Abstract
Quasidifferentials are studied with the theory of maximal normal operators. The quasidifferential of a normally quasidifferentiable function is a pair of upper and lower semicontinuous operators, which are special maximal normal operators. The function sum of the upper and lower semicontinuous operators is the Clarke subdifferential of this function. Basic calculus and minimal forms of quasidifferentials are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley & Sons, New York, 1983).
V.F. Dem'yanov and L.V. Vasil'ev,Nondifferentiable Optimization (Optimization Software, New York, 1985).
V.F. Dem'yanov and L.C.W. Dixon,Quasidifferential Calculus (North-Holland, Amsterdam, 1986).
R. Mifflin, “Semismooth and semiconvex functions in constrained optimization”,SIAM Journal on Control and Optimizaion 15 (1977) 959–972.
L. Qi, “The maximal normal operator space and integration of subdifferentials of nonconvex functions”,Nonlinear Analysis 13 (1989) 1003–1011.
L. Qi, “Maximal normal operators and the second-order derivatives of nonsmooth functions”, Industrial Engineering Technical Report 88-3, University of Wisconsin-Madison (Madison, 1988).
L. Qi, “Semismoothness and decomposition of maximal normal operators”,Journal of Mathematical Analysis and Applications 146 (1990) 271–279.
R.T. Rockafellar,The Theory of Subgradients and Its Applications to Problems of Optimization: Convex and Nonconvex Functions (Helderman-Verlag, Berlin, 1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Qi, L. Quasidifferentials and maximal normal operators. Mathematical Programming 49, 263–271 (1990). https://doi.org/10.1007/BF01588791
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01588791