Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Mean value property and subdifferential criteria for lower semicontinuous functions


Authors: Didier Aussel, Jean-Noël Corvellec and Marc Lassonde
Journal: Trans. Amer. Math. Soc. 347 (1995), 4147-4161
MSC: Primary 49J52; Secondary 46N10, 47H99, 58C20
DOI: https://doi.org/10.1090/S0002-9947-1995-1307998-0
MathSciNet review: 1307998
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the "smooth" variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, conemonotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.


References [Enhancements On Off] (What's this?)

  • [1] E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31-47. MR 0231199 (37:6754)
  • [2] D. Aussel, J.-N. Corvellec, and M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, J. Convex Analysis 1 (1994), 1-7. MR 1363111 (97e:49012)
  • [3] J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527. MR 902782 (88k:49013)
  • [4] J. M. Borwein and H. Strojwas, Proximal analysis and boundaries of closed sets in Banach space, Part II: Applications, Canad. J. Math. 39 (1987), 428-472. MR 899844 (88f:46034)
  • [5] L. Caklovic, S. Li, and M. Willem, A note on Palais-Smale condition and coercivity, Differential and Integral Equations 3 (1990), 799-800. MR 1044221 (90m:58027)
  • [6] F. H. Clarke, Optimization and nonsmooth analysis, Wiley-Interscience, New York, 1983. MR 709590 (85m:49002)
  • [7] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Introduction to nonsmooth analysis, book in preparation.
  • [8] F. H. Clarke, R. J. Stern, and P. R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993), 1167-1183. MR 1247540 (94j:49018)
  • [9] R. Correa, A. Jofré, and L. Thibault, Characterization of lower semicontinuous convex functions, Proc. Amer. Math. Soc. 116 (1992), 67-72. MR 1126193 (92k:49027)
  • [10] -, Subdifferential monotonicity as characterization of convex functions, Numer. Funct. Anal. Optim. 15 (1994), 531-535. MR 1281560 (95d:49029)
  • [11] D. G. Costa and E. A. de B. e Silva, The Palais-Smale condition versus coercivity, Nonlinear Anal. 16 (1991), 371-381. MR 1093847 (92i:58033)
  • [12] M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487-502. MR 732102 (86a:35031)
  • [13] M. Degiovanni, A. Marino, and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal. 9 (1985), 1401-1443. MR 820649 (87h:35147)
  • [14] R. Deville, G. Godefroy, and V. Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), 197-212. MR 1200641 (94b:49010)
  • [15] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. MR 0346619 (49:11344)
  • [16] D. Goeleven, A note on Palais-Smale condition in the sense of Szulkin, Differential and Integral Equations 6 (1993), 1041-1043. MR 1230479 (94h:58039)
  • [17] A. D. Ioffe, Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389-416. MR 719677 (84m:49029)
  • [18] -, Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal. 8 (1984), 517-539. MR 741606 (85k:46049)
  • [19] -, Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175-192. MR 1063554 (91i:46045)
  • [20] D. T. Luc, Characterizations of quasiconvex functions, Bull. Austral. Math. Soc. 48 (1993), 393-405. MR 1248043 (94h:90033)
  • [21] -, On the maximal monotonicity of subdifferentials, Acta Math. Vietnam. 18 (1993), 99-106. MR 1248886 (95d:49028)
  • [22] L. McLinden, An application of Ekeland's theorem to minimax problems, Nonlinear Anal. 6 (1982), 189-196. MR 651700 (83e:49044)
  • [23] J.-P. Penot, Sous-différentiels de fonctions numériques non convexes, C.R. Acad. Sci. Paris 278 (1974), 1553-1555. MR 0352978 (50:5464)
  • [24] -, On the mean value theorem, Optimization 19 (1988), 147-156. MR 948386 (89i:26005)
  • [25] R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Math., vol. 1364, 2nd ed., Springer-Verlag, Berlin, 1993. MR 1238715 (94f:46055)
  • [26] R. A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Anal. 14 (1990), 305-317. MR 1040008 (91b:90155)
  • [27] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216. MR 0262827 (41:7432)
  • [28] -, Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331-355. MR 548983 (80j:46070)
  • [29] -, Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257-280. MR 571922 (81f:49006)
  • [30] -, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424-436. MR 629642 (83m:90088)
  • [31] S. Simons, The least slope of a convex function and the maximal monotonicity of its subdifferential, J. Optim. Theory Appl. 71 (1991), 127-136. MR 1131453 (92k:49031)
  • [32] L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), 33-58. MR 1312029 (95i:49032)
  • [33] J. S. Treiman, Generalized gradients, Lipschitz behavior and directional derivatives, Canad. J. Math. 37 (1985), 1074-1084. MR 828835 (87h:90263)
  • [34] D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. 12 (1988), 1413-1428. MR 972409 (89k:58034)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49J52, 46N10, 47H99, 58C20

Retrieve articles in all journals with MSC: 49J52, 46N10, 47H99, 58C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1307998-0
Keywords: Nonsmooth analysis, renorming, variational principle, subdifferential, mean value theorem, Lipschitz behavior, quasiconvexity, convexity
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society