The difference-of-convex (DC) program is an important model in nonconvex optimization due to its structure, which encompasses a wide range of practical applications. In this paper, we aim to tackle a generalized class of DC programs, where the objective function is formed by summing a possibly nonsmooth nonconvex function and a differentiable nonconvex function with Lipschitz continuous gradient, and then subtracting a nonsmooth continuous convex function. We develop a proximal splitting algorithm that utilizes proximal evaluation for the concave part and Douglas–Rachford splitting for the remaining components. The algorithm guarantees subsequential convergence to a critical point of the problem model. Under the widely used Kurdyka–Łojasiewicz property, we establish global convergence of the full sequence of iterates and derive convergence rates for both the iterates and the objective function values, without assuming the concave part is differentiable. The performance of the proposed algorithm is tested on signal recovery problems with a nonconvex regularization term and exhibits competitive results compared to notable algorithms in the literature on both synthetic data and real-world data.

中文翻译：

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一种用于广义直流编程的近端分裂算法，适用于信号恢复

凸差 （DC） 程序因其结构而成为非凸优化中的重要模型，它包含了广泛的实际应用。在本文中，我们旨在解决一类广义的直流程序，其中目标函数是通过用 Lipschitz 连续梯度将一个可能不平滑的非凸函数和一个可微非凸函数相加，然后减去一个非光滑的连续凸函数来形成的。我们开发了一种近端分裂算法，该算法对凹部分利用近端评估，对其余组件利用 Douglas-Rachford 分裂。该算法保证后续收敛到问题模型的关键点。在广泛使用的 Kurdyka-Łojasiewicz 属性下，我们建立了迭代完整序列的全局收敛，并推导出迭代和目标函数值的收敛率，而不假设凹部分是可微的。所提出的算法的性能在具有非凸正则化项的信号恢复问题上进行了测试，与文献中关于合成数据和真实世界数据的著名算法相比，表现出有竞争力的结果。