The Kuhn-Tucker conditions, also called the Karush-Kuhn-Tucker conditions, characterize candidate solutions of a constrained problem in optimization
theory. For the problem of minimizing 
subject to 
and 
, they assert the existence of multipliers 
and 
such that
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The first relation requires the gradient of the Lagrangian function to vanish. The next two express satisfaction of the original constraints
and nonnegativity of the inequality multipliers. The last means that each inequality
is active or has zero multiplier. Under standard regularity hypotheses, these conditions
are necessary for a local minimum. When 
and the 
are convex functions,
they are also sufficient for a global minimum.
