Self-concordant function

In optimization, a self-concordant function is a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which

$|f'''(x)| \leq 2 f''(x)^{3/2}.$

A function $g(x) : \mathbb{R}^n \rightarrow \mathbb{R}$ is self-concordant if its restriction to any arbitrary line is self-concordant. ^[1]

History

The self-concordant functions are introduced by Yurii Nesterov and Arkadi Nemirovski in their 1994 book.^[2]

Properties

Self concordance is preserved under addition, affine transformations, and scalar multiplication by a value greater than one.

Applications

Among other things, self-concordant functions are useful in the analysis of Newton's method. Self-concordant barrier functions are used to develop the barrier functions used in interior point methods for convex and nonlinear optimization.

References

1. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
2. Nesterov, Yurii; Arkadii, Nemirovskii (1995). Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics. ISBN 0898715156.