Convergence analysis of block majorize-minimize subspace approach

We consider the minimization of a differentiable Lipschitz gradient but non necessarily convex, function F defined on $${\mathbb {R}}^N$$ . We propose an accelerated gradient descent approach which combines three strategies, namely (i) a variable metric derived from the majorization-minimization principle; (ii) a subspace strategy incorporating information from the past iterates; (iii) a block alternating update. Under the assumption that F satisfies the Kurdyka–Łojasiewicz property, we give conditions under which the sequence generated by the resulting block majorize-minimize subspace algorithm converges to a critical point of the objective function, and we exhibit convergence rates for its iterates.