Abstract
This paper first shows that for any p ∈ (1, 2) there exists a continuously differentiable function f on lp (and Lp) such that the proximal subdifferential of f is empty everywhere, and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces lp and LP with p ∈ (1, 2). On the other hand, this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2, which conclude all Hilbert spaces and all the classical spaces lp and Lp with p ∈ (2, +∞). In particular, in such a smooth space, we provide the proximal subdifferential rules for sum functions, product functions, composite functions and supremum functions, which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces. Some of our main results are new even in the Hilbert space case. As applications, we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential.
Similar content being viewed by others
References
Borwein, J. M.: Future challenges for variational analysis, In: Variational Analysis and Generalized Differentiation in Optimization and Control, Springer Optim. Appl., Vol. 47 (Burachik and Yao eds.), Springer, New York, 2010, 95–107
Borwein, J. M., Ioffe, A.: Proximal analysis in smooth spaces. Set-valued Anal., 4, 1–24 (1996)
Borwein, J. M., Zhu, Q. J.: A survey of subdifferential calculus with applications. Nonlinear Anal. TMA, 38, 687–773 (1999)
Clarke, F. H., Ledyaev, Y. S., Wolenski, P. R.: Proximal analysis and minimization principles. J. Math. Anal. Appl., 196, 722–735 (1995)
Clarke, F. H., Ledyaev, Y. S., Stern, R., et al.: Nonsmooth Analysis and Control Theory, Springer, New York, 1998
Correa, R., Hantoute A., Correa, M. A.: Weaker conditions for subdifferential calculus of convex functions. J. Funct. Anal., 271, 1177–1212 (2016)
Correa, R., Hantoute, A., Lopez, M. A.: Valadier-like formulas for the supremum function II: the compactly indexed case. J. Convex Anal., 26, 299–324 (2019)
Goberna, A., Lopez, M. A. eds.: Semi-infinite Programming–Recent Advances, Kluwer, Boston, 2001
Hantoute, A., Lopez, M. A., Zalinescu, C.: Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions, SIAM J. Optim., 19, 863–882 (2008)
Hettich, R., Kortanek, K. O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev., 35, 380–429 (1993)
Hiriart-Urruty, J.-B.: Convex analysis and optimization in the past 50 years: some snapshots. In: Constructive Nonsmooth Analysis and Related Topics. Springer Optimization and Its Applications, Vol. 87, Springer, New York, 2014, 245–253
Ioffe, A. D., Levin, V. L.: Subdifferentials of convex functions (In Russian). Tr. Moskov Mat. Obshch, 26, 3–73 (1972)
Ioffe, A. D., Tikhomirov, V. H.: Theory of Extremal Problems, Studies in Mathematics and Its Applications, Vol. 6., North-Holland, Amsterdam, 1979
Lindenstrauss, J., Tzafriri, D. L.: Classical Banach Spaces II, Springer, New York, 1979
Lopez, M. A., Still, G.: Semi-infinite programming. Euro. J. Oper. Res., 180, 491–518 (2007)
Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, Springer, Berlin, 2006
Mordukhovich, B. S., Nghia, T. T. A.: Subdifferentials of nonconvex superum functions and their applications to semi-infinite and infinite programs with Lipschitz data, SIAM J. Optim., 23, 403–431 (2013)
Preiss, D.: Differentiability of Lipschitz functions on Banach spaces. J. Funct. Anal., 91, 312–345 (1990)
Rockafellar, R. T., Wets, R. J. B.: Variational Analysis, Springer, Heidelberg, 1998
Rolewicz, S.: Paraconvex analysis. Control Cybernet., 34, 951–965 (2005)
Shapiro, A.: Semi-infinite programming, duality, discretization and optimality conditions, Optimization, 58, 133–161 (2009)
Schirotzek, W.: Nonsmooth Analysis, Springer, Berlin, 2007
Valadier, M.: Sous-differentiels d’une borne superieure et d’une somme continue de fonctions convexes. C. R. Acad. Sci. Paris Ser. A-B, 268, A39–A42 (1969)
Xu, Z. B., Roach, G. F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl., 157, 189–210 (1991)
Zalinescu, C.: Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002
Zheng, X. Y., Ng, K. F.: Proximal normal cone analysis on smooth Banach spaces and applications. SIAM J. Optim., 24, 363–384 (2014)
Zhu, Q. J.: Clarke–Ledyaev mean value inequality in smooth Banach spaces. Nonlinear Anal., TMA, 32, 315–324 (1998)
Zhu, Q. J.: The equivalence of several basic theorems for subdifferentials. Set-Valued Anal., 6, 171–185 (1998)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
Supported by the National Natural Science Foundation of P. R. China (Grant No. 12171419)
Rights and permissions
About this article
Cite this article
Zheng, X.Y. Variational Analysis Based on Proximal Subdifferential on Smooth Banach Spaces. Acta. Math. Sin.-English Ser. 40, 595–618 (2024). https://doi.org/10.1007/s10114-023-2439-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-2439-5