Solving a class of asymmetric variational inequalities by a new alternating direction method.

*(English)*Zbl 0959.49009Summary: 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.

##### MSC:

49J40 | Variational inequalities |

65K10 | Numerical optimization and variational techniques |

90C20 | Quadratic programming |

90C25 | Convex programming |

##### Keywords:

convergence properties; variational inequalities; alternating direction method; convex quadratic programming
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\textit{S. Wang} et al., Comput. Math. Appl. 40, No. 8--9, 927--937 (2000; Zbl 0959.49009)

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