Is there a curse of dimensionality for contraction fixed points in the worst case?

摘要

This paper analyzes the complexity of the contraction fixed point problems: compute an ε-approximation to the fixed point V~* = Γ(V~*) of a contraction mapping Γ that maps a Banach space B_d of continuous functions of d variables into itself. We focus on quasi linear contractions where Γ is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Γ, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an ε-approximation to a fixed point V~* ∈ B_d increases exponentially in d. We show that the curse of dimensionality disappears if the domain of Γ has additional special structure. We identify a particular type of special structure for which the problem is strongly tractable even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an ε-approximation of V~* is bounded by Cε~(-p) where C and p are constants independent of d. We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent p = 1 is nearly achieved.