A block coordinate variable metric forward-backward algorithm

A number of recent works have emphasized the prominent role played by the Kurdyka-Aojasiewicz inequality for proving the convergence of iterative algorithms solving possibly nonsmooth/nonconvex optimization problems. In this work, we consider the minimization of an objective function satisfying this property, which is a sum of two terms: (i) a differentiable, but not necessarily convex, function and (ii) a function that is not necessarily convex, nor necessarily differentiable. The latter function is expressed as a separable sum of functions of blocks of variables. Such an optimization problem can be addressed with the Forward-Backward algorithm which can be accelerated thanks to the use of variable metrics derived from the Majorize-Minimize principle. We propose to combine the latter acceleration technique with an alternating minimization strategy which relies upon a flexible update rule. We give conditions under which the sequence generated by the resulting Block Coordinate Variable Metric Forward-Backward algorithm converges to a critical point of the objective function. An application example to a nonconvex phase retrieval problem encountered in signal/image processing shows the efficiency of the proposed optimization method.