一类单调变分不等式的非精确交替方向法

An Inexact Alternating Direction Method for Solving a Class of Monotone Variational Inequalities

交替方向法适合于求解大规模问题．该文对于一类变分不等式提出了一种新的交替方向法．在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成．基于子问题的精确求解,该文证明了算法的收敛性．进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题．在一定的非精确条件下,算法的收敛性得以证明．

Alternating direction methods are suitable ones for solving large-scale problems. This paper presents a new alternating direction method for a class of variational inequalities. At each iterations the proposed subproblem consists of a strongly monotonic linear variational inequality and a well-conditioned system of nonlinear equations, which is easily to be solved. The convergence theorem of the proposed method is proved based on the exact solution of the subproblem. Furthermore, the authors develop the proposed alternating direction method as an inexact method, which only needs to solve the subproblem inexactly. Under some inexact conditions, the convergence of inexact alternating direction method is proved too.