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.{{cite book |first=Akira |last=Takayama |authorlink=Akira Takayama |title=Mathematical Economics |url=https://archive.org/details/mathematicalecon00taka |url-access=registration |location=New York |publisher=Cambridge University Press |year=1985 |pages=[https://archive.org/details/mathematicalecon00taka/page/90 90–112] |isbn=0-521-31498-4 }} 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:

:$$

\begin{align}

\text{minimize } & f(x) \, \\

\text{subject to: } & g_i(x) \le 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, $\backslash mathcal\{I\}$, $\backslash mathcal\{A\}$ and $\backslash mathcal\{E\}$ 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 $\backslash lambda=[\backslash lambda\_0,\; \backslash lambda\; \_1,\; \backslash lambda\; \_2,\backslash dots,\backslash lambda\; \_n]$ such that:

:$\backslash begin\{cases\}$

\lambda_0 \nabla f(x^*) + \sum\limits_{i\in \mathcal{A}} \lambda_i \nabla g_i(x^*) + \sum\limits_{i\in \mathcal{E}} \lambda_i \nabla h_i (x^*) =0\\[10pt]

\lambda_i \ge 0,\ i\in \mathcal{A}\cup\{0\} \\[10pt]

\exists i\in \left( \{0,1,\ldots ,n\} \backslash \mathcal{I} \right) \left( \lambda_i \ne 0 \right)

\end{cases}

$\backslash lambda\_0>0$ if the $\backslash nabla\; g\_i\; (i\backslash in\backslash mathcal\{A\})$ and $\backslash nabla\; h\_i\; (i\backslash in\backslash mathcal\{E\})$ 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 $\backslash lambda\_0\; >\; 0$. When $\backslash lambda\_0=0$, 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.{{cn|date=June 2019}}