Beppo-Levi space

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:

${\begin{aligned}{\text{minimize }}&f(x)\,\\{\text{subject to: }}&g_{i}(x)\geq 0,\ i\in \left\{1,\dots ,m\right\}\\&h_{j}(x)=0,\ j\in \left\{m+1,\dots ,n\right\}\end{aligned}}$

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 indicesTemplate: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:

${\begin{cases}\lambda _{0}\nabla f(x^{*})=\sum \limits _{i\in {\mathcal {I}}'}\lambda _{i}\nabla g_{i}(x^{*})+\sum \limits _{i\in {\mathcal {E}}}\lambda _{i}\nabla h_{i}(x^{*})\\[10pt]\lambda _{i}\geq 0,\ i\in {\mathcal {I}}'\\[10pt]\exists i\in \left(\{0,1,\ldots ,n\}\backslash {\mathcal {I}}\right)\left(\lambda _{i}\neq 0\right)\end{cases}}$

$\lambda _{0}=0$ 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

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