A class of customized proximal point algorithms for linearly constrained convex optimization

A Family of Multi-Parameterized Proximal Point Algorithms

“…Obviously, the efficiency of ALM heavily depends on the solvability of the x-subproblem, that is, whether or not the core x-subproblem has closed-form solution. Unfortunately, in many real applications [1,3,8,9], the coefficient matrix A is not identity matrix (or does not satisfies AA T = I m ), which makes it difficult even infeasible for solving this subproblem of ALM. To overcome such difficulty, Yang and Yuan [14] proposed a linearized ALM aiming at linearizing the x-subproblem such that its closed-form solution can be easily derived.…”

General parameterized proximal point algorithm with applications in statistical learning

“…Recently, Cai et al [2] designed a PPA with a relaxation step for the model (1.1) with p = 2, whose global convergence and the worst-case sub-linear convergence rate were analyzed in detail. More recently, by introducing some parameters to the metric proximal matrix, an extended parameterized PPA based on [15] was developed for the two block separable convex programming [1], whose effectiveness and robustness was demonstrated by testing a sparse vector optimization problem in the statistical learning compared with two popular algorithms.…”

A parameterized proximal point algorithm for separable convex optimization

“…Cai et al [5] also proposed a R-PPA for solving (1) and analyzed its global convergence with a worst-case linear convergence rate. Based on the results of [5,11,15], Ma and Ni [23] recently revisited the application of PPA for solving the basis pursuit and matrix completion problem. Our new proposed Parameterized PPA (P-PPA) can be actually regarded as more general extensions of the algorithms developed in [15] and [23] which does not make use of the separable structure of the objective function in (1).…”

“…Major contributions of this paper are summarized in the following. Firstly, the proximal matrix in our proposed P-PPA is more general and flexible than those in the previous work [15,23], due to more induced parameters to take consideration of the problem structure instead of a unique objective function. Secondly, by properly choosing the algorithm parameters, the new P-PPA could significantly outperform some state-of-the-art methods, such as ADMM [1] and R-PPA [11], for solving the separable convex optimization, especially when the problem size is large and high accurate solutions are required.…”