Variance-Based Subgradient Extragradient Method for Stochastic Variational Inequality Problems

In this paper, we propose a variance-based subgradient extragradient algorithm with line search for stochastic variational inequality problems by aiming at robustness with respect to an unknown Lipschitz constant. This algorithm may be regarded as an integration of a subgradient extragradient algori...

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Veröffentlicht in:	Journal of scientific computing Jg. 89; H. 1; S. 4
Hauptverfasser:	Yang, Zhen-Ping, Zhang, Jin, Wang, Yuliang, Lin, Gui-Hua
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.10.2021 Springer Nature B.V
Schlagworte:	Algebra Algorithms Approximation Computational Mathematics and Numerical Analysis Convergence Iterative methods Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Numerical analysis Random variables Regularization methods Robustness (mathematics) Theoretical Stochastic variational inequality 65K15 Bounded proximal error bound 90C33 Variance reduction Convergence rate Stochastic approximation 90C15 Subgradient extragradient algorithm 62L20
ISSN:	0885-7474, 1573-7691
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Abstract	In this paper, we propose a variance-based subgradient extragradient algorithm with line search for stochastic variational inequality problems by aiming at robustness with respect to an unknown Lipschitz constant. This algorithm may be regarded as an integration of a subgradient extragradient algorithm for deterministic variational inequality problems and a stochastic approximation method for expected values. At each iteration, different from the conventional variance-based extragradient algorithms to take projection onto the feasible set twicely, our algorithm conducts a subgradient projection which can be calculated explicitly. Since our algorithm requires only one projection at each iteration, the computation load may be reduced. We discuss the asymptotic convergence, the sublinear convergence rate in terms of the mean natural residual function, and the optimal oracle complexity for the proposed algorithm. Furthermore, we establish the linear convergence rate with finite computational budget under both the strongly Minty variational inequality and the error bound condition. Preliminary numerical experiments indicate that the proposed algorithm is competitive with some existing methods.
AbstractList	In this paper, we propose a variance-based subgradient extragradient algorithm with line search for stochastic variational inequality problems by aiming at robustness with respect to an unknown Lipschitz constant. This algorithm may be regarded as an integration of a subgradient extragradient algorithm for deterministic variational inequality problems and a stochastic approximation method for expected values. At each iteration, different from the conventional variance-based extragradient algorithms to take projection onto the feasible set twicely, our algorithm conducts a subgradient projection which can be calculated explicitly. Since our algorithm requires only one projection at each iteration, the computation load may be reduced. We discuss the asymptotic convergence, the sublinear convergence rate in terms of the mean natural residual function, and the optimal oracle complexity for the proposed algorithm. Furthermore, we establish the linear convergence rate with finite computational budget under both the strongly Minty variational inequality and the error bound condition. Preliminary numerical experiments indicate that the proposed algorithm is competitive with some existing methods.
ArticleNumber	4
Author	Zhang, Jin Yang, Zhen-Ping Lin, Gui-Hua Wang, Yuliang
Author_xml	– sequence: 1 givenname: Zhen-Ping surname: Yang fullname: Yang, Zhen-Ping organization: School of Mathematics, Jiaying University, School of Management, Shanghai University – sequence: 2 givenname: Jin surname: Zhang fullname: Zhang, Jin email: zhangj9@sustech.edu.cn organization: Department of Mathematics, Southern University of Science and Technology, National Center for Applied Mathematics Shenzhen – sequence: 3 givenname: Yuliang surname: Wang fullname: Wang, Yuliang organization: Research Center for Mathematics, Beijing Normal University at Zhuhai, Division of Science and Technology, BNU-HKBU United International College – sequence: 4 givenname: Gui-Hua surname: Lin fullname: Lin, Gui-Hua email: guihualin@shu.edu.cn organization: School of Management, Shanghai University
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Keywords	Stochastic variational inequality 65K15 Bounded proximal error bound 90C33 Variance reduction Convergence rate Stochastic approximation 90C15 Subgradient extragradient algorithm 62L20
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Snippet	In this paper, we propose a variance-based subgradient extragradient algorithm with line search for stochastic variational inequality problems by aiming at...
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SubjectTerms	Algebra Algorithms Approximation Computational Mathematics and Numerical Analysis Convergence Iterative methods Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Numerical analysis Random variables Regularization methods Robustness (mathematics) Theoretical
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Title	Variance-Based Subgradient Extragradient Method for Stochastic Variational Inequality Problems
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