Residual regularization path-following methods for linear complementarity problems

TL;DR

This paper introduces a residual regularization path-following method with trust-region updates for linear complementarity problems, improving robustness and efficiency over existing solvers, especially for dense cases.

Contribution

It proposes a novel residual regularization path-following method with trust-region strategy, eliminating the need for traditional assumptions and outperforming state-of-the-art solvers in speed and robustness.

Findings

The new method is robust and efficient for dense linear complementarity problems.

It is faster and more robust than PATH and MILES solvers.

Computational time is reduced to about 1/3 to 1/10 of PATH for dense cases.

Abstract

In this article, we consider the residual regularization path-following method with the trust-region updating strategy for the linear complementarity problem. This time-stepping selection based on the trust-region updating strategy overcomes the shortcoming of the line search method, which consumes the unnecessary trial steps in the transient-state phase. In order to improve the robustness of the path-following method, we use the residual regularization parameter to replace the traditional complementarity regularization parameter. Moreover, we prove the global convergence of the new method under the standard assumptions without the traditional assumption condition of the priority to feasibility over complementarity. Numerical results show that the new method is robust and efficient for the linear complementarity problem, especially for the dense cases. And it is more robust and faster than some state-of-the-art solvers such as the built-in subroutines PATH and MILES of the GAMS v28.2 (2019) environment. The computational time of the new method is about 1/3 to 1/10 of that of PATH for the dense linear complementarity problem.