---
_id: '8077'
abstract:
- lang: eng
  text: The projection methods with vanilla inertial extrapolation step for variational
    inequalities have been of interest to many authors recently due to the improved
    convergence speed contributed by the presence of inertial extrapolation step.
    However, it is discovered that these projection methods with inertial steps lose
    the Fejér monotonicity of the iterates with respect to the solution, which is
    being enjoyed by their corresponding non-inertial projection methods for variational
    inequalities. This lack of Fejér monotonicity makes projection methods with vanilla
    inertial extrapolation step for variational inequalities not to converge faster
    than their corresponding non-inertial projection methods at times. Also, it has
    recently been proved that the projection methods with vanilla inertial extrapolation
    step may provide convergence rates that are worse than the classical projected
    gradient methods for strongly convex functions. In this paper, we introduce projection
    methods with alternated inertial extrapolation step for solving variational inequalities.
    We show that the sequence of iterates generated by our methods converges weakly
    to a solution of the variational inequality under some appropriate conditions.
    The Fejér monotonicity of even subsequence is recovered in these methods and linear
    rate of convergence is obtained. The numerical implementations of our methods
    compared with some other inertial projection methods show that our method is more
    efficient and outperforms some of these inertial projection methods.
acknowledgement: The authors are grateful to the two anonymous referees for their
  insightful comments and suggestions which have improved the earlier version of the
  manuscript greatly. The first author has received funding from the European Research
  Council (ERC) under the European Union Seventh Framework Programme (FP7 - 2007-2013)
  (Grant agreement No. 616160).
article_processing_charge: No
article_type: original
author:
- first_name: Yekini
  full_name: Shehu, Yekini
  id: 3FC7CB58-F248-11E8-B48F-1D18A9856A87
  last_name: Shehu
  orcid: 0000-0001-9224-7139
- first_name: Olaniyi S.
  full_name: Iyiola, Olaniyi S.
  last_name: Iyiola
citation:
  ama: 'Shehu Y, Iyiola OS. Projection methods with alternating inertial steps for
    variational inequalities: Weak and linear convergence. <i>Applied Numerical Mathematics</i>.
    2020;157:315-337. doi:<a href="https://doi.org/10.1016/j.apnum.2020.06.009">10.1016/j.apnum.2020.06.009</a>'
  apa: 'Shehu, Y., &#38; Iyiola, O. S. (2020). Projection methods with alternating
    inertial steps for variational inequalities: Weak and linear convergence. <i>Applied
    Numerical Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/j.apnum.2020.06.009">https://doi.org/10.1016/j.apnum.2020.06.009</a>'
  chicago: 'Shehu, Yekini, and Olaniyi S. Iyiola. “Projection Methods with Alternating
    Inertial Steps for Variational Inequalities: Weak and Linear Convergence.” <i>Applied
    Numerical Mathematics</i>. Elsevier, 2020. <a href="https://doi.org/10.1016/j.apnum.2020.06.009">https://doi.org/10.1016/j.apnum.2020.06.009</a>.'
  ieee: 'Y. Shehu and O. S. Iyiola, “Projection methods with alternating inertial
    steps for variational inequalities: Weak and linear convergence,” <i>Applied Numerical
    Mathematics</i>, vol. 157. Elsevier, pp. 315–337, 2020.'
  ista: 'Shehu Y, Iyiola OS. 2020. Projection methods with alternating inertial steps
    for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics.
    157, 315–337.'
  mla: 'Shehu, Yekini, and Olaniyi S. Iyiola. “Projection Methods with Alternating
    Inertial Steps for Variational Inequalities: Weak and Linear Convergence.” <i>Applied
    Numerical Mathematics</i>, vol. 157, Elsevier, 2020, pp. 315–37, doi:<a href="https://doi.org/10.1016/j.apnum.2020.06.009">10.1016/j.apnum.2020.06.009</a>.'
  short: Y. Shehu, O.S. Iyiola, Applied Numerical Mathematics 157 (2020) 315–337.
date_created: 2020-07-02T09:02:33Z
date_published: 2020-11-01T00:00:00Z
date_updated: 2023-08-22T07:50:43Z
day: '01'
ddc:
- '510'
department:
- _id: VlKo
doi: 10.1016/j.apnum.2020.06.009
ec_funded: 1
external_id:
  isi:
  - '000564648400018'
file:
- access_level: open_access
  checksum: 87d81324a62c82baa925c009dfcb0200
  content_type: application/pdf
  creator: dernst
  date_created: 2020-07-02T09:08:59Z
  date_updated: 2020-07-14T12:48:09Z
  file_id: '8078'
  file_name: 2020_AppliedNumericalMath_Shehu.pdf
  file_size: 2874203
  relation: main_file
file_date_updated: 2020-07-14T12:48:09Z
has_accepted_license: '1'
intvolume: '       157'
isi: 1
language:
- iso: eng
month: '11'
oa: 1
oa_version: Submitted Version
page: 315-337
project:
- _id: 25FBA906-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '616160'
  name: 'Discrete Optimization in Computer Vision: Theory and Practice'
publication: Applied Numerical Mathematics
publication_identifier:
  issn:
  - 0168-9274
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Projection methods with alternating inertial steps for variational inequalities:
  Weak and linear convergence'
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 157
year: '2020'
...