# 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.

## Properties

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

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## 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.

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## References

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