Outer approximated projection and contraction method for solving variational inequalities

V. A. Uzor; O. T. Mewomo; T. O. Alakoya; A. Gibali

doi:10.1186/s13660-023-03043-8

Outer approximated projection and contraction method for solving variational inequalities

V. A. Uzor, O. T. Mewomo^*, T. O. Alakoya, A. Gibali

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

18 Downloads (Pure)

Abstract

In this paper we focus on solving the classical variational inequality (VI) problem. Most common methods for solving VIs use some kind of projection onto the associated feasible set. Thus, when the involved set is not simple to project onto, then the applicability and computational effort of the proposed method could be arguable. One such scenario is when the given set is represented as a finite intersection of sublevel sets of convex functions. In this work we develop an outer approximation method that replaces the projection onto the VI’s feasible set by a simple, closed formula projection onto some “superset”. The proposed method also combines several known ideas such as the inertial technique and self-adaptive step size. Under standard assumptions, a strong minimum-norm convergence is proved and several numerical experiments validate and exhibit the performance of our scheme.

Original language	English
Article number	141
Number of pages	28
Journal	Journal of Inequalities and Applications
Volume	2023
DOIs	https://doi.org/10.1186/s13660-023-03043-8
Publication status	Published - 02 Nov 2023
Externally published	Yes

Bibliographical note

Funding Information:
The first author is funded by International Mathematical Union Breakout Graduate Fellowship (IMU-BGF). The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903) and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa (Grant Number 2022-087-OPA). The third author is funded by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship.

Funding Information:
The authors sincerely thank the anonymous referees for their careful reading, constructive comments, and useful suggestions that improved the manuscript. The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903) and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa (Grant Number 2022-087-OPA). The third author is wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS and NRF.

Publisher Copyright:
© 2023, The Author(s).

Keywords

Inertial technique
Non-Lipschitz variational inequalities
Projection and contraction method
Relaxation technique
Self-adaptive step size
Sublevel sets

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.1186/s13660-023-03043-8Licence: CC BY

Outer approximated projection and contraction method for solving variational inequalities
Copyright 2023 the authors. This is an open access article published under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited.
Final published version, 1.86 MBLicence: CC BY

Cite this

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title = "Outer approximated projection and contraction method for solving variational inequalities",

abstract = "In this paper we focus on solving the classical variational inequality (VI) problem. Most common methods for solving VIs use some kind of projection onto the associated feasible set. Thus, when the involved set is not simple to project onto, then the applicability and computational effort of the proposed method could be arguable. One such scenario is when the given set is represented as a finite intersection of sublevel sets of convex functions. In this work we develop an outer approximation method that replaces the projection onto the VI{\textquoteright}s feasible set by a simple, closed formula projection onto some “superset”. The proposed method also combines several known ideas such as the inertial technique and self-adaptive step size. Under standard assumptions, a strong minimum-norm convergence is proved and several numerical experiments validate and exhibit the performance of our scheme.",

keywords = "Inertial technique, Non-Lipschitz variational inequalities, Projection and contraction method, Relaxation technique, Self-adaptive step size, Sublevel sets",

author = "Uzor, {V. A.} and Mewomo, {O. T.} and Alakoya, {T. O.} and A. Gibali",

note = "Funding Information: The first author is funded by International Mathematical Union Breakout Graduate Fellowship (IMU-BGF). The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903) and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa (Grant Number 2022-087-OPA). The third author is funded by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. Funding Information: The authors sincerely thank the anonymous referees for their careful reading, constructive comments, and useful suggestions that improved the manuscript. The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903) and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa (Grant Number 2022-087-OPA). The third author is wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS and NRF. Publisher Copyright: {\textcopyright} 2023, The Author(s).",

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AU - Mewomo, O. T.

AU - Alakoya, T. O.

AU - Gibali, A.

N1 - Funding Information: The first author is funded by International Mathematical Union Breakout Graduate Fellowship (IMU-BGF). The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903) and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa (Grant Number 2022-087-OPA). The third author is funded by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. Funding Information: The authors sincerely thank the anonymous referees for their careful reading, constructive comments, and useful suggestions that improved the manuscript. The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903) and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa (Grant Number 2022-087-OPA). The third author is wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS and NRF. Publisher Copyright: © 2023, The Author(s).

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Y1 - 2023/11/2

N2 - In this paper we focus on solving the classical variational inequality (VI) problem. Most common methods for solving VIs use some kind of projection onto the associated feasible set. Thus, when the involved set is not simple to project onto, then the applicability and computational effort of the proposed method could be arguable. One such scenario is when the given set is represented as a finite intersection of sublevel sets of convex functions. In this work we develop an outer approximation method that replaces the projection onto the VI’s feasible set by a simple, closed formula projection onto some “superset”. The proposed method also combines several known ideas such as the inertial technique and self-adaptive step size. Under standard assumptions, a strong minimum-norm convergence is proved and several numerical experiments validate and exhibit the performance of our scheme.

AB - In this paper we focus on solving the classical variational inequality (VI) problem. Most common methods for solving VIs use some kind of projection onto the associated feasible set. Thus, when the involved set is not simple to project onto, then the applicability and computational effort of the proposed method could be arguable. One such scenario is when the given set is represented as a finite intersection of sublevel sets of convex functions. In this work we develop an outer approximation method that replaces the projection onto the VI’s feasible set by a simple, closed formula projection onto some “superset”. The proposed method also combines several known ideas such as the inertial technique and self-adaptive step size. Under standard assumptions, a strong minimum-norm convergence is proved and several numerical experiments validate and exhibit the performance of our scheme.

KW - Inertial technique

KW - Non-Lipschitz variational inequalities

KW - Projection and contraction method

KW - Relaxation technique

KW - Self-adaptive step size

KW - Sublevel sets

U2 - 10.1186/s13660-023-03043-8

DO - 10.1186/s13660-023-03043-8

M3 - Article

AN - SCOPUS:85175860120

SN - 1025-5834

VL - 2023

JO - Journal of Inequalities and Applications

JF - Journal of Inequalities and Applications

M1 - 141

ER -

Outer approximated projection and contraction method for solving variational inequalities

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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