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:

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

where ƒ is the function to be minimized, $g_i$ the inequality constraints and $h_j$ the equality constraints, and where, respectively, $\mathcal{ℐ}$, ${\mathcal{ℐ}}^{\prime }$ 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:

$\{\begin{array}{l}{\lambda }_0\mathrm{\nabla }f(x^{*})=\sum _{i\in {\mathcal{ℐ}}^{\prime }}{\lambda }_i\mathrm{\nabla }{g}_i(x^{*})+\sum _{i\in \mathcal{ℰ}}{\lambda }_i\mathrm{\nabla }{h}_i(x^{*})\\ {\lambda }_i\ge 0,i\in {\mathcal{ℐ}}^{\prime }\\ \mathrm{\exists }i\in (\{0,1,\dots ,n\}\mathrm{\setminus }\mathcal{ℐ})({\lambda }_i\ne 0)\end{array}$

${\lambda }_0=0$ iff the $\mathrm{\nabla }{g}_i(i\in {\mathcal{ℐ}}^{\prime })$ and $\mathrm{\nabla }{h}_i(i\in \mathcal{ℰ})$ are linearly dependent and ${\lambda }_i\ne 0,\mathrm{\forall }i\in {\mathcal{ℐ}}^{\prime }\cup \mathcal{ℰ}$, 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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