# Self-concordant function

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

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

or, equivalently, a function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ that, wherever ${\displaystyle f''(x)>0}$, satisfies

${\displaystyle \left|{\frac {d}{dx}}{\frac {1}{\sqrt {f''(x)}}}\right|\leq 1}$

and which satisfies ${\displaystyle f'''(x)=0}$ elsewhere.

More generally, a multivariate function ${\displaystyle f(x):\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is self-concordant if

${\displaystyle \left.{\frac {d}{d\alpha }}\nabla ^{2}f(x+\alpha y)\right|_{\alpha =0}\preceq 2{\sqrt {y^{T}\nabla ^{2}f(x)\,y}}\,\nabla ^{2}f(x)}$

or, equivalently, 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.