In this paper, acceleration of gradient methods for convex optimization problems with weak levels of convexity and smoothness is considered. Starting from the universal fast gradient method which was designed to be an optimal method for weakly smooth problems whose gradients are Hölder continuous, its momentum is modified appropriately so that it can also accommodate uniformly convex and weakly smooth problems. Different from the existing works, fast gradient methods proposed in this paper do not use the restarting technique but use momentums that are suitably designed to reflect both the uniform convexity and weak smoothness information of the target energy function. Both theoretical and numerical results that support the superiority of the proposed methods are presented.

在本文中，考虑了具有弱凸度和平滑度的凸优化问题的梯度加速方法。从普遍快速梯度法开始，该方法被设计为梯度是Hölder连续的弱光滑问题的最佳方法，其动量被适当地修改，以便它也可以适应均匀凸和弱光滑问题。与现有工作不同，本文提出的快速梯度方法不使用重新启动技术，而是使用经过适当设计的动量来反映目标能量函数的均匀凸度和弱平滑度信息。提出了支持所提出方法的优越性的理论和数值结果。