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

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: t...

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Vydáno v:Journal of optimization theory and applications Ročník 203; číslo 2; s. 1127 - 1159
Hlavní autoři: Wang, Ziyuan, Themelis, Andreas, Ou, Hongjia, Wang, Xianfu
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.11.2024
Témata:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Engineering
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theory of Computation
49J53
Relative smoothness
49J52
Forward–reflected–backward splitting
Inertia
Mordukhovich limiting subdifferential
90C26
Bregman distance
Nonsmooth nonconvex optimization
ISSN:0022-3239, 1573-2878
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Popis
Shrnutí: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-024-02383-9