# 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.[1] 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, the inequality constraints and the equality constraints, and where, respectively, , and are the indices set of inactive, active and equality constraints and is an optimal solution of , then there exists a non-zero vector such that:

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 , 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.

## References

- Takayama, Akira (1985).
*Mathematical Economics*. New York: Cambridge University Press. pp. 90–112. ISBN 0-521-31498-4.

## Further reading

- Rau, Nicholas (1981). "Lagrange Multipliers".
*Matrices and Mathematical Programming*. London: Macmillan. pp. 156–174. ISBN 0-333-27768-6.