representation of subdifferentials of directionally Lipschitz functions
Authors:
Alejandro Jofré and Lionel Thibault
Journal:
Proc. Amer. Math. Soc. 110 (1990), 117123
MSC:
Primary 90C48; Secondary 46G05, 49A52, 58C20, 90C25
MathSciNet review:
1015680
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Abstract 
References 
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Additional Information
Abstract: Subdifferentials of convex functions and some regular functions are expressed in terms of limiting gradients at points in a given dense subset of .
 [1]
J. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed set in space II: applications (to appear).
 [2]
S. ChuChung, Remarques sur le gradient généralisé, J. Math. Pures Appl. 61 (1982), 301310. MR 690398 (84d:58008)
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F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247262. MR 0367131 (51:3373)
 [4]
, Nonsmooth analysis and optimization, WileyInterscience, New York, 1983.
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R. Correa and A. Jofre, Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory Appl. (to appear). MR 993912 (90h:49009)
 [6]
R. Correa and L. Thibault, Subdifferential analysis of bivariate separately regular functions, J. Math. Anal. Appl. (to appear). MR 1052052 (91b:49018)
 [7]
J. R. Giles, Convex analysis with application in the differentiation of convex functions, Pitman, 1982. MR 650456 (83g:46001)
 [8]
J. B. HiriartUrruty, A note on the mean value theorem for convex functions, Boll. Un. Mat. Ital. B 17 (1980), 765775. MR 580556 (82b:26007)
 [9]
R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control. Optim. 15 (1977), 959972. MR 0461556 (57:1541)
 [10]
R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, NJ, 1970.
 [11]
, Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257280. MR 571922 (81f:49006)
 [12]
, Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331355. MR 548983 (80j:46070)
 [13]
, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424436. MR 629642 (83m:90088)
 [14]
L. Thibault, On generalized differentials and subdifferentials of Lipschitz vectorvalued functions, Nonlinear. Anal. 6 (1982), 10371053. MR 678055 (85e:58020)
 [15]
J. S. Treiman, Clarke's gradients and epsilonsubgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 6578. MR 819935 (87d:90188)
 [1]
 J. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed set in space II: applications (to appear).
 [2]
 S. ChuChung, Remarques sur le gradient généralisé, J. Math. Pures Appl. 61 (1982), 301310. MR 690398 (84d:58008)
 [3]
 F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247262. MR 0367131 (51:3373)
 [4]
 , Nonsmooth analysis and optimization, WileyInterscience, New York, 1983.
 [5]
 R. Correa and A. Jofre, Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory Appl. (to appear). MR 993912 (90h:49009)
 [6]
 R. Correa and L. Thibault, Subdifferential analysis of bivariate separately regular functions, J. Math. Anal. Appl. (to appear). MR 1052052 (91b:49018)
 [7]
 J. R. Giles, Convex analysis with application in the differentiation of convex functions, Pitman, 1982. MR 650456 (83g:46001)
 [8]
 J. B. HiriartUrruty, A note on the mean value theorem for convex functions, Boll. Un. Mat. Ital. B 17 (1980), 765775. MR 580556 (82b:26007)
 [9]
 R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control. Optim. 15 (1977), 959972. MR 0461556 (57:1541)
 [10]
 R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, NJ, 1970.
 [11]
 , Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257280. MR 571922 (81f:49006)
 [12]
 , Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331355. MR 548983 (80j:46070)
 [13]
 , Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424436. MR 629642 (83m:90088)
 [14]
 L. Thibault, On generalized differentials and subdifferentials of Lipschitz vectorvalued functions, Nonlinear. Anal. 6 (1982), 10371053. MR 678055 (85e:58020)
 [15]
 J. S. Treiman, Clarke's gradients and epsilonsubgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 6578. MR 819935 (87d:90188)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010156803
PII:
S 00029939(1990)10156803
Keywords:
Directionally Lipschitz functions,
pseudoregular functions,
weak Asplund spaces
Article copyright:
© Copyright 1990
American Mathematical Society
