Fritz John conditions
The Fritz John conditions (abbr. FJ conditions), in mathematics, are a necessary condition for a solution in nonlinear programming to be optimal. They are used as lemma in the proof of the Karush–Kuhn–Tucker conditions.
We consider the following optimization problem:
where ƒ is the function to be minimized, the inequality constraints and the equality constraints, and where, respectively, , and are the indices[disambiguation needed] set of inactive, active and equality constraints and is an optimal solution of , then there exists a non-zero number and a non-zero vector such that: