In this paper, we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. The equivalence between the classical and simplified Riemannian versions of the DCA is established. We also prove that under mild assumptions the Riemannian version of the DCA is well defined and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Some duality relations between the DC problem and its dual are also established. To illustrate the algorithm’s effectiveness, some numerical experiments are presented.

在本文中，我们提出了凸差算法（DCA）的黎曼版本来解决涉及凸差（DC）函数的最小化问题。建立了 DCA 的经典版本和简化黎曼版本之间的等价性。我们还证明，在温和的假设下，黎曼版本的 DCA 得到了很好的定义，并且所提出的方法生成的序列的每个聚类点（如果有的话）都是目标 DC 函数的临界点。DC问题及其对偶之间也建立了一些对偶关系。为了说明算法的有效性，给出了一些数值实验。