Kuhn-Tucker conditions

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Let be a complete finite-dimensional Riemannian manifold, let be a convex objective function and let be a totally convex subset described by a system of inequalities , , where are concave functions (cf. Concave function). The solution of the convex programming problem is completely characterized by a saddle-point theorem initially stated on [a2] by H.W. Kuhn and A.W. Tucker.

Suppose and are of class , and let . Introduce the Lagrange function , , . A point is the optimal solution of the convex programming problem if and only if there exists a such that the so-called Kuhn–Tucker conditions hold:

These equations provide a criterion for testing whether a solution which has been found by other methods is in fact an optimal solution.

The Kuhn–Tucker theorem has been generalized in various directions:

to necessary or sufficient conditions, or both types, for an extremum of a function subject to equality or inequality constraints [a1];

to changing the geometrical structure of the underlying manifold [a1], [a3].


[a1] K.J. Arrow, L. Hurwicz, H. Uzawa, "Studies in linear and nonlinear programming" , Stanford Univ. Press (1958)
[a2] H.W. Kuhn, A.W. Tucker, "Nonlinear programming" , Proc. Second Berkeley Symp. Math. Stat. Probab. , California Univ. Press (1951) pp. 481–492
[a3] C. Udrişte, "Convex functions and optimization methods on Riemannian manifolds" , Kluwer Acad. Publ. (1994)
How to Cite This Entry:
Kuhn-Tucker conditions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by C. UdriÅŸte (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article