Multi-block Convex Minimization Research Articles

Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights

In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) method for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: (1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; (2) cyclic BCD may outperform RPBCD for “nice” problems, and RPBCD should be applied with caution when solving general convex optimization problems especially with a few blocks.

Convergence Analysis of L-ADMM for Multi-block Linear-Constrained Separable Convex Minimization Problem

We focus on the convergence analysis of the extended linearized alternating direction method of multipliers (L-ADMM) for solving convex minimization problems with three or more separable blocks in the objective functions. Previous convergence analysis of the L-ADMM needs to reduce the multi-block convex minimization problems to two blocks by grouping the variables. Moreover, there has been no rate of convergence analysis for the L-ADMM. In this paper, we construct a counter example to show the failure of convergence of the extended L-ADMM. We prove the convergence and establish the sublinear convergence rate of the extended L-ADMM under the assumptions that the proximal gradient step sizes are smaller than certain values, and any two coefficient matrices in linear constraints are orthogonal.

A distributed Douglas-Rachford splitting method for multi-block convex minimization problems

A customized Douglas-Rachford splitting method (DRSM) was recently proposed to solve two-block separable convex optimization problems with linear constraints and simple abstract constraints. The algorithm has advantage over the well-known alternating direction method of multipliers (ADMM), the dual application of DRSM to the two-block convex minimization problem, in the sense that the subproblems can have larger opportunity of possessing closed-form solutions since they are unconstrained. In this paper, we further study along this way by considering the primal application of DRSM for the general case m?3, i.e., we consider the multi-block separable convex minimization problem with linear constraints where the objective function is separable into m individual convex functions without coupled variables. The resulting method fully exploits the separable structure and enjoys decoupled subproblems which can be solved simultaneously. Both the exact and inexact versions of the new method are presented in a unified framework. Under mild conditions, we manage to prove the global convergence of the algorithm. Preliminary numerical experiments for extracting the background from corrupted surveillance video verify the encouraging efficiency of the new algorithm.

The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desir...