一个抽象收敛框架及其在惯性不精确向前向后方法中的应用
An abstract convergence framework with application to inertial inexact forward--backward methods
摘要：
本文介绍了一种新的抽象下降方案，适用于最小化正常和下连续函数。建议的抽象计划概括了一组属性，这些属性对于设计用于非光滑非凸最优化问题的几个一阶方法的收敛性至关重要。如果目标函数满足Kurdyka-Lojasiewicz性质，则这样的性质保证了整个迭代序列收敛到一个稳定点。抽象框架允许设计新的算法。我们提出了两个惯性型算法的主要迭代更新步骤与实现不精确的标准。第一种算法i $^2$ Piano通过调整局部Lipschitz常数来利用大步长。第二个算法，iPila，克服了基于线搜索的方法的主要缺点，通过强制下降，而不是目标函数上的价值函数。这两种算法都有可能通过利用惯性特征来逃避局部最小值（或静止点）。此外，他们被证明享有充分的收敛保证的抽象下降计划，这是最好的，我们可以期望在这样一个一般的非光滑非凸优化设置使用一阶方法。所提出的算法的效率上证明了两个示例性的图像去模糊问题，在那里我们可以欣赏的好处，沿下降方向内的惯性计划执行lineswitch。
英文摘要：
In this paper we introduce a novel abstract descent scheme suited for the minimization of proper and lower semicontinuous functions. The proposed abstract scheme generalizes a set of properties that are crucial for the convergence of several first-order methods designed for nonsmooth nonconvex optimization problems. Such properties guarantee the convergence of the full sequence of iterates to a stationary point, if the objective function satisfies the Kurdyka-Lojasiewicz property. The abstract framework allows for the design of new algorithms. We propose two inertial-type algorithms with implementable inexactness criteria for the main iteration update step. The first algorithm, i$^2$Piano, exploits large steps by adjusting a local Lipschitz constant. The second algorithm, iPila, overcomes the main drawback of line-search based methods by enforcing a descent only on a merit function instead of the objective function. Both algorithms have the potential to escape local minimizers (or stationary points) by leveraging the inertial feature. Moreover, they are proved to enjoy the full convergence guarantees of the abstract descent scheme, which is the best we can expect in such a general nonsmooth nonconvex optimization setup using first-order methods. The efficiency of the proposed algorithms is demonstrated on two exemplary image deblurring problems, where we can appreciate the benefits of performing a linesearch along the descent direction inside an inertial scheme.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
