Abstract

This paper is firstly concerned with the modulus of metric regularity of intersection mappings. We consider a finite collection of set-valued mappings and analyze the relationship between the regularity moduli of these mappings (specifically, the maximum of them) and the regularity modulus of the associated intersection mapping. As an application we derive the Lipschitz modulus of the feasible set mapping associated with linear systems of (possibly) infinitely many linear inequalities and finitely many equations. Previously we characterize the metric regularity of such systems. Specifically, we consider an intersection mapping which obeys the strategy of splitting equations into inequalities, and then we apply preliminary results for inequality systems.

Author information

Contact details are reproduced from the original publication and may be historical.

María Josefa Cánovas

Center of Operations Research, Miguel Hernández University, 03202 Elche - Alicante, Spain

canovas@umh.es

Francisco José Gómez-Senent

Center of Operations Research, Miguel Hernández University, 03202 Elche - Alicante, Spain

paco.gomez@umh.es

Juan Parra

Center of Operations Research, Miguel Hernández University, 03202 Elche - Alicante, Spain

parra@umh.es

Keywords

  • Metric regularity
  • linear systems
  • intersection mappings
  • linear regularity

Mathematics Subject Classification

65F22, 90C34, 90C05, 15A39, 49J53

References

  1. H. H. Bauschke, J. M. Borwein, W. Li: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization, Math. Program., Ser. A 86 (1999) 135–160.https://doi.org/10.1007/s101070050083Google Scholar
  2. M. J. Cánovas, A. L. Dontchev, M. A. López, J. Parra: Metric regularity of semi-infinite constraint systems, Math. Program., Ser. B 104 (2005) 329–346.https://doi.org/10.1007/s10107-005-0618-zGoogle Scholar
  3. M. J. Cánovas, F. J. Gómez-Senent, J. Parra: Stability of systems of linear equations and inequalities: Distance to ill-posedness and metric regularity, Optimization 56 (2007) 1–24.https://doi.org/10.1080/02331930600815926Google Scholar
  4. M. J. Cánovas, F. J. Gómez-Senent, J. Parra: Regularity modulus of arbitrarily perturbed linear inequality systems, J. Math. Anal. Appl. 343 (2008) 315–327.https://doi.org/10.1016/j.jmaa.2008.01.021Google Scholar
  5. M. J. Cánovas, F. J. Gómez-Senent, J. Parra: Linear regularity, equirregularity, and intersection mappings for convex semi-infinite inequality systems, Math. Program., Ser. B 123 (2010) 33–60.https://doi.org/10.1007/s10107-009-0312-7Google Scholar
  6. M. J. Cánovas, M. A. López, J. Parra, F. J. Toledo: Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems, Math. Program., Ser. A 103 (2005) 95–126.https://doi.org/10.1007/s10107-004-0519-6Google Scholar
  7. A. L. Dontchev, A. S. Lewis, R. T. Rockafellar: The radius of metric regularity, Trans. Amer. Math. Soc. 355 (2003) 493–517.https://doi.org/10.1090/S0002-9947-02-03088-XGoogle Scholar
  8. M. A. Goberna, M. A. López: Linear Semi-Infinite Optimization, John Wiley & Sons, Chichester (1998).Google Scholar
  9. R. Henrion, D. Klatte: Metric regularity of the feasible set mapping in semi-infinite optimization, Appl. Math. Optimization 30 (1994) 103–109.https://doi.org/10.1007/BF01261993Google Scholar
  10. A. D. Ioffe: Metric regularity and subdifferential calculus, Usp. Mat. Nauk 55(3) (333) 103–162 (in Russian); Russ. Math. Surv. 55(3) (2000) 501–558 (in English).https://doi.org/10.1070/RM2000v055n03ABEH000292Google Scholar
  11. D. Klatte, B. Kummer: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications, Kluwer, Dordrecht (2002).Google Scholar
  12. J.-E. Martínez-Legaz, A. Rubinov, I. Singer: Downward sets and their separation and approximation properties, J. Glob. Optim. 23 (2002) 111–137.https://doi.org/10.1023/A:1015583411806Google Scholar
  13. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  15. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1997).https://doi.org/10.1007/978-3-642-02431-3Google Scholar

Suggested citation

M. J. Cánovas, F. J. Gómez-Senent, J. Parra. “Regularity Modulus of Intersection Mappings. Application to the Stability of Equations via Splitting into Inequalities.” Journal of Convex Analysis 19 (2012), No. 4, 913–926. https://doi.org/10.68381/jca19048

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.