# Self concordance

### From Glossary

Properties of a function that yield nice performance of Newton's method used for line search when optimizing a barrier function. Specifically, let B be a barrier function for with strict interior Let x be in S and let d be a direction vector in such that the line segment is in S for where Then, define by:

(while noting that F depends on x and d). The function F is
*self-concordant* if it is convex in C^{3} and
satisfies the following for all x and d:

One calls F *k-self-concordant* in an open convex domain if

The logarithmic barrier function, associated with linear programming, is self-concordant with
This further extends naturally to functions in