  This paper develops an adaptive Proximal Alternating Direction Method of
Multipliers (P-ADMM) for solving linearly-constrained, weakly convex, composite
optimization problems. This method is adaptive to all problem parameters,
including smoothness and weak convexity constants. It is assumed that the
smooth component of the objective is weakly convex and possibly nonseparable,
while the non-smooth component is convex and block-separable. The proposed
method is tolerant to the inexact solution of its block proximal subproblem so
it does not require that the non-smooth component has easily computable block
proximal maps. Each iteration of our adaptive P-ADMM consists of two steps: (1)
the sequential solution of each block proximal subproblem, and (2) adaptive
tests to decide whether to perform a full Lagrange multiplier and/or penalty
parameter update(s). Without any rank assumptions on the constraint matrices,
it is shown that the adaptive P-ADMM obtains an approximate first-order
stationary point of the constrained problem in a number of iterations that
matches the state-of-the-art complexity for the class of P-ADMMs. The two
proof-of-concept numerical experiments that conclude the paper suggest our
adaptive P-ADMM enjoys significant computational benefits.
