Inexact Splitting Methods For Solving Structured Variational Inequalities

Variational inequality?VI? problem is a class of important problems in the field of optimization and many problems, which arise from applications of field, can change to variational inequality problems, such as, convex programming problems,complementarity problems, fixed point problems, and traffic equilibrium problems, etc.Nowadays, there are a series of algorithms for solving variational inequality problems,e.g., proximal point algorithm?PPA?, projection contraction method, augmented Lagrangian method,alternating direction method?ADM?. These algorithms are widely used in handling statistical learning,signal processing, traffic optimization, and matrix optimization.With the rapid development of information technology, studying on large-scale problems with special structure has become an important research hotspot in the fields of mathematical programming. Therefore, the purpose of this paper is to design effective algorithms to solve this kind of special problems. The main research work of this paper is as follows:?1? For variational inequality problem with separable structure, Chen proposed an inexact alternating direction method?IADM? in [44]. When the dimensionality of data is tremendous large, parallel splitting method?PSM? is more efficient than ADM, so based on IADM, we proposed a new inexact parallel splitting method?NIPSM?, and apply it to solve some applications in traffic equilibrium problems. The characteristic of this new algorithm is solving the sub variational inequalities in Jacobi type, and instead of solving the sub-VIs exactly, we add an inexact term to solve them inexactly. Then, we get a predictor and correct them to approximate the sub-VIs' real solutions. So, it is also a prediction-correction method. Convergence of the new method is proved under mild assumptions and some numerical results demonstrate that the new method NIPSM is efficient.?2? Because of the similar structure between IADM and NIPSM, we propose a new algorithm for solving the unified structure. The convergence and effectiveness of the new method is proved under mild assumptions.?3? Still based on the reference [44], we proposed a new method, which use a linear combination of two directions d₁?w^k,???^k? and d₂?w^k,???^k? as a new direction in the correction step for correcting the predictor in order to approximate the real solutions.The convergence and effectiveness of the new algorithm are proved.