Linear Algebra Enhancements to the PATH Solver

This research aims enhancing the efficiency and reliability of PATH, the most widely used solver for mixed complementarity problems. A key component of the PATH algorithm is solving a series of linear complementary subproblems with a pivotal scheme. Improving the efficiency of the linear system routines (factor, solve, and update) required by the pivotal method is the critical computational issue. We incorporate two new options besides the default LUSOL package in PATH for such functionalities. One of the options employs the UMFPACK package for factor and solve operations, together with an implementation of a stable and efficient block-LU updating scheme, which leads to a significantly more effective version of PATH for solving many large-scale sparse systems. The other option exploits the COIN-OR utilities enhanced by adapting the linear refinements and scaling schemes used in the COIN-LP routines, which is effective in solving smaller-scale systems but less competitive on large-scale cases.