Primal Central Paths and Riemannian Distances for Convex Sets

被引：0

作者：

Y. Nesterov

A. Nemirovski

机构：

[1] Catholic University of Louvain,CORE

[2] Technion-Israel Institute of Technology,undefined

来源：

Foundations of Computational Mathematics | 2008年 / 8卷

关键词：

Riemannian geometry; Convex optimization; Structural optimization; Interior-point methods; Path-following methods; Self-concordant functions; Polynomial-time methods; 52A41; 53C22; 68Q25; 90C22; 90C25; 90C51; 90C60;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

In this paper, we study the Riemannian length of the primal central path in a convex set computed with respect to the local metric defined by a self-concordant function. Despite some negative examples, in many important situations, the length of this path is quite close to the length of a geodesic curve. We show that in the case of a bounded convex set endowed with a ν-self-concordant barrier, the length of the central path is within a factor O(ν1/4) of the length of the shortest geodesic curve.

引用

页码：533 / 560

页数：27

共 50 条

[21] A primal-dual operation on sets linked with closed convex relaxation processes
Volle, M.
[J]. TOP, 2012, 20 (02) : 534 - 546
[22] Embedding four-directional paths on convex point sets
[J]. Aichholzer, Oswin, 1600, Springer Verlag (8871):
[23] Iterative projection onto convex sets using multiple Bregman distances
Byrne, C
[J]. INVERSE PROBLEMS, 1999, 15 (05) : 1295 - 1313
[24] CENTRAL DECOMPOSITIONS FOR COMPACT CONVEX-SETS
ELLIS, AJ
[J]. COMPOSITIO MATHEMATICA, 1975, 30 (03) : 211 - 219
[25] Sampling and Optimization on Convex Sets in Riemannian Manifolds of Non-Negative Curvature
Goyal, Navin
Shetty, Abhishek
[J]. CONFERENCE ON LEARNING THEORY, VOL 99, 2019, 99
[26] REMARK ON THE HELLY NUMBER FOR STRONGLY CONVEX-SETS ON RIEMANNIAN-MANIFOLDS
KLEINJOHANN, N
[J]. MANUSCRIPTA MATHEMATICA, 1981, 34 (01) : 27 - 29
[27] Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
Alexander V. Kolesnikov
Emanuel Milman
[J]. Calculus of Variations and Partial Differential Equations, 2016, 55
[28] Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds
Tsuji, Hiroshi
[J]. ANALYSIS AND GEOMETRY IN METRIC SPACES, 2021, 9 (01): : 219 - 253
[29] Riemannian metrics on convex sets with applications to Poincare and log-Sobolev inequalities
Kolesnikov, Alexander V.
Milman, Emanuel
[J]. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2016, 55 (04)
[30] Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection
Zambotti, L
[J]. PROBABILITY THEORY AND RELATED FIELDS, 2002, 123 (04) : 579 - 600

← 1 2 3 4 5 →