# 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:

\begin{align} \text{minimize } & f(x) \, \\ \text{subject to: } & g_i(x) \ge 0,\ i \in \left \{1,\dots,m \right \}\\ & h_j(x) = 0, \ j \in \left \{m+1,\dots,n \right \} \end{align}

where ƒ is the function to be minimized, $g_i$ the inequality constraints and $h_j$ the equality constraints, and where, respectively, $\mathcal{I}$, $\mathcal{I'}$ and $\mathcal{E}$ are the indices[disambiguation needed] set of inactive, active and equality constraints and $x^*$ is an optimal solution of $f$, then there exists a non-zero number $\lambda _0$ and a non-zero vector $\lambda=[\lambda _1, \lambda _2,\dots,\lambda _n]$ such that:

$\begin{cases} \lambda_0 \nabla f(x^*) = \sum\limits_{i\in \mathcal{I}'} \lambda_i \nabla g_i(x^*) + \sum\limits_{i\in \mathcal{E}} \lambda_i \nabla h_i (x^*) \\[10pt] \lambda_i \ge 0,\ i\in \mathcal{I}' \\[10pt] \exists i\in \left( \{0,1,\ldots ,n\} \backslash \mathcal{I} \right) \left( \lambda_i \ne 0 \right) \end{cases}$

$\lambda_0=0$ iff the $\nabla g_i (i\in\mathcal{I}')$ and $\nabla h_i (i\in\mathcal{E})$ are linearly dependent and $\lambda_i\neq 0,\,\forall i\in\mathcal{I}'\cup\mathcal{E}$, i.e. if the constraint qualifications do not hold.

Named after Fritz John, these conditions are equivalent to the Karush–Kuhn–Tucker conditions in the case $\lambda_0 = 1$.

## References

• WANG Yiju, XIU Naihua. 非线性规划的理论和方法 (The theory and methods of the Non-linear programming) (in Chinese). p. 4. ISBN 978-7-5369-3825-0.