Continuous Homotopies for the Linear Complementarity Problem

There are various formulations of the linear complementarity problem as a Kakutani fixed point problem, a constrained optimization, or a nonlinear system of equations. These formulations have remained a curiosity since not many people seriously thought that a linear combinatorial problem should be converted to a nonlinear problem. Recent advances in homotopy theory and new mathematical software capabilities such as HOMPACK indicate that continuous nonlinear formulations of linear and combinatorial problems may not be far-fetched. Several different types of continuous homotopies for the linear complementarity problem are presented and analyzed here, with some numerical results. The homotopies with the best theoretical properties (global convergence and no singularities along the zero curve) turn out to also be the best in practice.