Oracle Algorithms for Fixed-Point Problems — An Axiomatic Approach

Abstract

We examine three problems which can be considered as “approximating a fixed point of a continuous function f mapping a closed n-cell into itself”, namely: P1) Determine whether a given set contains a fixed point of f; P2) find a point x_o such that ‖ x_o − f (x_o) ‖ ≤ ε; P3) find a subset of diameter ≤ ε that contains a fixed point of f. It is shown that none of these problems can be solved by any algorithm which may evaluate the function for an arbitrary, but bounded, number of points but which does not require any other, analytical knowledge about the continuous function. To obtain these results, we give a formal, axiomatic definition of an “oracle algorithm”, a concept which can also be used to obtain several other lower bounds in complexity theory. This definition is based on the idea that the action of the algorithm at any step — the determination either of the next argument x for which f shall be evaluated or of the final solution — depends on all the “previous informations” about f, i.e. on the function evaluations in all previous steps.

Supported by Sonderforschungsbereich 21 (DFG), Institut für ökonometrie und Operations Research, Bonn