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The alternating direction method of multipliers (ADMM) proposed by Glowinskiand Marrocco is a benchmark algorithm for two-block separable convexoptimization problems with linear equality constraints. It has been modified,specified, and generalized from various perspectives to tackle more concrete orcomplicated application problems. Despite its versatility and phenomenalpopularity, it remains unknown whether or not the ADMM can be extended toseparable convex optimization problems with linear inequality constraints. Inthis paper, we lay down the foundation of how to extend the ADMM to two-blockand multiple-block (more than two blocks) separable convex optimizationproblems with linear inequality constraints. From a high-level andmethodological perspective, we propose a unified framework of algorithmicdesign and a roadmap for convergence analysis in the context of variationalinequalities, based on which it is possible to design a series of concreteADMM-based algorithms with provable convergence in the prediction-correctionstructure. The proposed algorithmic framework and roadmap for convergenceanalysis are eligible to various convex optimization problems with differentdegrees of separability, in which both linear equality and linear inequalityconstraints can be included. The analysis is comprehensive yet can be presentedby elementary mathematics, and hence generically understandable.

Extensions of ADMM for Separable Convex Optimization Problems with Linear Equality or Inequality Constraints