Abstract
Most abstract multiplier rules in the literature are based on the tangential approximation at a point to some set in a Banach space. The present paper is concerned with the study of a generalized tangent cone, which is a tangential approximation to that set at a common point of two sets. The new notion of tangent cone generalizes previous concepts of tangent cones. This generalized tangent cone is used to characterize the optimality conditions for a simultaneous maximization and minimization problem. The paper is of theoretical character; practical applications are not found so far.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abadie, J.,Problémes d'Optimisation, Institut Blaise Pascal, Paris, France, 1965.
Varaiya, P. P.,Nonlinear Programming in Banach Space, SIAM Journal on Applied Mathematics, Vol. 15, pp. 284–294, 1967.
Guignard, M.,Generalized Kuhn-Tucker Conditions for Mathematical Programming Problems in a Banach Space, SIAM Journal on Control and Optimization, Vol. 7, pp. 232–241, 1969.
Borwein, J.,Weak Tangent Cones and Optimization in a Banach Space, SIAM Journal on Control and Optimization, Vol. 16, pp. 512–522, 1978.
Salinetti, G., andWets, R. J. B.,On the Convergence of Closed-Valued Measurable Multifunctions, Transactions of the American Mathematical Society, Vol. 266, pp. 275–289, 1981.
Dancs, S.,Tangent and Generalized Tangent Cones in Finite-Dimensional Spaces: A Unified Approach (to appear).
Rockafellar, R. T.,Clarke's Tangent Cones and Boundaries of Closed Sets in R n, Nonlinear Analysis: Theory, Methods and Applications, Vol. 3, pp. 145–154, 1979.
Hestenes, M. R.,Optimization Theory: The Finite-Dimensional Case, John Wiley and Sons, New York, New York, 1975.
Abadie, J.,On the Kuhn-Tucker Theorem in Nonlinear Programming, Nonlinear Programming, Edited by J. Abadie, North-Holland, Amsterdam, Holland, 1967.
Bazaraa, M. S., Goode, J. J., andNashed, M. Z.,On the Cones of Tangents with Applications to Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 13, pp. 389–426, 1974.
Flett, T. M.,On Differentiation in Normed Vector Spaces, Journal of the London Mathematical Society, Vol. 42, pp. 523–533, 1967.
Bouligand, G.,Introduction a la Geometrie Infinitesimale Directe, Gauthier-Villars, Paris, France, 1932.
Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1984.
Penot, J. P.,Open Mappings Theorem and Linearization Stability, Numerical Functional Analysis and Optimization, Vol. 1, pp. 21–35, 1985.
Author information
Authors and Affiliations
Additional information
Communicated by P. Varaiya
Rights and permissions
About this article
Cite this article
Dancs, S. Generalized tangent cone and an optimization problem in a normed space. J Optim Theory Appl 67, 43–55 (1990). https://doi.org/10.1007/BF00939734
Issue Date:
DOI: https://doi.org/10.1007/BF00939734