  In this paper we propose a proximal algorithm for minimizing an objective
function of two block variables consisting of three terms: 1) a smooth
function, 2) a nonsmooth function which is a composition between a strictly
increasing, concave, differentiable function and a convex nonsmooth function,
and 3) a smooth function which couples the two block variables. We propose a
variable metric composite proximal alternating linearized minimization (CPALM)
to solve this class of problems. Building on the powerful Kurdyka-\L ojasiewicz
property, we derive the convergence analysis and establish that each bounded
sequence generated by CPALM globally converges to a critical point. We
demonstrate the CPALM method on parallel magnetic resonance image
reconstruction problems. The obtained numerical results shows the viability and
effectiveness of the proposed method.
