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.
No self-concordant barrier interior point method is strongly polynomial
自协调障碍理论能否为线性规划提供一种具有强多项式复杂性的内点方法是一个有待解决的问题。在对数障碍的特殊情况下，在[Allamigeon，Benchimol，Gaubert和Joswig，SIAM J. on Applied Algebra and Geometry，2018]中显示答案是否定的。本文证明了自协调障碍内点方法都不是强多项式的。这个结果是通过建立，在参数族的凸优化问题，中心路径的对数极限退化到分段线性曲线，独立的障碍函数的选择。我们提供了一个显式线性规划，它与Klee-Minty反例属于同一类，即，在具有$2n$约束的维度$n$中，其中迭代次数为$Ω(2^n)$。
Polynomial iteration complexity of a path-following smoothing Newton method for symmetric cone programming
Institute of Mathematics, Hebei University of Technology（河北工业大学数学研究所）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and the University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院、中国科学院大学数学科学学院）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
