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Transactions of the American Mathematical Society

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Submonotone subdifferentials of Lipschitz functions


Author: Jonathan E. Spingarn
Journal: Trans. Amer. Math. Soc. 264 (1981), 77-89
MSC: Primary 26B25; Secondary 47H05, 49A51, 58C20, 90C25
DOI: https://doi.org/10.1090/S0002-9947-1981-0597868-8
MathSciNet review: 597868
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Abstract | References | Similar Articles | Additional Information

Abstract: The class of "lowwer-$ {C^1}$" functions, that is functions which arise by taking the maximum of a compactly indexed family of $ {C^1}$ functions, is characterized in terms of properties of the generalized subdifferential. A locally Lipschitz function is shown to be lower-$ {C^1}$ if and only if its subdifferential is "strictly submonotone". Other properties of functions with "submonotone" subdifferentials are investigated.


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

  • [1] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247-262. MR 0367131 (51:3373)
  • [2] -, Generalized gradients of Lipschitz functionals, Adv. in Math. (to appear). MR 616160 (83a:26031)
  • [3] G. Lebourg, Valeur moyenne pour gradient généralisé, C. R. Acad. Sci. Paris Ser. A 281 (1975), 795-797. MR 0388097 (52:8934)
  • [4] R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optimization 15 (1977), 6. MR 0461556 (57:1541)
  • [5] G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341-346. MR 0169064 (29:6319)
  • [6] B. N. Pshenichnyi, Necessary conditions for an extremum, Marcel Dekker, New York, 1971. MR 0276845 (43:2585)
  • [7] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1972. MR 1451876 (97m:49001)
  • [8] -, The multiplier method of Hestenes and Powell applied to convex programming, J. Optimization Theory Appl. 12 (1973), 6. MR 0334953 (48:13271)
  • [9] -, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1 (1976). MR 0418919 (54:6954)
  • [10] -, The theory of subgradients and its applications to problems of optimization, Lecture Notes, Univ. of Montreal, Feb.-March, 1978.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0597868-8
Keywords: Submonotone mapping, generalized gradient, lower-$ {C^1}$ function, nondifferentiable optimization
Article copyright: © Copyright 1981 American Mathematical Society

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