  The proximal alternating linearized minimization method (PALM) suits well for
solving block-structured optimization problems, which are ubiquitous in real
applications. In the cases where subproblems do not have closed-form solutions,
e.g., due to complex constraints, infeasible subsolvers are indispensable,
giving rise to an infeasible inexact PALM (PALM-I). Numerous efforts have been
devoted to analyzing feasible PALM, while little attention has been paid to
PALM-I. The usage of PALM-I thus lacks theoretical guarantee. The essential
difficulty of analyses consists in the objective value nonmonotonicity induced
by the infeasibility. We study in the present work the convergence properties
of PALM-I. In particular, we construct a surrogate sequence to surmount the
nonmonotonicity issue and devise an implementable inexact criterion. Based upon
these, we manage to establish the stationarity of any accumulation point and,
moreover, show the iterate convergence and the asymptotic convergence rates
under the assumption of the Lojasiewicz property. The prominent advantages of
PALM-I on CPU time are illustrated via numerical experiments on problems
arising from quantum physics and 3D anisotropic frictional contact.
