An inexact augmented Lagrangian multiplier method for solving quadratic complementary problems: An adapted algorithmic framework combining specific resolution techniques

Abstract

In this paper, an inexact augmented Lagrangian multiplier method (ALM) is designed for solving the quadratic complementarity problem (QCP). The primary goal is proposing an algorithm exploring the structures of QCPs and incorporating them into the implementation. In the implementation, an inexact proximal alternating minimization method is employed to solve the augmented Lagrangian subproblems. As a result, all the subproblems are either strongly convex quadratic programs or linear equations with positive semidefinite symmetric coefficient matrices, which in turn can either be solved with explicitly formula solutions, or by an accelerated proximal gradient method, or a conjugate gradient method. Convergence rates of these subproblems are given, and global convergence of the ALM algorithm is established under mild assumptions. Preliminary numerical experiments show that the algorithm is feasible and efficient.