Projectively self-concordant barriers

TL;DR

This paper introduces the concept of projectively self-concordant barriers, linking projective and affine structures, and demonstrates their potential to improve the efficiency of interior-point methods in convex programming.

Contribution

It defines projective self-concordance, relates it to affine self-concordance, and shows how it can lead to tighter estimates and faster convergence in interior-point algorithms.

Findings

Tighter estimates for interior-point methods using projective self-concordance.

Projective self-concordance applies naturally to conic programming barriers.

Potential for improved parameter tuning and faster convergence in convex optimization.

Abstract

Self-concordance is the most important property required for barriers in convex programming. It is intrinsically linked to the affine structure of the underlying space. Here we introduce an alternative notion of self-concordance which is linked to the projective structure. A function on a set $X \subset A^n$ in an $n$-dimensional affine space is projectively self-concordant if and only if it can be extended to an affinely self-concordant logarithmically homogeneous function on the conic extension $K \subset V^{n+1}$ of $X$ in the $(n+1)$-dimensional vector space obtained by homogenization of $A^n$. The feasible sets in conic programs, notably linear and semi-definite programs, are naturally equipped with projectively self-concordant barriers. However, the interior-point methods used to solve these programs employ only affine self-concordance. We show that estimates used in the analysis of interior-point methods are tighter for projective self-concordance, in particular inner and outer approximations of the set. This opens the way to a better tuning of parameters in interior-points algorithms to allow larger steps and hence faster convergence. Projective self-concordance is also a useful tool in the theoretical analysis of logarithmically homogeneous barriers on cones.