Linearized Alternating Direction Method of Multipliers with Adaptive Stepsize

The linearized alternating direction method of multipliers is a versatile tool for solving a wide range of constrained optimization problems. However, its performance depends strongly on the user-defined linearization parameter. We study the linearized alternating direction method of multipliers which boosts performance by adaptively tuning the linearization parameter to achieve fast convergence without user oversight. We provide theoretical conditions that guarantee convergence of the proposed algorithm under very general circumstances and present a worst-case $$\mathcal {O}(1/k)$$ convergence rate result in both ergodic and non-ergodic senses for the proposed algorithm, where k is the iteration counter. Numerical results on several applications in medical image processing demonstrate its fast practical convergence.