Abstract
Two differences of convex compact sets in ℜm× n are proposed. In the light of these differences, representations of the Clarke generalized Jacobian and the B-differential via the quasidifferential are developed for a certain class of functions. These representations can be used to calculate the Clarke generalized Jacobian and the B-differential via the quasidifferential.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Demyanov, V. F., On a Relation between the Clarke Subdifferential and Quasidifferential, Vestnik Leningrad University, Vol. 13, pp. 183–189, 1981.
Demyanov, V. F., and Sutti, C., Representation of the Clarke Subdifferential for a Regular Quasidifferentiable Function, Journal of Optimization Theory and Applications, Vol. 87, pp. 553–561, 1995.
Gao, Y., Demyanov Difference of Two Sets and Optimality Conditions of Lagrange Multipliers Type for Constrained Quasidifferentiable Optimization, Journal of Optimization Theory and Applications, Vol. 104, pp. 377–394, 2000.
Qi, L., Quasidifferentials and Maximal Normal Operators, Mathematical Programming, Vol. 49, pp. 263–271, 1991.
Rubinov, A. M., Differences of Convex Compact Sets and Their Applications in Nonsmooth Analysis, Nonsmooth Optimization Methods and Applications, Edited by F. Giannessi, Gordon and Breach Science Publishers, Singapore, Republic of Singapore, pp. 366–378, 1992.
Rubinov, A. M., and Akhundov, I. S., Difference of Convex Compact Sets in the Sense of Demyanov and Its Applications in Nonsmooth Analysis, Optimization, Vol. 23, pp. 179–188, 1992.
Rubinov, A. M., and Vladimirov, A. A., Differences of Convex Compacta and Metric Spaces of Convex Compacta with Applications: A Survey, Quasidifferentiability and Related Topics, Edited by V. F. Demyanov and A. M. Rubinov, Kluwer Academic Publishers, Dordrecht, Holland, pp. 263–296, 2000.
Qi, L., Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operations Research, Vol. 18, pp. 227–253, 1993.
Demyanov, V. F., and Rubinov, A. M., Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main, Germany, 1995.
Demyanov, V. F., Stavroulakis, G. E., Polyakova, L. N., and Panagiotopoulous, P. D., Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering, and Economics, Kluwer Academic Publishers, Dordercht, Holland, 1996.
Pallaschke, D., Recht, P., and Urbanski, R., On Locally Lipschitz Quasidifferential Functions in Banach Spaces, Optimization, Vol. 17, pp. 287–295, 1986.
Clarke, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, NY, 1983.
Hiriart-urruty, J. B., and Lemarechal, C., Convex Analysis and Minimization Algorithms, Springer Verlag, Berlin, Germany, 1993.
Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.
Chaney, R. W., Piecewise C k Functions in Nonsmooth Analysis, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 15, pp. 649–660, 1990.
Bartels, S. G., Kuntz, L., and Scholtes, S., Continuous Selections of Linear Functions and Nonsmooth Critical Point Theory, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 24, pp. 385–407, 1995.
Pang, J. S., and Ralph, D., Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps, Mathematics of Operations Research, Vol. 21, pp. 401–426, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gao, Y. Representation of the Clarke Generalized Jacobian via the Quasidifferential. Journal of Optimization Theory and Applications 123, 519–532 (2004). https://doi.org/10.1007/s10957-004-5721-4
Issue Date:
DOI: https://doi.org/10.1007/s10957-004-5721-4