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About Regularity of Collections of Sets

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Abstract

The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular.

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References

  1. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis, Wiley, New York (1984)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. 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  MATH  MathSciNet  Google Scholar 

  4. 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)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Burke, J.V., Deng, S.: Weak sharp minima revisited. I. Basic theory, Control Cybernet. 31(3), 439–469 (2002). Well-posedness in optimization and related topics (Warsaw, 2001)

    MATH  MathSciNet  Google Scholar 

  7. 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  MATH  MathSciNet  Google Scholar 

  8. Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming, SIAM J. Control Optim. 31(5), 1340–1359 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dmitruk, A.V., Milyutin, A.A., Osmolovsky, N.P.: Lyusternik's theorem and the theory of extrema, Russian Math. Surveys 35, 11–51 (1980)

    Article  Google Scholar 

  10. Dubovitskii, A.Y., Milyutin, A.A.: Extremum problems in the presence of restrictions, U.S.S.R. Comp. Maths. Math. Phys. 5, 1–80 (1965)

    Article  MATH  Google Scholar 

  11. Ioffe, A.D.: Metric regularity and subdifferential calculus, Russian Math. Surveys 55, 501–558 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Klatte, D., Kummer, B.: Nonsmooth equations in optimization: Regularity, calculus, methods and applications, Kluwer, Dordrecht (2002)

  13. Kruger, A.Y.: On calculus of strict \(\varepsilon\)-semidifferentials, Dokl. Akad. Nauk Belarusi 40(4), 34–39 (1996). In Russian

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  15. Kruger, A.Y.: Strict (\(\varepsilon,\delta\))-semidifferentials and extremality of sets and functions, Dokl. Nats. Akad. Nauk Belarusi 44(4), 21–24 (2000). In Russian

    MathSciNet  Google Scholar 

  16. Kruger, A.Y.: Strict (\(\varepsilon,\delta\))-semidifferentials and extremality conditions, Optimization 51, 539–554 (2002).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. Kruger, A.Y.: Stationarity and regularity of real-valued functions, Appl. Comput. Math. (2006). To appear

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

    MATH  MathSciNet  Google Scholar 

  22. Mordukhovich, B.S.: The extremal principle and its applications to optimization and economics. In: Rubinov, A. Glover, B. (eds.) Optimization and Related Topics, Applied Optimization, vol. 47, pp. 343–369. Kluwer, Dordrecht (2001)

    Google Scholar 

  23. Mordukhovich, B.S.: Variational analysis and generalized differentiation, I: Basic theory, II: Applications, Springer, Berlin Heidelberg New York (2005)

  24. Mordukhovich, B.S., Shao, Y.: Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124, 197–205 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edition. Lecture Notes in Mathematics, vol. 1364, Springer, Berlin Heidelberg New York (1993)

  29. Polyak, B.T.: Introduction to optimization. Optimization Software Inc. Publications Division, New York (1987). Translated from Russian

  30. Robinson, S.M.: Regularity and stability for convex multivalued functions, Math. Oper. Res. 1(2), 130–143 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Springer, Berlin Heidelberg New York (1998)

    MATH  Google Scholar 

  32. Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

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Authors and Affiliations

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

    Alexander Ya. Kruger

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  1. Alexander Ya. Kruger
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Correspondence to Alexander Ya. Kruger.

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Kruger, A.Y. About Regularity of Collections of Sets. Set-Valued Anal 14, 187–206 (2006). https://doi.org/10.1007/s11228-006-0014-8

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  • DOI: https://doi.org/10.1007/s11228-006-0014-8

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