Self-adaptive alternating direction method of multiplier for a fourth order variational inequality

We propose an alternating direction method of multiplier for approximation solution of the unilateral obstacle problem with the biharmonic operator. We introduce an auxiliary unknown and augmented Lagrangian functional to deal with the inequality constrained, and we deduce a constrained minimization problem that is equivalent to a saddle-point problem. Then the alternating direction method of multiplier is applied to the corresponding problem. By using iterative functions, a self-adaptive rule is used to adjust the penalty parameter automatically. We show the convergence of the method and give the penalty parameter approximation in detail. Finally, the numerical results are given to illustrate the efficiency of the proposed method.