Primal Central Paths and Riemannian Distances for Convex Sets

Y. Nesterov; A. Nemirovski

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Primal Central Paths and Riemannian Distances for Convex Sets

机译：凸集的原始中心路径和黎曼距离

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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(ν 1/4) of the length of the shortest geodesic curve.

机译：在本文中，我们研究了凸集中原始中心路径的黎曼长度，该凸集是相对于由自协调函数定义的局部度量计算的。尽管有一些负面的例子，但在许多重要情况下，该路径的长度非常接近测地曲线的长度。我们表明，在有界凸集具有ν-自协调障碍的情况下，中心路径的长度在最短测地曲线的长度的因数O（ν1/4 ）之内。

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来源
《Foundations of Computational Mathematics》 |2008年第5期|533-560|共28页
作者
Y. Nesterov; A. Nemirovski;
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作者单位

CORE Catholic University of Louvain 34 voie du Roman Pays 1348 Louvain-la-Neuve Belgium;

Technion-Israel Institute of Technology Haifa Israel;

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正文语种 eng
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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;

机译：黎曼几何;凸优化;结构优化;内点法;路径跟踪法;自协调函数;多项式时间法;52A41;53C22;68Q25;90C22;90C25;90C51;90C60;

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1. Primal central paths and Riemannian distances for convex sets [J] . Nesterov Y., Nemirovski A. Foundations of computational mathematics . 2008,第5期

机译：凸集的原始中心路径和黎曼距离
2. A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms [J] . Monteiro RDC, Tsuchiya T Mathematical Programming . 2008,第1期

机译：中心路径曲率积分及其与原对偶路径跟随LP算法的迭代复杂度的关系的强边界
3. A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms [J] . Renato D. C. Monteiro, Takashi Tsuchiya Mathematical Programming . 2008,第1期

机译：中心路径曲率积分及其与原对偶路径跟随LP算法的迭代复杂度的关系的强边界
4. A new primal-dual path-following interior-point algorithm for convex quadratic semidefinite optimization with full Nesterov-Todd step [C] . Yujing Yue, Guoqiang Wang 2012 IEEE International Conference on Oxide Materials for Electronic Engineering. . 2012

机译：带有完整Nesterov-Todd步骤的凸二次半定优化的新的原始-对偶路径跟踪内点算法
5. Primal-Dual Active-Set Methods for Convex Quadratic Optimization with Applications [D] . Han, Zheng 2015

机译：凸二次对偶二次优化集及其应用
6. Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces [O] . Binayak S. Choudhury, Pranati Maity, Nikhilesh Metiya, -1

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7. Primal central paths and Riemannian distances for convex sets [O] . Nesterov, Yurii, Nemirovski, A. 2008

机译：凸集的原始中心路径和黎曼距离

Primal Central Paths and Riemannian Distances for Convex Sets

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