Exploiting structure in piecewise-linear homotopy algorithms for solving equations

AbstractWe consider the recent algorithms for computing fixed points or zeros of continuous functions fromRn to itself that are based on tracing piecewise-linear paths in triangulations. We investigate the possible savings that arise when these fixed-point algorithms with their usual triangulations are applied to computing zeros of functionsf with special structure:f is either piecewise-linear in certain variables, separable, or has Jacobian with small bandwidth. Each of these structures leads to a property we call modularity; the algorithmic path within a simplex can be continued into an adjacent simplex without a function evaluation or linear programming pivot. Modularity also arises without any special structure onf from the linearity of the function that is deformed tof. In the case thatf is separable we show that the path generated by Kojima's algorithm with the homotopyH2 coincides with the path generated by the standard restart algorithm of Merrill when the usual triangulations are employed. The extra function evaluations and linear programming steps required by the standard algorithm can be avoided by exploiting modularity.