随机算法分布的收敛性:部分可分离优化的案例

IF 2.2 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Mathematical Programming Pub Date : 2024-07-27 DOI:10.1007/s10107-024-02124-w
D. Russell Luke
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引用次数: 0

摘要

我们对求解部分分块优化问题的分块随机算法进行了马尔可夫链分析。对于有关这些方法的大量文献,我们的主要贡献是对随机算法迭代后面的马尔可夫算子和分布的说明,特别是马尔可夫算子的正则性和相应马尔可夫链分布的收敛率。这提供了序列矩的详细特征,而不仅仅是预期行为。这也是一个案例研究,说明随机化如何恢复算法的有利特性,而仅部分信息的迭代会破坏这些特性。我们在非凸(以及作为特例的凸)、非光滑优化的前向后向算法和道格拉斯-拉赫福德算法的随机顺时针实现上证明了这一点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Convergence in distribution of randomized algorithms: the case of partially separable optimization

We present a Markov-chain analysis of blockwise-stochastic algorithms for solving partially block-separable optimization problems. Our main contributions to the extensive literature on these methods are statements about the Markov operators and distributions behind the iterates of stochastic algorithms, and in particular the regularity of Markov operators and rates of convergence of the distributions of the corresponding Markov chains. This provides a detailed characterization of the moments of the sequences beyond just the expected behavior. This also serves as a case study of how randomization restores favorable properties to algorithms that iterations of only partial information destroys. We demonstrate this on stochastic blockwise implementations of the forward–backward and Douglas–Rachford algorithms for nonconvex (and, as a special case, convex), nonsmooth optimization.

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来源期刊
Mathematical Programming
Mathematical Programming 数学-计算机:软件工程
CiteScore
5.70
自引率
11.10%
发文量
160
审稿时长
4-8 weeks
期刊介绍: Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.
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