| Author: | Qu, Xin |
| Title: | An extra gradient Anderson-accelerated algorithm for pseudomonotone variational inequalities and its extension to stochastic problems |
| Advisors: | Chen, Xiaojun (AMA) |
| Degree: | Ph.D. |
| Year: | 2026 |
| Department: | Department of Applied Mathematics |
| Pages: | xx, 103 pages : color illustrations |
| Language: | English |
| Abstract: | Nonmonotone variational inequalities represent a challenging class of problems that arise in various applications, including economics, engineering, and network analysis, where traditional monotonicity assumptions do not hold. The study of algorithms for nonmonotone variational inequalities has consistently been a central topic in optimization. This thesis aims to design efficient algorithms for solving pseudomonotone variational inequality problems, which are a special class of nonmonotone variational inequalities. The content is mainly divided into the following two aspects. In the first part of the thesis, we propose an extra gradient Anderson-accelerated algorithm for solving pseudomonotone variational inequalities, which uses the extra gradient scheme with line search to guarantee the global convergence and Anderson acceleration to achieve fast convergent rate. We prove that the sequence generated by the proposed algorithm from any initial point converges to a solution of the pseudomonotone variational inequality problem without assuming the Lipschitz continuity and contractive condition, which are used for convergence analysis of the extra gradient method and Anderson-accelerated method, respectively in existing literatures. Subsequently, under the condition that the operator is locally Lipschitz continuous, we provide the convergence rate concerning the residual function. Additionally, we propose an improved version of the above algorithm, aiming to enhance convergence efficiency by reducing the number of projections onto the feasible set. Numerical experiments, particular emphasis on Harker-Pang problems, fractional programming problems, nonlinear complementarity problems, PDE problems with free boundary and linear complementarity problems, are conducted to validate the effectiveness and good performance of the proposed algorithm comparing with the existing extra gradient method and Anderson-accelerated method. In the second part of the thesis, we propose a stochastic extra gradient Anderson-accelerated algorithm for solving pseudomonotone stochastic variational inequalities. This algorithm is developed based on the extra gradient method and Anderson-accelerated method, utilizing a stochastic approximation approach. By incorporating a line search technique to against the unknown Lipschitz constants, the proposed stochastic algorithm is designed. We prove that the sequence generated by the proposed stochastic algorithm converges almost surely to a solution of the pseudomonotone stochastic variational inequality problem. Additionally, we establish the sublinear convergence rate of the proposed stochastic algorithm in terms of the mean residual function, along with its optimal oracle complexity. Finally, we validate the effectiveness of the proposed stochastic algorithm by some experiments. |
| Rights: | All rights reserved |
| Access: | open access |
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