A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
Authors:
J. M. Borwein and D. Preiss
Journal:
Trans. Amer. Math. Soc. 303 (1987), 517527
MSC:
Primary 49A27; Secondary 46B20, 46G05, 49A51, 49A52, 58C20, 90C25, 90C48
MathSciNet review:
902782
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Abstract 
References 
Similar Articles 
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Abstract: We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadraticlike function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.
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 [2]
, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), 952. MR 825383 (87m:58018)
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J. M. Borwein and J. R. Giles, The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987), 371381. MR 887515 (88m:49013)
 [4]
J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach space. Part 1: Theory, Canad. Math. J. 38 (1986), 431452. MR 833578 (87h:90258)
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, Proximal analysis and boundaries of closed sets in Banach space, Part 2: Applications, Canad. Math. J (to appear).
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, Subderivatives and nonsmooth analysis (to appear).
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F. H. Clarke, Optimization and nonsmooth analysis, Canadian Math. Soc. Series, Wiley, 1983. MR 709590 (85m:49002)
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M. Coban and P. S. Kenderov, Dense Gateaux differentiability of the supnorm in and the topological properties of , C.R. Acad. Bulgare Sci. 38 (1985), 16031604. MR 837262 (87h:46070)
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M. M. Day, Normed linear spaces, 3rd ed., SpringerVerlag, 1973. MR 0344849 (49:9588)
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J. Diestel, Geometry of Banach spacesSelected topics, Lecture Notes in Math., vol. 485, SpringerVerlag, 1975. MR 0461094 (57:1079)
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I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324353. MR 0346619 (49:11344)
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, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 443474. MR 526967 (80h:49007)
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I. Ekeland and G. Lebourg, Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193216. MR 0431253 (55:4254)
 [14]
M. Fabian, Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. 51 (1985), 113126. MR 788852 (86j:46017)
 [15]
M. Fabian, J. H. M. Whitfield and V. Zizler, Norms with locally Lipschitz derivatives, Israel J. Math. 44 (1983), 262276. MR 693663 (84i:46028)
 [16]
M. Fabian and N. V. Zhivkov, A characterization of Asplund spaces with the help of local supports of Ekeland and Lebourg, C.R. Acad. Bulgare Sci. 38 (1985), 671674. MR 805439 (87e:46021)
 [17]
J. R. Giles, Convex analysis with application in differentiation of convex functions, Pitman Research Notes in Math., 58, Pitman, 1982. MR 650456 (83g:46001)
 [18]
D. G. Larman and R. R. Phelps, Gâteaux differentiability of convex functions on Banach spaces, J. London Math. Soc. 20 (1979), 115127. MR 545208 (80m:46017)
 [19]
E. B. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120126. MR 0293394 (45:2471)
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J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces, SpringerVerlag, 1978. MR 540367 (81c:46001)
 [21]
G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326350. MR 0394135 (52:14940)
 [22]
D. Preiss, Frèchet derivatives of Lipschitz functions (to appear).
 [23]
R. T. Rockafellar, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424436. MR 629642 (83m:90088)
 [24]
F. Sullivan, Nearly smooth norms on Banach spaces, Rev. Roumaine Math. Pures Appl. 21 (1981), 10531057. MR 627473 (82i:46031)
 [25]
J. S. Treiman, Clarke's gradients and epsilonsubgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 6678. MR 819935 (87d:90188)
 [1]
 J. M. Borwein, Weak local supportability and applications to approximation, Pacific J. Math. 82 (1979), 323338. MR 551692 (81h:46016)
 [2]
 , Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), 952. MR 825383 (87m:58018)
 [3]
 J. M. Borwein and J. R. Giles, The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987), 371381. MR 887515 (88m:49013)
 [4]
 J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach space. Part 1: Theory, Canad. Math. J. 38 (1986), 431452. MR 833578 (87h:90258)
 [5]
 , Proximal analysis and boundaries of closed sets in Banach space, Part 2: Applications, Canad. Math. J (to appear).
 [6]
 , Subderivatives and nonsmooth analysis (to appear).
 [7]
 F. H. Clarke, Optimization and nonsmooth analysis, Canadian Math. Soc. Series, Wiley, 1983. MR 709590 (85m:49002)
 [8]
 M. Coban and P. S. Kenderov, Dense Gateaux differentiability of the supnorm in and the topological properties of , C.R. Acad. Bulgare Sci. 38 (1985), 16031604. MR 837262 (87h:46070)
 [9]
 M. M. Day, Normed linear spaces, 3rd ed., SpringerVerlag, 1973. MR 0344849 (49:9588)
 [10]
 J. Diestel, Geometry of Banach spacesSelected topics, Lecture Notes in Math., vol. 485, SpringerVerlag, 1975. MR 0461094 (57:1079)
 [11]
 I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324353. MR 0346619 (49:11344)
 [12]
 , Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 443474. MR 526967 (80h:49007)
 [13]
 I. Ekeland and G. Lebourg, Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193216. MR 0431253 (55:4254)
 [14]
 M. Fabian, Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. 51 (1985), 113126. MR 788852 (86j:46017)
 [15]
 M. Fabian, J. H. M. Whitfield and V. Zizler, Norms with locally Lipschitz derivatives, Israel J. Math. 44 (1983), 262276. MR 693663 (84i:46028)
 [16]
 M. Fabian and N. V. Zhivkov, A characterization of Asplund spaces with the help of local supports of Ekeland and Lebourg, C.R. Acad. Bulgare Sci. 38 (1985), 671674. MR 805439 (87e:46021)
 [17]
 J. R. Giles, Convex analysis with application in differentiation of convex functions, Pitman Research Notes in Math., 58, Pitman, 1982. MR 650456 (83g:46001)
 [18]
 D. G. Larman and R. R. Phelps, Gâteaux differentiability of convex functions on Banach spaces, J. London Math. Soc. 20 (1979), 115127. MR 545208 (80m:46017)
 [19]
 E. B. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120126. MR 0293394 (45:2471)
 [20]
 J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces, SpringerVerlag, 1978. MR 540367 (81c:46001)
 [21]
 G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326350. MR 0394135 (52:14940)
 [22]
 D. Preiss, Frèchet derivatives of Lipschitz functions (to appear).
 [23]
 R. T. Rockafellar, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424436. MR 629642 (83m:90088)
 [24]
 F. Sullivan, Nearly smooth norms on Banach spaces, Rev. Roumaine Math. Pures Appl. 21 (1981), 10531057. MR 627473 (82i:46031)
 [25]
 J. S. Treiman, Clarke's gradients and epsilonsubgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 6678. MR 819935 (87d:90188)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198709027827
PII:
S 00029947(1987)09027827
Keywords:
Weak Asplund spaces,
subderivatives,
renorms,
nonsmooth analysis,
Ekeland's principle,
proximal normals
Article copyright:
© Copyright 1987
American Mathematical Society
