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Title
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High-order methods beyond the classical complexity bounds : inexact high-order proximal-point methods
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Author
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Abstract
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| We introduce a Bi-level OPTimization (BiOPT) framework for minimizing the sum of two convex functions, where one of them is smooth enough. The BiOPT framework offers three levels of freedom: (i) choosing the order p of the proximal term; (ii) designing an inexact pth-order proximal-point method in the upper level; (iii) solving the auxiliary problem with a lower-level non-Euclidean method in the lower level. We here regularize the objective by a th-order proximal term (for arbitrary integer ) and then develop the generic inexact high-order proximal-point scheme and its acceleration using the standard estimating sequence technique at the upper level. This follows at the lower level with solving the corresponding pth-order proximal auxiliary problem inexactly either by one iteration of the pth-order tensor method or by a lower-order non-Euclidean composite gradient scheme. Ultimately, it is shown that applying the accelerated inexact pth-order proximal-point method at the upper level and handling the auxiliary problem by the non-Euclidean composite gradient scheme lead to a 2q-order method with the convergence rate (for and the iteration counter k), which can result to a superfast method for some specific class of problems. |
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Language
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English
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Source (journal)
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Mathematical programming. - Amsterdam, 1971, currens
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Publication
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Heidelberg
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Springer heidelberg
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2024
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ISSN
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0025-5610
[print]
1436-4646
[online]
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DOI
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10.1007/S10107-023-02041-4
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Volume/pages
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208
:1-2
(2024)
, p. 365-407
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ISI
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001136171900002
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Full text (Publisher's DOI)
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Full text (open access)
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