The Primal-Dual Active Set Method for a Class of Nonlinear Problems with T-Monotone Operators

Xiahui He,Peng Yang

doi:10.1155/2019/2912301

Abstract

The family of primal-dual active set methods is drawing more attention in scientific and engineering applications due to its effectiveness and robustness for variational inequality problems. In this work, we introduce and study a primal-dual active set method for the solution of the variational inequality problems with T-monotone operators. We show that the sequence generated by the proposed method globally and monotonously converges to the unique solution of the variational inequality problem. Moreover, the convergence rate of the proposed scheme is analyzed under the framework of the algebraic setting; i.e., the established convergence results show that the iteration number of the methods is bounded by the number of the unknowns. Finally, numerical results show that the efficiency can be achieved by the primal-dual active set method.

Highlights

The variational inequality problem associated with Tmonotone operators has many applications, e.g., the diffusion problem involving Michaelis-Menten or second-order irreversible reactions; see, for example, [1,2,3,4,5,6,7] and the references therein for details
A major advantage of the Schwarz methods is amenable to implement and its convergence rate will not be deteriorated with the refinement of the mesh size when it is applied to the system arising from the discretization of partial differential equations (PDEs)
We get the equivalent relation between the primal-dual active set method and Howard’s algorithm introduced in [9, 28, 29] for the variational inequality problem with T-monotone operators, show the convergence theorem of Howard’s algorithm, and obtain the convergence theorem of the primal-dual active set method

Summary

Introduction

The variational inequality problem associated with Tmonotone operators has many applications, e.g., the diffusion problem involving Michaelis-Menten or second-order irreversible reactions; see, for example, [1,2,3,4,5,6,7] and the references therein for details. The convergence of this kind of methods depends crucially on the choice of the relaxation parameter Another popular approach for the solution of variational inequality problems is the Schwarz algorithm [1, 7, 11,12,13,14], which is based on the framework of domain decomposition methods [15,16,17]. Using the equivalence of the proposed algorithms for variational inequality problems with T-monotone operators, we give a simple proof for the global monotone convergence of the primal-dual active set method and conclude that the primal-dual active set method converges in no more than n + 1 iterations, where n is the size of the solution vector.

Preliminaries

Primal-Dual Active Set Method for the Variational Inequality Problem

Convergence Results

Numerical Experiments

Concluding Remarks