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 farfetched. 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.

Probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one. These methods are theoretically powerful, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical. The concomitant numerical linear algebra problems deal with rectangular matrices, and good algorithms require a delicate balance (not always achieved) of accuracy, robustness, and efficiency in both space and time. The author's experience with globally convergent homotopy algorithms is surveyed here, and some of the linear algebra difficulties for dense and sparse problems are discussed.