Abstract
In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R n. In this context, we establish a subdifferential characterization and show that qualified convexly C 1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C 1 in the context of certain uniformly convex spaces.
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References
Attouch, H.: Variational Convergence for Functions and Operators, Pitman, London, 1984.
Bernard, F. and Thibault, L.: Prox-regular functions in Hilbert spaces, submitted.
Bernard, F. and Thibault, L.: Uniform prox-regularity of functions and epigraphs in Hilbert spaces, submitted.
Borwein, J. M. and Giles, J. R.: The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987), 371-381.
Bougeard, M.: Contributions à la théorie de Morse en dimension finie, Thesis, Université Paris IX Dauphine, 1978.
Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
Clarke, F. H., Ledyaev, Yu. S., Stern, J. R. and Wolenski, R. P.: Nonsmooth Analysis and Control Theory, Springer, New York, 1997.
Combari, C., Elhilali Alaoui, A., Levy, A., Poliquin, R. and Thibault, L.: Convex composite functions in Banach spaces and the primal lower nice property, Proc. Amer. Math. Soc. 126 (1998), 3701-3708.
Combari, C., Poliquin, R. A. and Thibault, L.: Convergence of subdifferentials of convexly composite functions, Canad. J. Math. 51 (1999), 250-265.
Combari, C. and Thibault, L.: Epi-convergence of convexly composite functions in Banach spaces, SIAM J. Optim., to appear.
Correa, R., Jofré, A. and Thibault, L.: Characterization of lower semicontinuous convex functions, Proc. Amer. Math. Soc. 116 (1992), 61-72.
Correa, R., Jofré, A. and Thibault, L.: Subdifferential monotonicity as characterization of convex functions, Numer. Funct. Anal. Optim. 15 (1994), 531-536.
Degiovanni, M., Marino, A. and Tosques, M.: Evolution equations with lack of convexity, Nonlinear Anal. Th. Meth. Appl. 9 (1985), 1401-1443.
Deville, R., Godefroy, G. and Zizler, V.: Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math., Longman, New York, 1993.
Ekeland, I.: Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474.
Ekeland, I. and Lasry, J.-M.: On the number of periodic trajectories for a Hamiltonian flow, Ann. of Math. 112 (1980), 293-319.
Ioffe, A. D.: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175-192.
Ivanov, M. and Zlateva, N.: On primal lower nice property, C.R. Acad. Bulgare Sci. 54(11) (2001), 5-10.
Janin, R.: Sur la dualité et la sensibilité dans les problèmes de programmes mathématiques, Thesis, Université Paris VI, 1974.
Levy, A. B., Poliquin, R. A. and Thibault, L.: Partial extension of Attouch's theorem with applications to proto-derivatives of subgradient mappings, Trans. Amer. Math. Soc. 347 (1995), 1269-1294.
Michel, Ph. and Penot, J.-P.: A generalized derivative for calm and stable functions, Differential Integral Equations 5 (1992), 433-454.
Mordukhovich, B. S.: Metric approximation and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526-530.
Mordukhovich, B. S. and Shao, Y.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 124 (1996), 1235-1280.
Moreau, J.-J.: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273-299.
Penot, J.-P. and Bougeard, M.: Approximation and decomposition properties of some classes of locally d.c. functions, Math. Programming 41 (1988), 195-227.
Poliquin, R. A.: Integration of subdifferentials of nonconvex functions, Nonlinear Anal. Th. Meth. Appl. 17 (1991), 385-398.
Poliquin, R. A. and Rockafellar, R. T.: Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996), 1805-1838.
Poliquin, R. A., Rockafellar, R. T. and Thibault, L.: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000), 5231-5249.
Rockafellar, R. T.: First and second order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), 75-107.
Rockafellar, R. T. and Wets, R. J.: Variational Analysis, Springer, 1998.
Thibault, L.: A note on the Zagrodny mean value theorem, Optimization 35 (1995), 127-130.
Thibault, L. and Zagrodny, D.: Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), 33-58.
Xu, Z.-B. and Roach, G. F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991), 189-210.
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Bernard, F., Thibault, L. Prox-Regularity of Functions and Sets in Banach Spaces. Set-Valued Analysis 12, 25–47 (2004). https://doi.org/10.1023/B:SVAN.0000023403.87092.a2
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DOI: https://doi.org/10.1023/B:SVAN.0000023403.87092.a2