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.
This paper shows that the self-concordance parameter of the universal barrier on any $n$-dimensional proper convex domain is upper bounded by $n$. This bound is tight and improves the previous $O(n)$ bound by Nesterov and Nemirovski. The key to our main result is a pair of new, sharp moment inequalities for $s$-concave distributions, which could be of independent interest.
Massachusetts Institute of Technology (MIT)（麻省理工学院）
For any convex body $K \subseteq \mathbb R^n$, S. Bubeck and R. Eldan introduced the entropic barrier on $K$ and showed that it is a $(1+o(1)) \, n$-self-concordant barrier. In this note, we observe that the optimal bound of $n$ on the self-concordance parameter holds as a consequence of the dimensional Brascamp-Lieb inequality.
It is an open question to determine if the theory of self-concordant barriers can provide an interior point method with strongly polynomial complexity in linear programming. In the special case of the logarithmic barrier, it was shown in [Allamigeon, Benchimol, Gaubert and Joswig, SIAM J. on Applied Algebra and Geometry, 2018] that the answer is negative. In this paper, we show that none of the self-concordant barrier interior point methods is strongly polynomial. This result is obtained by establishing that, on parametric families of convex optimization problems, the log-limit of the central …
Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized self-concordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in…
Quantum relative entropies are jointly convex functions of two positive definite matrices that generalize the Kullback-Leibler divergence and arise naturally in quantum information theory. In this paper, we prove self-concordance of natural barrier functions for the epigraphs of various quantum relative entropies and divergences. Furthermore we show that these barriers have optimal barrier parameter. These barriers allow convex optimization problems involving quantum relative entropies to be directly solved using interior point methods for non-symmetric cones, avoiding the approximations and l…
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）
The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In this paper, we study the tropicalization of the central path associated with the entropic barrier, i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamig…
We consider the problem of linear regression where the $\ell_2^n$ norm loss (i.e., the usual least squares loss) is replaced by the $\ell_p^n$ norm. We show how to solve such problems up to machine precision in $O^*(n^{|1/2 - 1/p|})$ (dense) matrix-vector products and $O^*(1)$ matrix inversions, or alternatively in $O^*(n^{|1/2 - 1/p|})$ calls to a (sparse) linear system solver. This improves the state of the art for any $p\not\in \{1,2,+\infty\}$. Furthermore we also propose a randomized algorithm solving such problems in {\em input sparsity time}, i.e., $O^*(Z + \mathrm{poly}(d))$ where $Z$ …
