Self-concordant function


In optimization, a self-concordant function is a function for which
or, equivalently, a function that, wherever, satisfies
and which satisfies elsewhere.
More generally, a multivariate function is self-concordant if
or, equivalently, if its restriction to any arbitrary line is self-concordant.

History

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

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.