Skip to main content
SpringerLink
Log in

Stationarity and Regularity of Infinite Collections of Sets

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces.

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

Access this article

Log in via an institution

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

Notes

  1. M. Théra, personal communication.

References

  1. Dubovitskii, A.Y., Miljutin, A.A.: Extremal problems with constraints. U.S.S.R. Comput. Math. Math. Phys. 5, 1–80 (1965)

    Article  Google Scholar 

  2. Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24(8), 684–687 (1980). In Russian

    MathSciNet  MATH  Google Scholar 

  3. Kruger, A.Y., Mordukhovich, B.S.: Generalized normals and derivatives and necessary conditions for an extremum in problems of nondifferentiable programming. Deposited in VINITI, I—no. 408-80, II—no. 494-80. Minsk (1980). In Russian

  4. Kruger, A.Y.: Generalized differentials of nonsmooth functions. Deposited in VINITI no. 1332-81. Minsk (1981). In Russian

  5. Mordukhovich, B.S., Shao, Y.: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces. Nonlinear Anal. 25(12), 1401–1424 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)

    Google Scholar 

  7. Bakan, A., Deutsch, F., Li, W.: Strong CHIP, normality, and linear regularity of convex sets. Trans. Am. Math. Soc. 357(10), 3831–3863 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program., Ser. A 86(1), 135–160 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bauschke, H.H., Borwein, J.M., Tseng, P.: Bounded linear regularity, strong CHIP, and CHIP are distinct properties. J. Convex Anal. 7(2), 395–412 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Burke, J.V., Deng, S.: Weak sharp minima revisited. II. Application to linear regularity and error bounds. Math. Program., Ser. B 104(2–3), 235–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borwein, J.M., Jofré, A.: A nonconvex separation property in Banach spaces. Math. Methods Oper. Res. 48(2), 169–179 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jeyakumar, V.: The strong conical hull intersection property for convex programming. Math. Program., Ser. A 106(1), 81–92 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kruger, A.Y.: On the extremality of set systems. Dokl. Nats. Akad. Nauk Belarusi 42(1), 24–28 (1998). In Russian

    MathSciNet  MATH  Google Scholar 

  15. Kruger, A.Y.: Strict (ε,δ)-semidifferentials and the extremality of sets and functions. Dokl. Nats. Akad. Nauk Belarusi 44(2), 19–22 (2000). In Russian

    MathSciNet  MATH  Google Scholar 

  16. Kruger, A.Y.: Strict (ε,δ)-subdifferentials and extremality conditions. Optimization 51(3), 539–554 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. 116(3), 3325–3358 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kruger, A.Y.: Weak stationarity: eliminating the gap between necessary and sufficient conditions. Optimization 53(2), 147–164 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kruger, A.Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1(1), 101–126 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14(2), 187–206 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13(6A), 1737–1785 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Li, C., Ng, K.F.: Constraint qualification, the strong CHIP, and best approximation with convex constraints in Banach spaces. SIAM J. Optim. 14(2), 584–607 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mordukhovich, B.S., Phan, H.M.: Rated extremal principles for finite and infinite systems. Optimization 60(7), 893–923 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mordukhovich, B.S., Phan, H.M.: Tangential extremal principles for finite and infinite systems of sets, II: Applications to semi-infinite and multiobjective optimization. To be published

  25. Ng, K.F., Yang, W.H.: Regularities and their relations to error bounds. Math. Program., Ser. A 99, 521–538 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9(1–2), 187–216 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, B.: The fuzzy intersection rule in variational analysis with applications. J. Math. Anal. Appl. 323(2), 1365–1372 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, B., Wang, D.: On the fuzzy intersection rule. Nonlinear Anal. 75(3), 1623–1634 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zheng, X.Y., Ng, K.F.: Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19(1), 62–76 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization. To be published

  31. Li, C., Ng, K.F., Pong, T.K.: The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 18(2), 643–665 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14(3), 757–772 (2003)

    Article  MathSciNet  Google Scholar 

  33. Zheng, X.Y., Wei, Z., Yao, J.C.: Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets. Nonlinear Anal. 73(2), 413–430 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cánovas, M.J., López, M.A., Mordukhovich, B.S., Parra, J.: Variational analysis in semi-infinite and infinite programming, II: necessary optimality conditions. SIAM J. Optim. 20(6), 2788–2806 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Dinh, N., Goberna, M.A., López, M.A., Son, T.Q.: New Farkas-type constraint qualifications in convex infinite programming. ESAIM Control Optim. Calc. Var. 13(3), 580–597 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I: Fundamentals. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305. Springer, Berlin (1993)

    Google Scholar 

  37. Li, W., Nahak, C., Singer, I.: Constraint qualifications for semi-infinite systems of convex inequalities. SIAM J. Optim. 11(1), 31–52 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming. Nonlinear Anal. 73(5), 1143–1159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. I. Sufficient optimality conditions. J. Optim. Theory Appl. 142(1), 147–163 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. II. Necessary optimality conditions. J. Optim. Theory Appl. 142(1), 165–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  42. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)

    Book  MATH  Google Scholar 

  43. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

Download references

Acknowledgements

The research was partially supported by the Australian Research Council, Project DP110102011 and the Spanish MTM2008-06695-C03(01). The first author is grateful to the University of Alicante for support and hospitality during his stay there in June 2011. The authors wish to thank Michel Théra and the anonymous referees for the careful reading of the paper and valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Centre for Informatics and Applied Optimization, School of Science, Information Technology and Engineering, University of Ballarat, POB 663, Ballarat, Vic, 3350, Australia

    Alexander Y. Kruger & Marco A. López

  2. Department of Statistics and Operations Research, University of Alicante, 03080, Alicante, Spain

    Marco A. López

Authors
  1. Alexander Y. Kruger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco A. López
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Y. Kruger.

Additional information

Communicated by Michel Théra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kruger, A.Y., López, M.A. Stationarity and Regularity of Infinite Collections of Sets. J Optim Theory Appl 154, 339–369 (2012). https://doi.org/10.1007/s10957-012-0043-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0043-4

Keywords

Search

Navigation