A Peaceman-Rachford splitting sequential quadratic programming method with double step-lengths for two-block nonconvex optimization

Abstract

In this paper, the decomposing and dimension-decreasing algorithms for large-scale two-block nonconvex optimization problems are discussed, and a Peaceman-Rachford (PR) splitting sequential quadratic programming (SQP) method with double step-lengths for the discussed problems is proposed. The main work and contributions are as follows. (1) Based on the idea of the PR splitting algorithm, the augmented Lagrangian quadratic programming (QP) in the classical SQP method is decomposed into two small-scale QPs. (2) The improved search directions are obtained by solving the two small-scale QPs. (3) By taking the augmented Lagrangian function as the merit function, Armijo line searches are performed along the two improved directions such that two step-lengths are yielded. Such a line search technique not only guarantees the global and strong convergence as well as reasonable iteration complexity of the method but also overcomes the Maratos effect under weak 条件. (4) A new symmetric update technique for the multiplier associated with the equality constraints is put forward. (5) Via a class of mathematical model and a kind of economic dispatch model of the power system as well as $\ell_2$ regularized binary classification problems, a large number of medium-scale numerical experiments of the proposed method, compared with three associated algorithms, are carried out. The numerical results show that the proposed method is effective and promising.