A Self-Adaptive Alternating Direction Multiplier Method for Variational Inequality in 2 Domains

YUAN Xingyue; CUI Xiyong; RAN Ruisheng; ZHANG Shougui

doi:10.21656/1000-0887.450171

Volume 46 Issue 7

Jul. 2025

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(7): 916-925

YUAN Xingyue, CUI Xiyong, RAN Ruisheng, ZHANG Shougui. A Self-Adaptive Alternating Direction Multiplier Method for Variational Inequality in 2 Domains[J]. Applied Mathematics and Mechanics, 2025, 46(7): 916-925. doi: 10.21656/1000-0887.450171

Citation:

PDF( 734 KB)

A Self-Adaptive Alternating Direction Multiplier Method for Variational Inequality in 2 Domains

doi: 10.21656/1000-0887.450171

1.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R.China
2.
CISDI Information Technology (Chongqing) Co., Ltd., Chongqing 401120, P.R.China
3.
School of Computer and Information Science, Chongqing Normal University, Chongqing 401331, P.R.China

Received Date: 2024-06-11
Rev Recd Date: 2024-11-09

Available Online: 2025-07-30

Publish Date: 2025-07-01

Abstract

Abstract

A self-adaptive alternating direction multiplier method was proposed for a class of contact problems defined in 2 domains. A minimization problem constrained by inequalities was obtained for the variational problem in 2 domains, then the problem was equivalent to a saddle-point problem through introduction of an auxiliary unknown on the contact boundary. The alternating direction multiplier method was applied to the saddle point problem for the numerical solution, with each iteration successively determining the auxiliary variable explicitly, solving a linear problem and updating the Lagrange multiplier. The self-adaptive alternating direction multiplier method was proposed to select the penalty parameter automatically, by means of a self-adaptive rule and iterative functions. The results prove the convergence and demonstrate the effectiveness of the proposed method.
- variational inequality,
- saddle point problem,
- alternating direction multiplier method,
- self-adaptive penalty parameter

FullText(HTML)

References(20)

References

[1]	TANG W J, HU H Y, SHEN L J. On the coupling of boundary integral and finite element methods for Signorini problems[J]. Journal of Computational Mathematics, 1998, 16 (6): 561-570.
[2]	DOSTÁL Z, NETO F A M G G, SANTOS S A. Duality-based domain decomposition with natural coarse-space for variational inequalities[J]. Journal of Computational and Applied Mathematics, 2000, 126 (1/2): 397-415.
[3]	DOSTÁL Z, HORÁK D. Theoretically supported scalable FETI for numerical solution of variational inequalities[J]. SIAM Journal on Numerical Analysis, 2007, 45 (2): 500-513.
[4]	PENG Z Y, LI D, ZHAO Y, et al. An accelerated subgradient extragradient algorithm for solving bilevel variational inequality problems involving non-Lipschitz operator[J]. Communications in Nonlinear Science and Numerical Simulation, 2023, 127 : 107549.
[5]	PENG Z Y, PENG Z Y, CAI G, et al. Inertial subgradient extragradient method for solving pseudomonotone variational inequality problems in Banach spaces[J]. Applicable Analysis, 2024, 103 (10): 1769-1789.
[6]	DOSTÁL Z, HORÁK D, STEFANICA D. A scalable FETI-DP algorithm for a semi-coercive variational inequality[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 196 (8): 1369-1379.
[7]	DOSTÁL Z, HORÁK D, STEFANICA D. A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface[J]. Journal of Computational and Applied Mathematics, 2009, 231 (2): 577-591.
[8]	DOSTÁL Z, HORÁK D, KRUŽÍK J, et al. Highly scalable hybrid domain decomposition method for the solution of huge scalar variational inequalities[J]. Numerical Algorithms, 2022, 91 (2): 773-801.
[9]	LEE J. Two domain decomposition methods for auxiliary linear problems of a multibody elliptic variational inequality[J]. SIAM Journal on Scientific Computing, 2013, 35 (3): A1350-A1375.
[10]	BOUCHALA J, DOSTÁL Z, SADOWSKÁ M. Theoretically supported scalable BETI method for variational inequalities[J]. Computing, 2008, 82 (1): 53-75.
[11]	IKRAM B, SAMIRA S. Generalized Schwarz algorithm for a class of variational inequalities[J]. Applied Mathematics & Information Sciences, 2021, 15 (1): 9-15.
[12]	CHORFI A, KOKO J. Alternating direction method of multiplier for the unilateral contact problem with an automatic penalty parameter selection[J]. Applied Mathematical Modelling, 2020, 78 : 706-723.
[13]	KOKO J. Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem[J]. Applied Mathematics Letters, 2009, 22 (10): 1534-1538.
[14]	ZHANG S G, GUO N X. Uzawa block relaxation method for free boundary problem with unilateral obstacle[J]. International Journal of Computer Mathematics, 2021, 98 (4): 671-689.
[15]	GLOWINSKI R. Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems[M]. Philadelphia: Society for Industrial and Applied Mathematics, 2015.
[16]	王霄婷, 龙宪军, 彭再云. 求解非单调变分不等式的一种二次投影算法[J]. 应用数学和力学, 2022, 43 (8): 927-934. doi: 10.21656/1000-0887.420414 WANG Xiaoting, LONG Xianjun, PENG Zaiyun. A double projection algorithm for solving non-monotone variational inequalities[J]. Applied Mathematics and Mechanics, 2022, 43 (8): 927-934. (in Chinese) doi: 10.21656/1000-0887.420414
[17]	张霖森, 程兰, 张守贵. 曲率障碍下四阶变分不等式的交替方向乘子法[J]. 应用数学和力学, 2023, 44 (5): 595-604. doi: 10.21656/1000-0887.430243 ZHANG Linsen, CHENG Lan, ZHANG Shougui. An alternating direction multiplier method for 4th-order variational inequalities with curvature obstacle[J]. Applied Mathematics and Mechanics, 2023, 44 (5): 595-604. (in Chinese) doi: 10.21656/1000-0887.430243
[18]	袁欣, 张守贵. 无摩擦弹性接触问题的自适应交替方向乘子法[J]. 应用数学和力学, 2023, 44 (8): 989-998. doi: 10.21656/1000-0887.440079 YUAN Xin, ZHANG Shougui. A self-adaptive alternating direction multiplier method for frictionless elastic contact problems[J]. Applied Mathematics and Mechanics, 2023, 44 (8): 989-998. (in Chinese) doi: 10.21656/1000-0887.440079
[19]	王欣, 郭科. 一类非凸优化问题广义交替方向法的收敛性[J]. 应用数学和力学, 2018, 39 (12): 1410-1425. WANG Xin, GUO Ke. Convergence of the generalized alternating direction method of multipliers for a class of nonconvex optimization problems[J]. Applied Mathematics and Mechanics, 2018, 39 (12): 1410-1425. (in Chinese)
[20]	ZHANG S G, YAN Y Y, RAN R S. Path-following and semismooth Newton methods for the variational inequality arising from two membranes problem[J]. Journal of Inequalities and Applications, 2019, 2019 (1): 1.