Submonotone subdifferentials of Lipschitz functions

Spingarn, Jonathan E.

doi:10.1090/S0002-9947-1981-0597868-8

Submonotone subdifferentials of Lipschitz functions
HTML articles powered by AMS MathViewer

by Jonathan E. Spingarn PDF

Trans. Amer. Math. Soc. 264 (1981), 77-89 Request permission

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

Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. MR 367131, DOI 10.1090/S0002-9947-1975-0367131-6
Frank H. Clarke, Generalized gradients of Lipschitz functionals, Adv. in Math. 40 (1981), no. 1, 52–67. MR 616160, DOI 10.1016/0001-8708(81)90032-3
Gérard Lebourg, Valeur moyenne pour gradient généralisé, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 19, Ai, A795–A797 (French, with English summary). MR 388097
Robert Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim. 15 (1977), no. 6, 959–972. MR 461556, DOI 10.1137/0315061
George J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346. MR 169064
B. N. Pshenichnyi, Necessary conditions for an extremum, Pure and Applied Mathematics, vol. 4, Marcel Dekker, Inc., New York, 1971. Translated from the Russian by Karol Makowski; Translation edited by Lucien W. Neustadt. MR 0276845
R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876
R. T. Rockafellar, The multiplier method of Hestenes and Powell applied to convex programming, J. Optim. Theory Appl. 12 (1973), 555–562. MR 334953, DOI 10.1007/BF00934777
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1 (1976), no. 2, 97–116. MR 418919, DOI 10.1287/moor.1.2.97

The theory of subgradients and its applications to problems of optimization