On the computational complexity of piecewise-linear homotopy algorithms

We show that piecewise-linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point. Our examples are based on an example of Murty exhibiting exponential growth for Lemke's algorithm for the linear complementarity problem.This research was supported in part by NSF grant ECS-7921279 and by a Guggenheim Fellowship.