论文标题
通过平滑功能的块结构的非平滑问题的前向后动力学方法
A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function
论文作者
论文摘要
在本文中，我们旨在最大程度地减少通过平滑耦合函数连接的单独变量中两个非平滑词（可能也可能也是非convex）函数的总和。为了解决这个问题，我们选择了一种连续的前向方法，并引入了一个动态系统，该系统是通过平滑耦合函数的部分梯度和两个非滑动函数的近端运算符进行配合的。此外，我们考虑了所得系统的内隐性速率。我们讨论了解决方案的存在和唯一性，并对目标函数的正则化满足kurdyka-lojasiewicz属性进行渐近分析对优化问题的临界点的渐近分析。我们进一步根据Lojasiewicz指数提供了解决方案轨迹的收敛速率。我们通过数值模拟得出了这项工作，这些模拟确认并验证了分析结果。
In this paper we aim to minimize the sum of two nonsmooth (possibly also nonconvex) functions in separate variables connected by a smooth coupling function. To tackle this problem we chose a continuous forward-backward approach and introduce a dynamical system which is formulated by means of the partial gradients of the smooth coupling function and the proximal point operator of the two nonsmooth functions. Moreover, we consider variable rates of implicitness of the resulting system. We discuss the existence and uniqueness of a solution and carry out the asymptotic analysis of its convergence behaviour to a critical point of the optimization problem, when a regularization of the objective function fulfills the Kurdyka-Lojasiewicz property. We further provide convergence rates for the solution trajectory in terms of the Lojasiewicz exponent. We conclude this work with numerical simulations which confirm and validate the analytical results.
