We investigate the convergence of the primal-dual algorithm for composite optimization problems when the objective functions are weakly convex. We introduce a modified duality gap function, which is a lower bound of the standard duality gap function. Under the sharpness condition of this new function, we identify the area around the set of saddle points where we obtain the convergence of the primal-dual algorithm. We give numerical examples and applications in image denoising and deblurring to demonstrate our results.

翻译：本文研究了复合优化问题中目标函数为弱凸函数时原始对偶算法的收敛性。我们引入了一种修正的对偶间隙函数，该函数是标准对偶间隙函数的下界。在此新函数的锐利条件下，我们确定了鞍点集周围的区域，并在该区域内证明了原始对偶算法的收敛性。我们通过图像去噪与去模糊的数值算例和应用验证了所得结果。