Methods for solving some complementarity problems which arise in engineering and economics will be described. The term ‘direct algorithms’ has been used to describe the methods, since they seek directly to discover the correct way to enforce the complementarity condition xiëfi(x)=0, for i=1,2,...,n. Such methods have great intuitive appeal for those not specializing in complementarity problems, since they are based on systematically determining which set of n equations in n unknowns needs to be satisfied in order to let every variable have physical significance (nonnegativity constraints). Some new classes of functions which are useful for applications of complementarity are introduced, and connections to the direct algorithms are shown. The methods have had several important applications recently. Among these, we consider two in detail: from mechanical engineering, the free boundary problem of elasto-hydrodynamic lubrication, and from regional economics, the general spatial price equilibrium problem.
