AI 中文总结
研究哈达玛流形上强拟凸平衡问题的近端点算法，结合惯性步与超松弛，利用定量方法论证收敛性，给出迭代点到解距离的收敛速率，将相关工作首次扩展到非线性设置并改进假设。
AI 中文摘要
我们研究了一种近端点型方法，用于逼近由哈达玛流形（即非正截面曲率的完备单连通黎曼流形）上的伪单调和强拟凸双函数产生的平衡问题的解。除了通常的近端点步，我们考虑的方法还结合了一个惯性步以及随后的超松弛，据我们所知，后者是首次在哈达玛流形的背景下进行处理。利用作者在先前工作中针对强拟凸优化的此类近端点方法所开发的定量方法，我们特别为该方法的收敛性提供了有效的论证，得出了迭代点到解的距离的明确、快速且非常一致的收敛速率。这些结果首次将Grad、Lara和Marcavillaca在有限维欧几里得空间上关于此类方法的先前工作扩展到了非线性设置，其中定量估计在欧几里得情形下也是新颖的。特别是，我们的有效方法允许对周围对象的假设进行细粒度的观察，从而使我们能够削弱甚至完全消除一些先前的假设。
英文摘要
We study a proximal point type method for approximating solutions to equilibrium problems generated by pseudomonotone and strongly quasiconvex bifunctions over Hadamard manifolds, that is complete simply connected Riemannian manifolds of nonpositive sectional curvature. Next to the usual proximal step, the method we consider also incorporates an inertia step together with a subsequent over-relaxation, the latter of which is treated in the context of Hadamard manifolds, to our knowledge, for the first time. Making use of a quantitative approach towards such proximal methods for strongly quasiconvex optimization developed by the authors in previous work, we in particular provide effective arguments for the convergence of the method, yielding explicit, fast and very uniform rates of convergence for the distance of the iterates towards the solution. These results extend previous work by Grad, Lara and Marcavillaca on such a method over finite-dimensional Euclidean spaces for the first time to a nonlinear setting, with the quantitative estimates already being novel in the Euclidean case. In particular, our effective approach allows for a fine-grained view on the assumptions on the surrounding objects, so that we are able to either weaken or even fully discharge some previous assumptions.
Comments31 pages
