The possibilities of exploiting the special structure of d.c. programs, which consist of optimising the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimisation problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. Convergence of the iterates is shown if this objective function satisfies the Kurdyka–Łojasiewicz property. In the last part, we apply the algorithm to an image processing model.

中文翻译：

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一种用于直流编程的通用双邻近梯度算法

利用 dc 程序的特殊结构（包括优化凸函数的差异）的可能性目前或多或少限于 Pham Dinh Tao 和 Le Thi Hoai An 在 1997 年提出的 DCA 变体。这些假设要么凸部分或凹部分，或两者都由它们的次梯度之一进行评估。在本文中，我们提出了一种算法，该算法允许通过它们的近点来评估凹面和凸面部分。此外，我们允许平滑部分，通过其梯度进行评估。本着原始对偶分裂算法的精神，凹部分可能是具有线性算子的凹函数的组合，但是，它们是单独计算的。对于该算法，我们表明每个聚类点都是优化问题的解决方案。此外，我们展示了与 Toland 对偶问题的联系，并证明了问题的原始对偶公式的目标函数值的下降属性。如果此目标函数满足 Kurdyka-Łojasiewicz 属性，则显示迭代收敛。在最后一部分，我们将该算法应用于图像处理模型。