# American Institute of Mathematical Sciences

• Previous Article
General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes
• JIMO Home
• This Issue
• Next Article
Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center
March  2022, 18(2): 773-794. doi: 10.3934/jimo.2020178

## Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems

 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 90 Medunsa 0204, South Africa

* Corresponding author: Lateef Olakunle Jolaoso

Received  March 2020 Revised  September 2020 Published  March 2022 Early access  December 2020

Using the concept of Bregman divergence, we propose a new subgradient extragradient method for approximating a common solution of pseudo-monotone and Lipschitz continuous variational inequalities and fixed point problem in real Hilbert spaces. The algorithm uses a new self-adjustment rule for selecting the stepsize in each iteration and also, we prove a strong convergence result for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. Finally, we provide some numerical examples to illustrate the performance and accuracy of our algorithm in finite and infinite dimensional spaces.

Citation: Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial and Management Optimization, 2022, 18 (2) : 773-794. doi: 10.3934/jimo.2020178
##### References:

show all references

##### References:
Example 1, Top Left: Case I; Top Right: Case II, Bottom Left: Case III, Bottom Right: Case IV
Example 2, Top Left: $m = 5$; Top Right: $m = 15$, Bottom: $m = 30$
Computation result for Example 1
 Algorithm 4 Algorithm 1 Algorithm 3 Case I Iter. 5 12 29 Time 0.6406 1.0043 0.7661 Case II Iter. 12 45 49 Time 3.0910 9.5282 3.3343 Case III Iter. 10 22 39 Time 1.1391 3.0101 1.7377 Case IV Iter. 13 56 53 Time 0.8596 3.9885 1.8918
 Algorithm 4 Algorithm 1 Algorithm 3 Case I Iter. 5 12 29 Time 0.6406 1.0043 0.7661 Case II Iter. 12 45 49 Time 3.0910 9.5282 3.3343 Case III Iter. 10 22 39 Time 1.1391 3.0101 1.7377 Case IV Iter. 13 56 53 Time 0.8596 3.9885 1.8918
Computation result for Example 2
 Algorithm 4 Algorithm 3 $m=5$ Iter. 7 11 Time 0.0036 0.0050 $m=15$ Iter. 8 13 Time 0.0052 0.0099 $m=30$ Iter. 8 27 Time 0.0255 0.0884
 Algorithm 4 Algorithm 3 $m=5$ Iter. 7 11 Time 0.0036 0.0050 $m=15$ Iter. 8 13 Time 0.0052 0.0099 $m=30$ Iter. 8 27 Time 0.0255 0.0884
 [1] Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial and Management Optimization, 2022, 18 (1) : 239-265. doi: 10.3934/jimo.2020152 [2] Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021046 [3] Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004 [4] Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043 [5] Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043 [6] Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022060 [7] Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022007 [8] Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial and Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045 [9] Nicholas Long. Fixed point shifts of inert involutions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [10] Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022045 [11] Raphaël Danchin, Piotr B. Mucha. Divergence. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1163-1172. doi: 10.3934/dcdss.2013.6.1163 [12] M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819 [13] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [14] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [15] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [16] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [17] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [18] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [19] Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021095 [20] Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 373-393. doi: 10.3934/naco.2021011

2020 Impact Factor: 1.801