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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 Holder continuous, its momentum is modified appropriately so that it can also accommodate uniformly convex and weakly smooth problems. Differently 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 the weak smoothness information of the target energy function. Both theoretical and numerical results that support the superiority of proposed methods are presented.
Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed with theoreti
The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the differentiable c
First-order methods (FOMs) have been widely used for solving large-scale problems. A majority of existing works focus on problems without constraint or with simple constraints. Several recent works have studied FOMs for problems with complicated func
Conditional gradient methods have attracted much attention in both machine learning and optimization communities recently. These simple methods can guarantee the generation of sparse solutions. In addition, without the computation of full gradients,