A Mirror Inertial Forward–Reflected–Backward Splitting: Convergence Analysis Beyond Convexity and Lipschitz Smoothness

Ziyuan Wang (), Andreas Themelis (), Hongjia Ou () and Xianfu Wang ()
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Ziyuan Wang: University of British Columbia
Andreas Themelis: Kyushu University
Hongjia Ou: Kyushu University
Xianfu Wang: University of British Columbia

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 2, No 3, 1127-1159

Abstract: Abstract This work investigates a Bregman and inertial extension of the forward–reflected–backward algorithm (Malitsky and Tam in SIAM J Optim 30:1451–1472, 2020) applied to structured nonconvex minimization problems under relative smoothness. To this end, the proposed algorithm hinges on two key features: taking inertial steps in the dual space, and allowing for possibly negative inertial values. The interpretation of relative smoothness as a two-sided weak convexity condition proves beneficial in providing tighter stepsize ranges. Our analysis begins with studying an envelope function associated with the algorithm that takes inertial terms into account through a novel product space formulation. Such construction substantially differs from similar objects in the literature and could offer new insights for extensions of splitting algorithms. Global convergence and rates are obtained by appealing to the Kurdyka–Łojasiewicz property.

Keywords: Nonsmooth nonconvex optimization; Forward–reflected–backward splitting; Inertia; Bregman distance; Mordukhovich limiting subdifferential; Relative smoothness; 90C26; 49J52; 49J53 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02383-9

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