Advertisement

Applied Mathematics & Optimization

, Volume 63, Issue 2, pp 239–256 | Cite as

On Implicit Active Constraints in Linear Semi-Infinite Programs with Unbounded Coefficients

  • M. A. Goberna
  • G. A. Lancho
  • M. I. Todorov
  • V. N. Vera de Serio
Article

Abstract

The concept of implicit active constraints at a given point provides useful local information about the solution set of linear semi-infinite systems and about the optimal set in linear semi-infinite programming provided the set of gradient vectors of the constraints is bounded, commonly under the additional assumption that there exists some strong Slater point. This paper shows that the mentioned global boundedness condition can be replaced by a weaker local condition (LUB) based on locally active constraints (active in a ball of small radius whose center is some nominal point), providing geometric information about the solution set and Karush-Kuhn-Tucker type conditions for the optimal solution to be strongly unique. The maintaining of the latter property under sufficiently small perturbations of all the data is also analyzed, giving a characterization of its stability with respect to these perturbations in terms of the strong Slater condition, the so-called Extended-Nürnberger condition, and the LUB condition.

Keywords

Linear semi-infinite programming Implicit active constraints Extended active constraints Locally upper bounded systems Strongly unique solution Stability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) zbMATHGoogle Scholar
  2. 2.
    Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-Linear Parametric Optimization. Birkhäuser, Basel (1983) zbMATHGoogle Scholar
  3. 3.
    Cánovas, M.J., Klatte, D., López, M.A., Parra, J.: Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. 18, 717–732 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Springer, Berlin (1993) zbMATHGoogle Scholar
  5. 5.
    Fajardo, M.D., López, M.A.: Locally Farkas-Minkowski systems in convex semi-infinite programming. J. Optim. Theory Appl. 103, 313–335 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, New York (1998) zbMATHGoogle Scholar
  7. 7.
    Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl. 17, 730–743 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Goberna, M.A., López, M.A., Todorov, M.I.: Extended active constraints in linear optimization with applications. SIAM J. Optim. 14, 608–619 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goberna, M.A., López, M.A., Todorov, M.I.: A sup-function approach to linear semi-infinite optimization. J. Math. Sci. 116, 3359–3368 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goberna, M.A., López, M.A., Todorov, M.I.: A generic result in linear semi-infinite optimization. Appl. Math. Optim. 48, 181–193 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Helbig, S., Todorov, M.I.: Unicity results for general linear semi-infinite optimization problems using a new concept of active constraints. Appl. Math. Optim. 38, 21–43 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hettich, R., Zencke, P.: Numerische Methoden der Approximation und Semi-infiniten Optimierung. Teubner, Stuttgart (1982) zbMATHGoogle Scholar
  13. 13.
    Lucchetti, R.: Convexity and Well-Posed Problems. Springer, New York (2006) zbMATHGoogle Scholar
  14. 14.
    Nürnberger, G.: Unicity in semi-infinite optimization. In: Brosowski, B., Deutsch, F. (eds.) Parametric Optimization and Approximation, pp. 231–247. Birkhäuser, Basel (1985) Google Scholar
  15. 15.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • M. A. Goberna
    • 1
  • G. A. Lancho
    • 2
  • M. I. Todorov
    • 3
    • 4
  • V. N. Vera de Serio
    • 5
  1. 1.Dep. of Statistics and Operations ResearchAlicante UniversityAlicanteSpain
  2. 2.Instituto de Física y MatemáticasUniversidad Tecnológica de MixtecaHuajuapan de LeonMexico
  3. 3.Dep. of Physics and MathematicsUDLASan Andrés CholulaMexico
  4. 4.IMI-BASSofiaBulgaria
  5. 5.Facultad de Ciencias Económicas, Instituto de Ciencias BásicasUniversidad Nacional de CuyoMendozaArgentina

Personalised recommendations