Projectively Self-Concordant Barriers

• 成果类型: Article; Early Access
• 署名作者: Hildebrand, Roland
• 署名单位: Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Inria; Moscow Institute of Physics & Technology
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1215
• 发表日期: 2022
• 关键词: interior-point methods

摘要

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 that is linked to the projective structure. A function on a set X in an 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 of X. The feasible sets in conic programs, notably linear and semidefinite programs, are naturally equipped with projectively self-concordant barriers. However, the interior-point methods used to solve these programs use 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.