Analysis of Fully Preconditioned Alternating Direction Method of Multipliers with Relaxation in Hilbert Spaces

Alternating direction method of multipliers is a powerful first-order method for non-smooth optimization problems including various applications in inverse problems and imaging. However, there is no clear result on the weak convergence of alternating direction method of multipliers in infinite-dimensional Hilbert spaces with relaxation. In this paper, by employing a kind of partial gap analysis, we prove the weak convergence of a general preconditioned and relaxed version in infinite-dimensional Hilbert spaces, with preconditioning for solving all the involved implicit equations under mild conditions. We also give the corresponding ergodic convergence rates respecting to the partial gap function. Furthermore, the connections between certain preconditioned and relaxed alternating direction method of multipliers and the corresponding Douglas-Rachford splitting methods are discussed. Numerical tests show the efficiency of the proposed overrelaxation variants with preconditioning.