### Exponential lower bounds for finding Brouwer fixed points

Michael D. Hirsch, Stephen Vavasis
1987 28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
The computation of Brouwer fixed points is a central tool in economic modeling. Although there have been several algorithms for computing a fixed point of a Brouwer map, starting with Scarf's algorithm of 1965, the question of worst-case complexity was not addressed. It has been conjectured that Scarf's algorithm has typical behavior that is polynomial in the dimension. Here we show that any algorithm for computing the Brouwer fixed point of a function based on function evaluations (a class
more » ... ations (a class that includes all known general purpose algorithms) must in the worst case perform a number of function evaluations that is exponential in both the number of digits of accuracy and the dimension. Our lower bounds are very close to the known upper bounds. 0 1989 Academic FY~SS hc. 380 HIRSCH, PAPADIMITRIOU, AND VAVASIS 384 HIRSCH, PAPADIMITRIOU, AND VAVASIS GEOMETRIC FACT 4. Let (r, 0) and (s, 4) be two points in R2 in a generalized polar coordinate system. Then /(r, 0) -(r, +)I % 2j(r, 0) -(\$7 &I. These next two facts concern the effect of making an interpolation. Assume that vector w is an interpolation of two vectors u and v, that is, it satisfies w = (1 -h)u + Au for some A E [0, I]. GEOMETRIC FACT 5. Suppose that u and v achieve their maximum absolute coordinate values ju( and Jv( at the same coordinate position i and with the same sign. Then Iw( 2 min(juj, Iv]).