Improved Pointwise Iteration-Complexity of A Regularized ADMM and of a Regularized Non-Euclidean HPE Framework

This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity of the latter method. Our analysis is based on first presenting and establishing the pointwise iteration-complexity of a regularized non-Euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error. It is then shown that the new method is a special instance of the latter framework where the sequence of summable errors is identically zero when the ADMM stepsize is less than one or a nontrivial sequence when the stepsize is in the interval [1, (1 + √ 5)/2). 2000 Mathematics Subject Classification: 47H05, 47J22, 49M27, 90C25, 90C30, 90C60, 65K10.