# An LQP-SQP alternating direction method for solving variational inequality problems with separable structure

Volume 10, Issue 12, pp 6246--6261 Publication Date: December 06, 2017       Article History
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### Authors

Adnan Alhomaidan - Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia Abdellah Bnouhachem - Laboratoire d'Ingénierie des Systèmes et Technologies de l'Information, Ibn Zohr University, Agadir, BP 1136, Morocco Abdul Latif - Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

### Abstract

In this paper, by combining the logarithmic-quadratic proximal (LQP) method and the square quadratic proximal (SQP) method, we propose an inexact alternating direction method for solving constrained variational inequalities $VI(S,f),$ where $S$ is a convex set with linear constraints. Under certain conditions, the global convergence of the proposed method is established. We show the O(1/t) convergence rate for the inexact LQP-SQP alternating direction method. To demonstrate the efficiency of the proposed method, we provide numerical results for traffic equilibrium problems.

### Keywords

• Proximal point algorithm
• variational inequality
• prediction-correction
• traffic equilibrium problems

•  90C33
•  49J405

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