Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Generalized tangent cone and an optimization problem in a normed space


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 to check access.


  1. 1.

    Abadie, J.,Problémes d'Optimisation, Institut Blaise Pascal, Paris, France, 1965.

  2. 2.

    Varaiya, P. P.,Nonlinear Programming in Banach Space, SIAM Journal on Applied Mathematics, Vol. 15, pp. 284–294, 1967.

  3. 3.

    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.

  4. 4.

    Borwein, J.,Weak Tangent Cones and Optimization in a Banach Space, SIAM Journal on Control and Optimization, Vol. 16, pp. 512–522, 1978.

  5. 5.

    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.

  6. 6.

    Dancs, S.,Tangent and Generalized Tangent Cones in Finite-Dimensional Spaces: A Unified Approach (to appear).

  7. 7.

    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.

  8. 8.

    Hestenes, M. R.,Optimization Theory: The Finite-Dimensional Case, John Wiley and Sons, New York, New York, 1975.

  9. 9.

    Abadie, J.,On the Kuhn-Tucker Theorem in Nonlinear Programming, Nonlinear Programming, Edited by J. Abadie, North-Holland, Amsterdam, Holland, 1967.

  10. 10.

    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.

  11. 11.

    Flett, T. M.,On Differentiation in Normed Vector Spaces, Journal of the London Mathematical Society, Vol. 42, pp. 523–533, 1967.

  12. 12.

    Bouligand, G.,Introduction a la Geometrie Infinitesimale Directe, Gauthier-Villars, Paris, France, 1932.

  13. 13.

    Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1984.

  14. 14.

    Penot, J. P.,Open Mappings Theorem and Linearization Stability, Numerical Functional Analysis and Optimization, Vol. 1, pp. 21–35, 1985.

Download references

Author information

Additional information

Communicated by P. Varaiya

Rights and permissions

Reprints 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).

Download citation

Key Words

  • Optimization theory
  • tangent cones
  • contingent cones
  • necessary conditions
  • abstract multiplier rules