization 60, 1209–1231 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cotrina, J., Karas, E.W., Ribeiro, A.A., Sosa, W., Yuan, J.Y.: Fenchel–Moreau conjugation for lower semi-continuous functions. Optimization 60, 1045–1057 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Moreau, J.-J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pure Appl. 49, 109–154 (1970)

    MATH  MathSciNet  Google Scholar 

  14. Balder, E.J.: An extension of duality–stability relations to nonconvex optimization problems. SIAM J. Control Optim. 15, 329–343 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  15. Burachik, R.S., Rubinov, A.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413–436 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dolecki, S., Kurcyusz, S.: On \(\varPhi \)-convexity in extremal problems SIAM. J. Control Optim. 16, 277–300 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  17. Flores-Bazán, F.: On a notion of subdifferentiability for non-convex functions. Optimization 33, 1–8 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1998)

  19. Iusem, A.N., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. Ser. B 116, 259–273 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Iusem, A.N., Kassay, G., Sosa, W.: An existence result for equilibrium problems with some surjectivity consequences. J. Convex Anal. 16, 807–826 (2009)

    MATH  MathSciNet  Google Scholar 

  21. Flores-Bazan, F., Carcamo, G.: A geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints. Math. Program. Ser. A 145, 263–290 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are thankful for the valuable suggestions given by the anonymous referees that improved the paper. Fernanda Raupp was partially supported by FAPERJ/CNPq through PRONEX 662199/2010-12 and CNPq Grant 311165/2013-3, whereas Wilfredo Sosa was partially supported by CNPq Grants 302074/2012-0 and 471168/2013-0.

Author information

Authors and Affiliations

  1. Universidad del Pacífico, Lima, Peru

    John Cotrina

  2. Departamento de Engenharia Industrial, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil

    Fernanda M. P. Raupp

  3. Laboratório Nacional de Computação Científica, Petrópolis, Brazil

    Fernanda M. P. Raupp

  4. Programa de Pós-Graduação em Economia, Universidade Católica de Brasília, Brasília, Brazil

    Wilfredo Sosa

Authors
  1. John Cotrina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernanda M. P. Raupp
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wilfredo Sosa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fernanda M. P. Raupp.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cotrina, J., Raupp, F.M.P. & Sosa, W. Semi-continuous quadratic optimization: existence conditions and duality scheme. J Glob Optim 63, 281–295 (2015). https://doi.org/10.1007/s10898-015-0306-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-015-0306-3

Keywords

Mathematics Subject Classification