Abstract
Certain useful basic results of the gradient (in the smooth case), the Clarkesubdifferential, the Michel–Penot subdifferential, which is also known asthe "small" subdifferential, and the directional derivative(in the nonsmooth case) are stated and discussed. One of the advantages ofthe Michel–Penot subdifferential is the fact that it is in general "smaller"than the Clarke subdifferential. In this paper it is shown that there existsubdifferentials which may be smaller than the Michel–Penot subdifferentialandwhich have certain useful calculus. It isfurther shown that in the case of quasidifferentiability, the Michel–Penotsubdifferential enjoys calculus whichhold for the Clarke subdifferential only in the regular case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Clarke, F.M. (1983), Optimization and Nonsmooth Analysis, Wiley Interscience, New York.
Crave, B.D., Ralph, D. and Glover, B.M. (1995), Small convex-valued subdifferentials in mathematical programming, Optimization, 33, pp.1–21.
Danskin, J. (1967) Theory of Max-Min and Its Application to Weapons Allocation Problems, Springer-Verlag, New York.
Demyanov, V.F. and Malozemov, V.N. (1994), Introduction to Minimax, Wiley Interscience, New York.
Demyanov, V.F. (1980) On relationship between the Clarke subdifferential and the quasidifferential, Vestnik of Leningrad University 13, pp. 18–24.
Demyanov, V.F. and Vasiliev, L.V. (1986), Nondifferentiable Optimization, Springer-Optimization Software, New York.
Demyanov, V.F. and Rubinov, A.M. (1995), Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt a/M
Demyanov, V.F. (1994), Convexification and concavification of a positively homogeneous function by the same family of linear functions, Università di Pisa, Report 3,208,802.
Ioffe, A.D. (1993), A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and non-functional constraints, Math. Progr., 58, pp. 137–145.
Jeyakumar, V. (1987), On optimality conditions in nonsmooth inequality constrained minimization, Numer. Funct. Anal. and Optim., 9(5 & 6), pp. 536–546.
Kuntz, L. and Scholtes, S. (1993), Constraint qualifications in quasidifferentiable optimization, Mathematical Programming, 60, pp. 339–347.
Michel. P. and Penot, J.-P. (1984), Calcus sous-differential pour les fonctions lipschitziennes et non-lipschitziennes, C.R. Acad. Sc. Paris, Ser.I 298, pp. 269–272.
Pschenichniy, B.N. (1980), Convex Analysis and Extremal Problems, Nauka, Moscow.
Rubinov, A.M. and Akhundov, I.S. (1992), Difference of compact sets in the sense of Demyanov and its application to nonsmooth analysis, Optimization, 23, pp. 179–188.
Treiman, J.S. (1988), Shrinking generalized gradients, Nonlinear Analysis TM & A., 12, pp. 1429–1449.
Rockafellar, R.T. (1970), Convex Analysis, Princeton University Press, Princeton N.J.
Studniarski, M. and Jeyakumar, V. (1993), A generalized mean-value theorem and optimality conditions in composite nonsmooth minimization, Nonlinear Analysis TM & A, 24(6), 883–894.
Xia, Z.-Q. (1987), On mean-value theorems in quasidifferential Calculus, J. Math. Res. and Exposition, D.I.T., Dalian (China) 4pp. 681–684.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Demyanov, V.F., Jeyakumar, V. Hunting for a Smaller Convex Subdifferential. Journal of Global Optimization 10, 305–326 (1997). https://doi.org/10.1023/A:1008246130864
Issue Date:
DOI: https://doi.org/10.1023/A:1008246130864