Inexact half-quadratic optimization for image reconstruction

We present new global convergence results for half-quadratic optimization in the context of image reconstruction. In particular, we do not assume that the inner optimization problem is solved exactly and we include the problematic cases where the objective function is nonconvex and has a continuum of stationary points. The inexact algorithm is modeled by a set-valued map defined from the majorization-minimization interpretation of half-quadratic optimization, and our main convergence results are based on the Kurdyka-Lojasiewicz inequality. We also propose a practical implementation that uses the conjugate gradient method and whose efficiency is illustrated by numerical experiments.