A probability-one homotopy algorithm for nonsmooth equations and mixed complementarity problems

Abstract. A probability-one homotopy algorithm for solving nonsmooth equations is described. This algorithm is able to solve problems involving highly nonlinear equations, where the norm of the residual has non-global local minima. The algorithm is based on constructing homotopy mappings that are smooth in the interior of their domains. The algorithm is specialized to solve mixed comple-mentarity problems through the use of MCP functions and associated smoothers. This specialized algorithm includes an option to ensure that all iterates remain feasible. Easily satisfiable sufficient conditions are given to ensure that the homotopy zero curve remains feasible, and global convergence properties for the MCP algorithm are developed. Computational results on the MCPLIB test library demonstrate the effectiveness of the algorithm