Our contribution deals with image restoration. The adopted approach consists in minimizing a penalized least squares (PLS) criterion. Here, we are interested in the search of efficient algorithms to carry out such a task. The minimization of PLS criteria can be addressed using a half-quadratic approach (HQ). However, the nontrivial inversion of a linear system is needed at each iteration. In practice, it is often proposed to approximate this inversion using a truncated preconditioned conjugate gradient (PCG) method. However, we point out that theoretical convergence is not proved for such approximate HQ algorithms, referred here as HQ+PCG. In the proposed contribution, we rely on a different scheme, also based on PCG and HQ ingredients and referred as PCG+HQ1D. General linesearch methods ensuring convergence of PCG type algorithms are difficult to code and
to tune. Therefore, we propose to replace the linesearch step by a truncated scalar HQ algorithm. Convergence is established for any finite number of HQ1D sub-iterations. Compared to the HQ+PCG approach, we show that our scheme is preferable on both the theoretical and practical grounds.