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 |
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01.11.2024
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| ISSN: | 0022-3239, 1573-2878 |
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Wang, Ziyuan Themelis, Andreas Ou, Hongjia Wang, Xianfu |
| Author_xml | – sequence: 1 givenname: Ziyuan surname: Wang fullname: Wang, Ziyuan organization: Department of Mathematics, Irving K. Barber Faculty of Science, University of British Columbia – sequence: 2 givenname: Andreas surname: Themelis fullname: Themelis, Andreas organization: Faculty of Information Science and Electrical Engineering (ISEE), Kyushu University – sequence: 3 givenname: Hongjia surname: Ou fullname: Ou, Hongjia organization: Faculty of Information Science and Electrical Engineering (ISEE), Kyushu University – sequence: 4 givenname: Xianfu surname: Wang fullname: Wang, Xianfu email: shawn.wang@ubc.ca organization: Department of Mathematics, Irving K. Barber Faculty of Science, University of British Columbia |
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| Keywords | 49J53 Relative smoothness 49J52 Forward–reflected–backward splitting Inertia Mordukhovich limiting subdifferential 90C26 Bregman distance Nonsmooth nonconvex optimization |
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| SubjectTerms | Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Engineering Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation |
| Title | A Mirror Inertial Forward–Reflected–Backward Splitting: Convergence Analysis Beyond Convexity and Lipschitz Smoothness |
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