Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution
In a Hilbertian framework, for the minimization of a general convex differentiable function f , we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of f with minimum norm. Our study is based on the non-autonomous versio...
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| Vydáno v: | Mathematical methods of operations research (Heidelberg, Germany) Ročník 99; číslo 3; s. 307 - 347 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2024
Springer Nature B.V Springer Verlag |
| Témata: | |
| ISSN: | 1432-2994, 1432-5217 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | In a Hilbertian framework, for the minimization of a general convex differentiable function
f
, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of
f
with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time
t
, is associated with the strongly convex function obtained by adding to
f
a Tikhonov regularization term with vanishing coefficient
ε
(
t
)
. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter
ε
(
t
)
. By adjusting the speed of convergence of
ε
(
t
)
towards zero, we will obtain both rapid convergence towards the infimal value of
f
, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of
f
. In particular, we obtain an improved version of the dynamic of Su-Boyd-Candès for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function
f
, we study the proximal algorithms in detail, and show that they benefit from similar properties. |
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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1432-2994 1432-5217 |
| DOI: | 10.1007/s00186-024-00867-y |