【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 ⊆ R n {\displaystyle F\subseteq \mathbf {R} ^{n}\,} 上の実数値関数 g {\displaystyle g\,} .
(1) 任意の x ¯ ∈ ∂ F {\displaystyle {\bar {x}}\in \partial F\,} に収束する F {\displaystyle F\,} の任意の点列 { x k } {\displaystyle \{x^{k}\}\,} に対し, k → ∞ {\displaystyle k\rightarrow \infty \,} で g ( x k ) → ∞ {\displaystyle g(x^{k})\rightarrow \infty \,} となる.
(2) 任意の x ∈ F {\displaystyle x\in F\,} において, 任意の方向 h ∈ R n {\displaystyle h\in \mathbf {R} ^{n}\,} に対して, 次が成り立つ.
| ∑ i , j , k ∂ 3 g ∂ x i ∂ x j ∂ x k ( x ) h i h j h k | ≤ 2 | ∑ i , j ∂ 2 g ∂ x i ∂ x j ( x ) h i h j | 3 / 2 , ( ∑ i ∂ g ∂ x i ( x ) h i ) 2 ≤ ν ∑ i , j ∂ 2 g ∂ x i ∂ x j ( x ) h i h j . {\displaystyle {\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}}\,}
上の実数値関数 
.
に収束する
の任意の点列 
に対し,
で 
となる.
において, 任意の方向 
に対して, 次が成り立つ.
![{\displaystyle {\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}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cc8d4b569300e906f37bd213294eebf4ca851a30)