We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward–backward algorithm, and the adaptive Douglas–Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

中文翻译：

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定点算法的圆锥平均性和收敛性分析

我们研究了平均非膨胀算子的圆锥扩展及其在定点算法收敛分析中的作用。系统地研究了圆锥平均算子的各种性质，特别是松弛、凸组合和组合下的稳定​​性。我们推导出广义单调算子的解算器的圆锥平均特性。然后利用这些属性来分析近端点算法、前向-后向算法和自适应 Douglas-Rachford 算法的收敛性。我们的研究统一、改进并为最近对这些主题的研究提供了新的思路。