Consider minimizingf onD which is diffeomorphic to a disk. Under a genericity assumption, the number of points onD satisfying the Kuhn-Tucker necessary conditions for minimum is odd. We give conditions which imply that a local minimum is global and a necessary and sufficient condition that a Kuhn-Tucker point is the solution. Convex transformable problems satisfy the latter condition.D may be of full dimension or be embedded on a manifold or it may be given by a system of concave inequalities.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Avriel, M.,Nonlinear Programming, Analysis and Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
Zang, I., andAvriel, M.,On Functions Whose Local Minima Are Global, Journal of Optimization Theory and Applications, Vol. 16, pp. 183–190, 1975.
Zang, I., Choo, E. V., andAvriel, M.,A Note on Functions Whose Local Minima Are Global, Journal of Optimization Theory and Applications, Vol. 18, pp. 555–559, 1976.
Smale, S.,Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold, a Review, Bulletin of the American Mathematical Society, Vol. 83, pp. 683–693, 1977.
Morse, M., andCairnes, S.,Critical Point Theory in Global Analysis and Differential Topology, Academic Press, New York, New York, 1969.
Ben-Tal, A.,On Generalized Means and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 21, pp. 1–13, 1977.
Smale, S.,Global Analysis and Economics, V: Pareto Theory with Constraints, Journal of Mathematical Economics, Vol. 1, pp. 213–221, 1974.
Milnor, J.,Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, Virginia, 1965.
Mas-Colell, A.,Homeomorphisms of Compact, Convex Sets and the Jacobian Matrix, SIAM Journal on Mathematical Analysis, Vol. 10, pp. 1105–1109, 1979.
Guillemin, V., andPollack, A.,Differential Topology, Prentice-Hall, Englewood Cliffs, New Jersey, 1974.
The work on which this paper is based was done while the author was visiting the Group for the Application of Mathematics and Statistics to Economics at the University of California, Berkeley, Spring 1981. The author would like to thank the Group, and Professor G. Debreu in particular, for the hospitality. He benefited from discussions with Professors S. Smale, A. Mas-Colell, K. Nishimura, and L. Chenault, among others. He also wishes to thank two referees of this journal for helpful comments.
Communicated by M. Avriel
About this article
Cite this article
Yun, K.K. Uniqueness conditions for Kuhn-Tucker points on a disk. J Optim Theory Appl 44, 701–721 (1984). https://doi.org/10.1007/BF00938403
- Kuhn-Tucker points
- local and global minima
- nonlinear programming
- Morse functions
- convex transformable programs