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Title
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A bregman forward-backward linesearch algorithm for nonconvex composite optimization : superlinear convergence to nonisolated local minima
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Author
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Abstract
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| We introduce BELLA, a locally superlinearly convergent Bregman forward-backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfies a relative smoothness condition and the other one is possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first- and second-order properties, which enjoys a nonlinear error bound when the objective function satisfies a Lojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along user-defined update directions and converges subsequentially to stationary points and globally under the Kurdyka-Lojasiewicz condition. Moreover, when the update directions are superlinear in the sense of Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I, Springer, New York, 2003], owing to the given nonlinear error bound unit stepsize is eventually always accepted and the algorithm attains superlinear convergence rates even when the limit point is a nonisolated minimum. |
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Language
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English
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Source (journal)
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SIAM journal on optimization / Society for Industrial and Applied Mathematics [Philadelphia, Pa] - Philadelphia, Pa
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Publication
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Philadelphia, Pa
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2021
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ISSN
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1052-6234
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DOI
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10.1137/19M1264783
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Volume/pages
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31
:1
(2021)
, p. 653-685
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ISI
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000636678300026
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Full text (Publisher's DOI)
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Full text (open access)
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