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The Alternating Direction Method of Multipliers (ADMM) has gained a lot ofattention for solving large-scale and objective-separable constrainedoptimization. However, the two-block variable structure of the ADMM stilllimits the practical computational efficiency of the method, because one bigmatrix factorization is needed at least once even for linear and convexquadratic programming. This drawback may be overcome by enforcing a multi-blockstructure of the decision variables in the original optimization problem.Unfortunately, the multi-block ADMM, with more than two blocks, is notguaranteed to be convergent. On the other hand, two positive developments havebeen made: first, if in each cyclic loop one randomly permutes the updatingorder of the multiple blocks, then the method converges in expectation forsolving any system of linear equations with any number of blocks. Secondly,such a randomly permuted ADMM also works for equality-constrained convexquadratic programming even when the objective function is not separable. Thegoal of this paper is twofold. First, we add more randomness into the ADMM bydeveloping a randomly assembled cyclic ADMM (RAC-ADMM) where the decisionvariables in each block are randomly assembled. We discuss the theoreticalproperties of RAC-ADMM and show when random assembling helps and when it hurts,and develop a criterion to guarantee that it converges almost surely. Secondly,using the theoretical guidance on RAC-ADMM, we conduct multiple numerical testson solving both randomly generated and large-scale benchmark quadraticoptimization problems, which include continuous, and binary graph-partition andquadratic assignment, and selected machine learning problems. Our numericaltests show that the RAC-ADMM, with a variable-grouping strategy, couldsignificantly improve the computation efficiency on solving most quadraticoptimization problems.

Managing Randomization in the Multi-Block Alternating Direction Method of Multipliers for Quadratic Optimization