论文标题
在一致的误差范围下的收敛分析
Convergence analysis under consistent error bounds
论文作者
论文摘要
我们介绍了一致的错误绑定功能的概念，该函数为多个凸集提供了一个统一的框架框架。该框架超出了经典的Lipschitzian和Hölderian误差的界限，其中包括指数锥体中发现的对数和熵误差边界。它还包括根据木锥理论获得的误差界限。我们的主要结果是，可行性问题的几种投影算法的收敛速率可以用潜在的一致误差绑定函数明确表示。另一个特征是使用卡拉马塔理论和规则变化的功能，这使我们能够对收敛速率进行推理，同时绕过某些复杂的表达式。最后，给出了锥形可行性问题的应用，我们表明许多算法根据问题的奇异程度明确地收敛率。
We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and Hölderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.
