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Error Bounds for the Difference of Two Convex Multifunctions

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Abstract

In this paper, we consider error bounds for DC multifunctions (difference of two convex multifunctions) with/without set constraints. We give some Robinson-Ursescu type results in Banach spaces. Using some techniques of convex analysis, we present some results on the existence of error bounds in terms of normal cone and coderivative.

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References

  1. 1.

    Amahroq, T., Penot, J.-P., Syam, A.: On the subdifferentiability of the difference of two functions and local minimization. Set-Valued Anal. 16(2), 413–427 (2008)

  2. 2.

    An, L.T.H., Pham, D.T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133(2), 23–46 (2005)

  3. 3.

    Aubin, J.-P.: Mutational and morphological analysis. Tools for Shape Evolution and Morphogenesis, Systems and Control: Foundations and Applications. Birkhäser, Boston (1999)

  4. 4.

    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2002)

  5. 5.

    Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of MODE-SMAI conference. ESAIM Proc. 13, 1–17 (2003)

  6. 6.

    Azé, D.: A unified theory for metric regularity of multifunctions. J. Convex Anal. 13(2), 225–252 (2006)

  7. 7.

    Bednarczuka, E.M., Krugerb, A.Y.: Error bounds for vector-valued functions: necessary and sufficient conditions. Nonlinear Anal. 75(3), 1124–1140 (2012)

  8. 8.

    Bosch, P., Jourani, A., Henrion, R.: Sufficient conditions for error bounds and applications. Appl. Math. Optim. 50(2), 161–181 (2004)

  9. 9.

    Clarke, F.H.: Optimization and nonsmooth analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, New York (1983)

  10. 10.

    Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12(1-2), 79–109 (2004)

  11. 11.

    Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set-valued Var. Anal. 18(2), 121–149 (2010)

  12. 12.

    Gautier, S.: Affine and eclipsing multifunctions. Numer. Funct. Anal. Optim. 11(8), 679–699 (1990)

  13. 13.

    Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21(3), 151–176 (2013)

  14. 14.

    Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13(2), 520–534 (2002)

  15. 15.

    Horst, R., Thoai, N.V.: DC programming: overview. J. Optim. Theory Appl. 103(1), 1–43 (1999)

  16. 16.

    Huang, H.: Coderivative conditions for error bounds of γ-paraconvex multifunctions. Set-Valued Var. Anal. 20(4), 567–579 (2012)

  17. 17.

    Huang, H., Li, R.X.: Global error bounds for γ-multifunctions. Set-Valued Var. Anal. 19(3), 487–504 (2011)

  18. 18.

    Ngai, H.V., Théra, M.: Error bounds and implict multifunction theorem in smooth Banach spaces and applications to optimization. Set-Valued Anal. 12, 195–223 (2004)

  19. 19.

    Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2–3), 199–227 (2008)

  20. 20.

    Jourani, A.: Weak regularity of functions and sets in Asplund spaces. Nonlinear Anal. 65(3), 660–676 (2006)

  21. 21.

    Le Thi, H.A., Pham Dinh, T., Ngai, H.V.: Exact penalty and error bounds in DC programming. J. Glob. Optim. 52(3), 509–535 (2012)

  22. 22.

    Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Properties of two DC algorithms for quadratic programming. J. Glob. Optim. 49(3), 481–495 (2010)

  23. 23.

    Lewis, A.S., Pang, J.S.: Error bound for convex inequality systems. In: Crouzeix, J.P., et al (eds.) Generalized convexity and generalized monotonicity. Proceedings of the 5th Symposium on Generalized Convexity, pp. 75–100. Kluwer, Dordrecht (1997)

  24. 24.

    Li, G., Mordukhovich, B.S.: Höder metric subregularity with applications to proximal point method. SIAM J. Optim. 22(4), 1655–1684 (2012)

  25. 25.

    Li, W., Singer, I.: Global error bounds for convex multifunctions and applications. Math. Oper. Res. 23(2), 443–462 (1998)

  26. 26.

    Martínez-Legaz, J.E., Penot, J.-P.: Regularization by erasement. Math. Scand. 98, 97–124 (2006)

  27. 27.

    Mordukhovich, B.S.: Variational analysis and generalized differentiation. I: Basic Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer-Verlag, Berlin (2006)

  28. 28.

    Mordukhovich, B.S., Nghia, T.T.A.: DC optimization approach to metric regularity of convex multifunctions with applications to infinite systems. J. Optim. Theory Appl. 155(3), 762–784 (2012)

  29. 29.

    Ng, K.F., Zheng, X.Y.: Characterizations of error bounds for convex multifunctions on Banach spaces. Math. Oper. Res. 29(1), 45–63 (2004)

  30. 30.

    Ngai, H.V., Kruger, A.Y., Thera, M.: Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim. 20(4), 2080–2096 (2010)

  31. 31.

    Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. B 79(1–3), 299–332 (1997)

  32. 32.

    Penot, J.P.: The directional subdifferential of the difference of two convex functions. J. Glob. Optim. 49(3), 505–519 (2011)

  33. 33.

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

  34. 34.

    Ursescu, C.: Multifunctions with convex closed graph. Czechoslov. Math. J. 25(100), 438–441 (1975)

  35. 35.

    Wu, Z., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program, Ser. A 92(2), 301–314 (2002)

  36. 36.

    Zălinescu, C.: A nonlinear extension of Hoffman’s error bounds for linear inequalities. Math. Oper. Res. 28(3), 524–532 (2003)

  37. 37.

    Zǎlinescu, C.: Convex analysis in general vector spaces. World Scientific, Singapore (2002)

  38. 38.

    Zheng, X.Y.: Error bounds for set inclusion. Sci. China, Ser. A 46(6), 750–763 (2003)

  39. 39.

    Zheng, X.Y., Ng, K.F.: The Fermat rule for multifunctions on Banach spaces. Math. Program. Ser. A 104(1), 69–90 (2005)

  40. 40.

    Zheng, X.Y., Ng, K.F.: Metric subregularity and calmness for nonconvex generalized equations. SIAM J. Optim. 20(5), 2119–2136 (2010)

  41. 41.

    Zheng, X.Y., Ng, K.F.: Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19(4), 62–76 (2008)

  42. 42.

    Zheng, X.Y., Yang, X.M., Teo, K.L.: Sharp minima for multiobjective optimization in Banach spaces. Set-Valued Anal. 14(1), 327–345 (2006)

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Correspondence to Hui Huang.

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Huang, H., Li, R. Error Bounds for the Difference of Two Convex Multifunctions. Set-Valued Var. Anal 22, 447–465 (2014). https://doi.org/10.1007/s11228-013-0271-2

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Keywords

  • Error bound
  • DC multifunction
  • Normal cone
  • Coderivative
  • Normed space

Mathematics Subject Classifications (2010)

  • 49J52
  • 90C29
  • 90C31