Bingsheng He The augmented Lagrangian method (ALM) provides a benchmark for tackling the canonical convex minimization problem with linear constraints. We consider a special case where the objective function is the sum of $m$ individual subfunctions without coupled variables. The recent study reveals that the direct extension of ALM for separable convex programming problems is not necessarily … Read more
The augmented Lagrangian method (ALM) is a benchmark for tackling the convex optimization problem with linear constraints; ALM and its variants for linearly equality-constrained convex minimization models have been well studied in the literatures. However, much less attention has been paid to ALM for efficiently solving the linearly inequality-constrained convex minimization model. In this paper, … Read more
An ADMM-based splitting method is proposed in [11] for solving convex minimization problems with linear constraints and multi-block separable objective functions; while a relatively large proximal parameter is required for theoretically ensuring the convergence. In this paper, we further study this method and find its optimal (smallest) proximal parameter. For succinctness, we focus on the … Read more
The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention from many areas because of its efficiency and easy implementation. To theoretically guarantee … Read more
The alternating direction method of multipliers (ADMM) is a popular method for the separable convex programming with linear constraints, and the proximal ADMM is its important variant. Previous studies show that the relaxation factor $\gamma\in (0, \frac{1+\sqrt{5}}{2})$ by Fortin and Glowinski for the ADMM is also valid for the proximal ADMM. In this paper, we … Read more
The augmented Lagrangian method (ALM) is fundamental for solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper, we show that it is not necessary … Read more
The augmented Lagrangian method (ALM) is fundamental for solving convex minimization models with linear constraints. When the objective function is separable such that it can be represented as the sum of more than one function without coupled variables, various splitting versions of the ALM have been well studied in the literature such as the alternating … Read more
The alternating direction method of multipliers (ADMM) is being widely used for various convex minimization models with separable structures arising in a variety of areas. In the literature, the proximalversion of ADMM which allows ADMM’s subproblems to be proximally regularized has been well studied. Particularly the linearized version of ADMM can be yielded when the … Read more
The primal-dual hybrid gradient method (PDHG) originates from the Arrow-Hurwicz method, and it has been widely used to solve saddle point problems, particularly in image processing areas. With the introduction of a combination parameter, Chambolle and Pock proposed a generalized PDHG scheme with both theoretical and numerical advantages. It has been analyzed that except for … Read more
Recently the alternating direction method of multipliers (ADMM) has been widely used for various applications arising in scientific computing areas. Most of these application models are, or can be easily reformulated as, linearly constrained convex minimization models with separable nonlinear objective functions. In this note we show that ADMM can also be easily used for … Read more