In this paper, we introduce and study $p$-uniform subsmoothness of a collection of infinitely many closed sets in a Banach space. Using variational analysis and techniques, we mainly study linear regularity for a collection of infinitely many closed sets satisfying $p$-uniform subsmoothness. The necessary or/and sufficient conditions on the linear regularity are obtained in this case. In particular, we extend the characterizations of linear regularity for a collection of infinitely many closed convex sets to the nonconvex setting.
References
R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. 0861.49015 10.1090/S0002-9947-96-01544-9 R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. 0861.49015 10.1090/S0002-9947-96-01544-9
F. Bernard and L. Thilbaut, Prox-regularity of functions and sets in Banach spaces, Set-Valued Anal., 12 (2004), 25-47. MR2069350 1046.49009 10.1023/B:SVAN.0000023403.87092.a2 F. Bernard and L. Thilbaut, Prox-regularity of functions and sets in Banach spaces, Set-Valued Anal., 12 (2004), 25-47. MR2069350 1046.49009 10.1023/B:SVAN.0000023403.87092.a2
R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, 1998. 0888.49001 R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, 1998. 0888.49001
R. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. 0960.49018 10.1090/S0002-9947-00-02550-2 R. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. 0960.49018 10.1090/S0002-9947-00-02550-2
D. Aussel, A. Daniilidis and L. Thibault, Subsmooth sets: Functional characterizations and related concepts, Trans. Amer. Math. Soc., 357 (2005), 1275-1301. MR2115366 1094.49016 10.1090/S0002-9947-04-03718-3 D. Aussel, A. Daniilidis and L. Thibault, Subsmooth sets: Functional characterizations and related concepts, Trans. Amer. Math. Soc., 357 (2005), 1275-1301. MR2115366 1094.49016 10.1090/S0002-9947-04-03718-3
X. Y. Zheng, Z. Wei and J.-C. Yao, Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets, Nonlinear Anal., 73 (2010), 413-430. 1188.90255 10.1016/j.na.2010.03.032 X. Y. Zheng, Z. Wei and J.-C. Yao, Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets, Nonlinear Anal., 73 (2010), 413-430. 1188.90255 10.1016/j.na.2010.03.032
H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426. 0865.47039 10.1137/S0036144593251710 H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426. 0865.47039 10.1137/S0036144593251710
G. Jameson, The duality of pair of wedges, Proc. Lond. Math. Soc., 24 (1972), 531- 547. 0233.46009 G. Jameson, The duality of pair of wedges, Proc. Lond. Math. Soc., 24 (1972), 531- 547. 0233.46009
H. Bauschke, J. M. Borwein and 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. 0998.90088 10.1007/s101070050083 H. Bauschke, J. M. Borwein and 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. 0998.90088 10.1007/s101070050083
K. F. Ng and W. H. Yang, Regularities and their relations to error bounds, Math. Program., Ser. A, 99 (2004), 521-538. 1077.90050 10.1007/s10107-003-0464-9 K. F. Ng and W. H. Yang, Regularities and their relations to error bounds, Math. Program., Ser. A, 99 (2004), 521-538. 1077.90050 10.1007/s10107-003-0464-9
C. Li, K. F. Ng and T. K. Pong, The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces, SIAM J. Optim., 18 (2007), 643-665. MR2338455 10.1137/060652087 C. Li, K. F. Ng and T. K. Pong, The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces, SIAM J. Optim., 18 (2007), 643-665. MR2338455 10.1137/060652087
X. Y. Zheng and K. F. Ng, Metric regularity and constraint qualification for convex inequalities on Banach spaces, SIAM. J. Optim., 15 (2003), 757-772. 1079.90103 10.1137/S1052623403423102 X. Y. Zheng and K. F. Ng, Metric regularity and constraint qualification for convex inequalities on Banach spaces, SIAM. J. Optim., 15 (2003), 757-772. 1079.90103 10.1137/S1052623403423102
X. Y. Zheng and K. F. Ng, Linear regularity for a collection of subsmooth sets in Banach spaces, SIAM J. Optim., 19 (2008), 62-76. 1190.90231 10.1137/060659132 X. Y. Zheng and K. F. Ng, Linear regularity for a collection of subsmooth sets in Banach spaces, SIAM J. Optim., 19 (2008), 62-76. 1190.90231 10.1137/060659132
X. Y. Zheng and K. F. Ng, Calmness for L-subsmooth multifunctions in Banach spaces, SIAM. J. Optim., 19 (2009), 1648-1673. MR2486044 10.1137/080714129 X. Y. Zheng and K. F. Ng, Calmness for L-subsmooth multifunctions in Banach spaces, SIAM. J. Optim., 19 (2009), 1648-1673. MR2486044 10.1137/080714129
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. 0582.49001 F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. 0582.49001
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Springer, 1998. 1047.49500 F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Springer, 1998. 1047.49500
R. R. Phelps, Convex functions, Monotone operators and Differentiability, Lecture Notes in Math. 1364, Springer, New York, 1989. 0658.46035 R. R. Phelps, Convex functions, Monotone operators and Differentiability, Lecture Notes in Math. 1364, Springer, New York, 1989. 0658.46035
M. M. Day, Normed Linear Spaces, Springer-Verlag, Berlin, 1962. 0100.10802 M. M. Day, Normed Linear Spaces, Springer-Verlag, Berlin, 1962. 0100.10802
