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.

翻译：本文提出了一种适用于真下半连续函数最小化的新型抽象下降方案。该抽象方案归纳了针对非光滑非凸优化问题设计的多种一阶方法收敛的关键性质。当目标函数满足Kurdyka-Lojasiewicz性质时，这些性质保证迭代全序列收敛至稳定点。该抽象框架可指导新算法设计，我们提出了两种具有可实施非精确性准则的惯性类算法用于主迭代更新步骤。第一个算法i$^2$Piano通过调节局部Lipschitz常数实现大步长，第二个算法iPila通过仅强制优值函数而非目标函数下降，克服了基于线搜索方法的主要缺陷。两种算法均可利用惯性特性逃离局部极小值（或稳定点），且被证明完全继承抽象下降方案的收敛保证——这是此类一般非光滑非凸优化框架下一阶方法所能达到的最佳效果。我们通过两个图像去模糊示例问题验证了所提算法的效率，在惯性方案中沿下降方向进行线搜索的优势得到了充分体现。