General parameterized proximal point algorithm with applications in statistical learning

In the literature, there are a few researches to design some parameters in the proximal point algorithm (PPA), especially for the multi-objective convex optimizations. Introducing some parameters to PPA can make it more flexible and attractive. Mainly motivated by our recent work [Bai et al. A parameterized proximal point algorithm for separable convex optimization. Optim Lett. (2017) doi:10.1007/s11590-017-1195-9], in this paper we develop a general parameterized PPA with a relaxation step for solving the multi-block separable structured convex programming. By making use of the variational inequality and some mathematical identities, the global convergence and the worst-case O(1/t) convergence rate of the proposed algorithm are established. Preliminary numerical experiments on solving a sparse matrix minimization problem from statistical learning validate that our algorithm is more efficient than several state-of-the-art algorithms.