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This work investigates a Bregman and inertial extension of theforward-reflected-backward algorithm [Y. Malitsky and M. Tam, SIAM J. Optim.,30 (2020), pp. 1451--1472] applied to structured nonconvex minimizationproblems under relative smoothness. To this end, the proposed algorithm hingeson two key features: taking inertial steps in the dual space, and allowing forpossibly negative inertial values. Our analysis begins with studying anassociated envelope function that takes inertial terms into account through anovel product space formulation. Such construction substantially differs fromsimilar objects in the literature and could offer new insights for extensionsof splitting algorithms. Global convergence and rates are obtained by appealingto the generalized concave Kurdyka-Lojasiewicz (KL) property, which allows usto describe a sharp upper bound on the total length of iterates. Finally, alinesearch extension is given to enhance the proposed method.

A mirror inertial forward-reflected-backward splitting: Global convergence and linesearch extension beyond convexity and Lipschitz smoothness