Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?
分析收缩不动点问题的计算复杂度，发现一般情形下存在维度诅咒，但若映射域具有特殊结构，问题可强可解，并给出资产定价等经济问题的例子。
This paper analyzes the complexity of the contraction fixed point problem: compute an e-approximation to the fixed point V * = Γ(V * ) of a contraction mapping r 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 e-approximation to a fixed point V * e 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 e-approximation of V * is bounded by Ce -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.