A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem


We investigate an algorithm of gradient type with a backward inertial step in connection with the minimization of a nonconvex differentiable function. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent. Finally, some numerical experiments are presented in order to compare our numerical scheme with some algorithms well known in the literature.

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The authors are thankful to three anonymous reviewers for remarks and suggestions which helped us to improve the quality of the paper.

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Correspondence to Szilárd Csaba László.

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This work was supported by a grant of Ministry of Research and Innovation, CNCS -UEFISCDI, project number PN-III-P1-1.1-TE-2016-0266, within PNCDI III.

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Alecsa, C.D., László, S.C. & Viorel, A. A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem. Numer Algor 84, 485–512 (2020). https://doi.org/10.1007/s11075-019-00765-z

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  • Inertial algorithm
  • Nonconvex optimization
  • Kurdyka-Łojasiewicz inequality
  • Convergence rate

Mathematics Subject Classification (2010)

  • 90C26
  • 90C30
  • 65K10