  The alternating direction method of multipliers (ADMM) is a powerful
splitting algorithm for linearly constrained convex optimization problems. In
view of its popularity and applicability, a growing attention is drawn towards
the ADMM in nonconvex settings. Recent studies of minimization problems for
noncovex functions include various combinations of assumptions on the objective
function including, in particular, a Lipschitz gradient assumption. We consider
the case where the objective is the sum of a strongly convex function and a
weakly convex function. To this end we present and study an adaptive version of
the ADMM which incorporates generalized notions of convexity and varying
penalty parameters adapted to the convexity constants of the functions. We
prove convergence of the scheme under natural assumptions. To this end we
employ the recent adaptive Douglas--Rachford algorithm by revisiting the well
known duality relation between the classical ADMM and the Douglas--Rachford
splitting algorithm, generalizing this connection to our setting. We illustrate
our approach by relating and comparing to alternatives, and by numerical
experiments on a signal denoising problem.
