A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique

In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the M...

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Vydáno v:Computational optimization and applications Ročník 87; číslo 2; s. 531 - 569
Hlavní autor: Karapetyants, Mikhail A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.03.2024
Springer Nature B.V
Témata:
Convergence
Convex analysis
Convex and Discrete Geometry
Convexity
Damping
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Regularization
Statistics
Strong convergence
65K05
Nonsmooth convex optimization
Moreau envelope
Proximal operator
Hessian-driven damping
Tikhonov regularization
37N40
90C25
Damped inertial dynamics
65K10
46N10
49M99
ISSN:0926-6003, 1573-2894
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Popis
Shrnutí:In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-023-00536-6