A New Projection-type Method with Nondecreasing Adaptive Step-sizes for Pseudo-monotone Variational Inequalities

Duong Viet Thong (), Phan Tu Vuong (), Pham Ky Anh () and Le Dung Muu ()
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Duong Viet Thong: Thu Dau Mot University
Phan Tu Vuong: University of Southampton
Pham Ky Anh: Vietnam National University, Hanoi
Le Dung Muu: TIMAS, Thang Long University

Networks and Spatial Economics, 2022, vol. 22, issue 4, No 4, 803-829

Abstract: Abstract We propose a new projection-type method with inertial extrapolation for solving pseudo-monotone and Lipschitz continuous variational inequalities in Hilbert spaces. The proposed method does not require the knowledge of the Lipschitz constant as well as the sequential weak continuity of the corresponding operator. We introduce a self-adaptive procedure, which generates dynamic step-sizes converging to a positive constant. It is proved that the sequence generated by the proposed method converges weakly to a solution of the considered variational inequality with the nonasymptotic O(1/n) convergence rate. Moreover, the linear convergence is established under strong pseudo-monotonicity and Lipschitz continuity assumptions. Numerical a exmples for solving a class of Nash–Cournot oligopolistic market equilibrium model and a network equilibrium flow problem are given illustrating the efficiency of the proposed method.

Keywords: Variational inequality; Projection methods; Pseudo-monotonicity; Lipschitz continuity; Convergence rate; 65Y05; 65K15; 68W10; 47H09; 47J25 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11067-022-09568-7

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