A second-order dynamical approach with variable damping to nonconvex smooth minimization

ABSTRACT

We investigate a second-order dynamical system with variable damping in connection with the minimization of a nonconvex differentiable function. The dynamical system is formulated in the spirit of the differential equation which models Nesterov's accelerated convex gradient method. We show that the generated trajectory converges to a critical point, if a regularization of the objective function satisfies the Kurdyka- Lojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Lojasiewicz exponent.

COMMUNICATED BY:

• Boris Mordukhovich

KEYWORDS:

• Second-order dynamical system
• nonconvex optimization
• Kurdyka–Łojasiewicz inequality
• convergence rate

AMS SUBJECT CLASSIFICATIONS:

• 90C26
• 90C30
• 65K10

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Erratum

Acknowledgements

The authors are thankful to an anonymous reviewer for comments and remarks which were helpful to gain a better insight into the asymptotic behaviour of the trajectories of the studied dynamical system.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Radu Ioan Bo http://orcid.org/0000-0002-4469-314X

Additional information

Funding

Radu Ioan Bo's research was partially supported by FWF (Austrian Science Fund) [project I 2419-N32] ; Ernö Robert Csetnek's research was supported by FWF (Austrian Science Fund) [project P 29809-N32] and Szilárd Csaba László's research was supported by a grant of Ministry of Research and Innovation - Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii (CNCS-UEFISCDI) [project number PN-III-P1-1.1-TE-2016-0266] within PNCDI III.