An SQP Regularization with Double Conjugate Gradient Implementation for Solving Nonlinear Complementarity Problems

Building upon the works proposed in [1] and [2], we introduce an advanced version of regularized proximal point methods to solve nonlinear complementarity problems (NCP). Our contribution is characterized by two key innovations. Firstly, we introduce an innovative square root quadratic term as part of the regularized subproblem framework, replacing the commonly used logarithmic quadratic term. Secondly, we implement the conjugate gradient algorithm in two stages: the intermediate step and the correction step. This dual approach employs two optimal descent directions with two step lengths to achieve multiplicative progress in each iteration, significantly accelerating convergence. We establish the global convergence of our innovative algorithm, under the condition that F exhibits monotonicity. Initial numerical experiments are presented to confirm the algorithm’s practical effectiveness.