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Solving a class of asymmetric variational inequalities by a new alternating direction method. (English) Zbl 0959.49009
Summary: The augmented Lagrangian method (also referred to as an alternating direction method) solves a class of Variational Inequalities (VI) via solving a series of sub-VI problems. The method is effective whenever the subproblems can be solved efficiently. However, the subproblem to be solved in each iteration of the augmented Lagrangian method itself is still a VI problem. It is essentially as difficult as the original one, the only difference is that the dimension of the subproblems is lower. In this paper, we propose a new alternating direction method for solving a class of monotone variational inequalities. In each iteration, the method solves a convex quadratic programming with simple constraints and a well-conditioned system of nonlinear equations. For such ‘easier’ subproblems, existing efficient numerical softwares are applicable. The effectiveness of the proposed method is demonstrated with an illustrative example.

49J40 Variational inequalities
65K10 Numerical optimization and variational techniques
90C20 Quadratic programming
90C25 Convex programming
Full Text: DOI
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