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A limiting example for the Local ``fuzzy'' sum rule in nonsmooth analysis
Author(s):
Jon
Vanderwerff;
Qiji
J.
Zhu
Journal:
Proc. Amer. Math. Soc.
126
(1998),
2691-2697.
MSC (1991):
Primary 26B05, 49J50, 49J52
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Abstract:
We show that assuming all the summand functions to be lower semicontinuous is not sufficient to ensure a (strong) fuzzy sum rule for subdifferentials in any infinite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-Ledyaev type mean value inequalities in the infinite dimensional setting.
References:
- [BI]
- J. M. Borwein and A. D. Ioffe, Proximal analysis in smooth spaces, CECM Research Report 93-04 (1993), Set-valued Analysis 4 (1996), 1-24.MR 96m:49028
- [BTZ]
- J. M. Borwein, J. S. Treiman and Q. J. Zhu, Necessary conditions for constrained optimization problems with semicontinuous and continuous data, CECM Research Report 95-51 (1995), Trans. Amer. Math. Soc., to appear. CMP 97:08
- [BZ]
- J. M. Borwein and Q. J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, CECM Research Report 94-12 (1994), SIAM J. Control and Optimization, 34 (1996), 1568-1591. MR 97g:49037
- [C]
- F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional conference series in applied mathematics, SIAM, Philadelphia, 1989. MR 91j:49001
- [CL]
- F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc., 344 (1994), 307-324. MR 94k:49016
- [C2]
- F. H. Clarke, Lecture Notes in Nonsmooth Analysis, unpublished.
- [DH]
- R. Deville and E. M. E. Haddad, The subdifferential of the sum of two functions in Banach spaces, I. first order case, J. Convex Analysis 3 (1996), 295-308. CMP 97:12
- [DI]
- R. Deville and M. Ivanov, Smooth variational principles with constraints, Math. Nachrichen, to appear.
- [D]
- J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, Springer-Verlag, 1984. MR 85i:46020
- [I1]
- A. D. Ioffe, On subdifferentiability spaces, Ann. New York Acad. Sci. 410 (1983), 107-112. MR 86g:90124
- [I2]
- A. D. Ioffe, Calulus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Analysis: TMA, 8 (1984), 517-539. MR 85k:46049
- [I3]
- A. D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc., 41 (1990), 175-192. MR 91i:46045
- [IR]
- A. D. Ioffe and T. R. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems, Calc. Var., 4 (1996), 59-87. MR 96k:49031
- [P]
- R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, No. 1364, Springer Verlag, N.Y., Berlin, Tokyo, 1988, Second Edition 1993. MR 90g:46063; MR 94f:46055
- [Z1]
- Q. J. Zhu, Subderivatives and their applications, Proceedings of the International Conference on Dynamic Systems, Springfield, MO, June 1996.
- [Z2]
- Q. J. Zhu, Clarke-Ledyaev mean value inequality in smooth Banach spaces, CECM Research Report 96-78 (1996), Nonlinear Analysis: TMA, to appear.
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Additional Information:
Jon
Vanderwerff
Affiliation:
Department of Mathematics, Walla Walla College, College Place, Washington 99324
Email:
vandjo@wwc.edu
Qiji
J.
Zhu
Affiliation:
Department of Mathematics & Statistics, Western Michigan University, Kalamazoo, Michigan 49008
Email:
zhu@math-stat.wmich.edu
DOI:
10.1090/S0002-9939-98-04365-2
PII:
S 0002-9939(98)04365-2
Received by editor(s):
January 30, 1997
Additional Notes:
The first author's research was partially supported by a Walla Walla College Faculty Development Grant.
The second author's work was partially supported by a grant from the Faculty Research and Creative Activities Support Fund, Western Michigan University.
Research for this note was completed while the authors were visiting Simon Fraser University. The authors thank J.M. Borwein and the CECM for their hospitality.
Communicated by:
Dale Alspach
Copyright of article:
Copyright
1998,
American Mathematical Society
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