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.


  1. 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.
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