The inertial proximal method is extended to minimize the sum of a series of separable nonconvex and possibly nonsmooth objective functions and a smooth nonseparable function (possibly nonconvex). Here, we propose two new algorithms. The first one is an inertial proximal coordinate subgradient algorithm, which updates the variables by employing the proximal subgradients of each separable function at the current point. The second one is an inertial proximal block coordinate method, which updates the variables by using the subgradients of the separable functions at the partially updated points. Global convergence is guaranteed under the Kurdyka–Łojasiewicz (KŁ) property and some additional mild assumptions. Convergence rate is derived based on the Łojasiewicz exponent. Two numerical examples are given to illustrate the effectiveness of the algorithms.

中文翻译：

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两种惯性近端坐标算法，用于一系列非平滑和非凸优化问题


惯性近似法被扩展为最小化一系列可分离的非凸和可能不光滑的目标函数与平滑不可分离函数（可能是非凸函数）之和。在这里，我们提出了两种新算法。第一个是惯性近端坐标次梯度算法，它通过在当前点采用每个可分离函数的近端次梯度来更新变量。第二种是惯性近端块坐标方法，该方法通过在部分更新的点处使用可分离函数的子梯度来更新变量。在 Kurdyka-Łojasiewicz （KŁ） 属性和一些额外的温和假设下保证了全局收敛。收敛速率是根据 Łojasiewicz 指数得出的。给出了两个数值示例来说明算法的有效性。