自己整合障壁関数

【じこせいごうしょうへきかんすう (self-concordant barrier function)】

以下の条件を満たす開凸領域 $F\subseteq \mathbf {R} ^{n}\,$ 上の実数値関数 $g\,$.

(1) 任意の 構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\bar {x}}\in \partial F\,} に収束する$F\,$ の任意の点列 $\{x^{k}\}\,$ に対し,構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k\rightarrow \infty \,} で 構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x^{k})\rightarrow \infty \,} となる.

(2) 任意の $x\in F\,$ において, 任意の方向 構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle h\in \mathbf {R} ^{n}\,} に対して, 次が成り立つ.

${\begin{array}{l}\displaystyle {\left|\sum _{i,j,k}{\frac {\partial ^{3}g}{\partial x_{i}\partial x_{j}\partial x_{k}}}(x)h_{i}h_{j}h_{k}\right|\leq }\\\ \ \ \ \ \ \ \ \ \ \displaystyle {2\left|\sum _{i,j}{\frac {\partial ^{2}g}{\partial x_{i}\partial x_{j}}}(x)h_{i}h_{j}\right|^{3/2},}\\[1.4em]\displaystyle {\left(\sum _{i}{\frac {\partial g}{\partial x_{i}}}(x)h_{i}\right)^{2}\leq \nu \sum _{i,j}{\frac {\partial ^{2}g}{\partial x_{i}\partial x_{j}}}(x)h_{i}h_{j}.}\end{array}}\,$