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, but they are relevant on their own. We consider the following optimization problem: where ƒ is the function to be minimized, g_i the inequality constraints and h_j the equality constraints, and where, respectively, \mathcal{I}, \mathcal{A} and \mathcal{E} are the indices sets of inactive, active and equality constraints and x^* is an optimal solution of f, then there exists a non-zero vector such that: \lambda_0>0 if the and are linearly independent or, more generally, when a constraint qualification holds. Named after Fritz John, these conditions are equivalent to the Karush–Kuhn–Tucker conditions in the case. When \lambda_0=0, the condition is equivalent to the violation of Mangasarian–Fromovitz constraint qualification (MFCQ). In other words, the Fritz John condition is equivalent to the optimality condition KKT or not-MFCQ.