Customized Douglas-Rachford splitting methods for structured inverse variational inequality problems

Abstract

Recently, structured inverse variational inequality (SIVI) problems have attracted much attention. In this paper, we propose new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to smaller and easier subproblems. The main computational tasks per iteration are the evaluations of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases where the resolvent operator is evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear rate of convergence, and linear rate of convergence results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods. 

Keywords:

• Inverse variational inequality
• Douglas-Rachford splitting method
• decoupling
• global convergence
• sublinear/linear rate of convergence

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (Grant Nos. 12131004, 12126603, 11871279), Grant 2021YFA1003600, and Youth Talent Development Special Project of Operations Research Society of China.

Notes on contributors

Y. N. Jiang

Y. N. Jiang received the Ph.D. in the School of Mathematical Sciences at Nanjing Normal University, Nanjing 210023, China. Then she carried out her postdoctoral research work in the Department of Mathematics at Nanjing University, Nanjing 210093, China. Her research interests involve optimization theory, algorithm design and applications of equilibrium control problems.

X. J. Cai

X. J. Cai is a Professor in the School of Mathematical Sciences at Nanjing Normal University, Nanjing 210023, China. Her research interests lie in optimization theory and algorithm, variational inequality, numerical optimization, and equilibrium problems in transportation network.

D. R. Han

D. R. Han is a Professor in the School of Mathematical Sciences at Beihang University, Beijing 100191, China. His research mainly focuses on numerical methods for large-scale optimization problems and variational inequality problems, as well as the application of optimization and variational inequality problems in transportation planning and magnetic resonance imaging.

J. F. Yang

J. F. Yang is a Professor in the Department of Mathematics at Nanjing University, Nanjing 210093, China. His research interests lie in the areas of in optimization, signal processing, machine learning and data science.