Integrability of subdifferentials of directionally Lipschitz functions
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- by Lionel Thibault and Nadia Zlateva PDF
- Proc. Amer. Math. Soc. 133 (2005), 2939-2948 Request permission
Abstract:
Using a quantitative version of the subdifferential characterization of directionally Lipschitz functions, we study the integrability of subdifferentials of such functions over arbitrary Banach space.References
- Jonathan M. Borwein and Warren B. Moors, Essentially smooth Lipschitz functions, J. Funct. Anal. 149 (1997), no. 2, 305–351. MR 1472362, DOI 10.1006/jfan.1997.3101
- Jonathan M. Borwein, Warren B. Moors, and Xianfu Wang, Generalized subdifferentials: a Baire categorical approach, Trans. Amer. Math. Soc. 353 (2001), no. 10, 3875–3893. MR 1837212, DOI 10.1090/S0002-9947-01-02820-3
- Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
- R. Correa and A. Jofré, Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory Appl. 61 (1989), no. 1, 1–21. MR 993912, DOI 10.1007/BF00940840
- R. Correa and L. Thibault, Subdifferential analysis of bivariate separately regular functions, J. Math. Anal. Appl. 148 (1990), no. 1, 157–174. MR 1052052, DOI 10.1016/0022-247X(90)90035-E
- Milen Ivanov and Nadia Zlateva, Abstract subdifferential calculus and semi-convex functions, Serdica Math. J. 23 (1997), no. 1, 35–58. MR 1610289
- Alejandro Jofré and Lionel Thibault, $D$-representation of subdifferentials of directionally Lipschitz functions, Proc. Amer. Math. Soc. 110 (1990), no. 1, 117–123. MR 1015680, DOI 10.1090/S0002-9939-1990-1015680-3
- René A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Anal. 14 (1990), no. 4, 305–317. MR 1040008, DOI 10.1016/0362-546X(90)90167-F
- R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 262827, DOI 10.2140/pjm.1970.33.209
- R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canadian J. Math. 32 (1980), no. 2, 257–280. MR 571922, DOI 10.4153/CJM-1980-020-7
- Lionel Thibault and Dariusz Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), no. 1, 33–58. MR 1312029, DOI 10.1006/jmaa.1995.1003
- —, Enlarged inclusion of subdifferentials, Canad. Math. Bull., to appear.
- Lionel Thibault and Nadia Zlateva, Integrability of subdifferentials of certain bivariate functions, Nonlinear Anal. 54 (2003), no. 7, 1251–1269. MR 1995929, DOI 10.1016/S0362-546X(03)00137-8
- Jay S. Treiman, Generalized gradients, Lipschitz behavior and directional derivatives, Canad. J. Math. 37 (1985), no. 6, 1074–1084. MR 828835, DOI 10.4153/CJM-1985-058-1
- Dariusz Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. 12 (1988), no. 12, 1413–1428. MR 972409, DOI 10.1016/0362-546X(88)90088-0
Additional Information
- Lionel Thibault
- Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France
- Email: thibault@math.univ-montp2.fr
- Nadia Zlateva
- Affiliation: Section of Operations Research, Department of Mathematics, Sofia University, 5, James Bourchier Blvd., 1164 Sofia, Bulgaria — and — Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France
- Email: zlateva@fmi.uni-sofia.bg, zlateva@math.univ-montp2.fr
- Received by editor(s): May 10, 2002
- Published electronically: May 13, 2005
- Additional Notes: The second author’s research was supported by a Marie Curie fellowship of the European Community programme Improving Human Potential under contract No. HPMF-CT-2001-01345
- Communicated by: Jonathan M. Borwein
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 2939-2948
- MSC (2000): Primary 49J52; Secondary 28B20
- DOI: https://doi.org/10.1090/S0002-9939-05-07883-4
- MathSciNet review: 2159772