Optimal bounds on finding fixed points of contraction mappings

Authors:
Ching-Lueh Chang;Yuh-Dauh Lyuu
Affiliations:
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan and Department of Finance, National Taiwan University, Taipei, Taiwan
Venue:
Theoretical Computer Science
Year:
2010

Citing 4
Cited 0

Exponential lower bounds for finding Brouwer fixed points

Journal of Complexity
On algorithms for discrete and approximate brouwer fixed points

Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Systems of equilibrium problems with applications to new variants of Ekeland's variational principle, fixed point theorems and parametric optimization problems

Journal of Global Optimization
The complexity of Tarski's fixed point theorem

Theoretical Computer Science

Quantified Score

Hi-index	5.23

Visualization

Abstract

The Banach fixed-point theorem states that a contraction mapping on a complete metric space has a unique fixed point. Given an oracle access to a finite metric space (M,d) and a contraction mapping f:M-M on it, we show that the fixed point of f can be found with an expected O(|M|) oracle queries. We also show that every randomized algorithm for finding a fixed point must make an expected @W(|M|) oracle queries to (M,d) and f for some finite metric space (M,d) and some contraction mapping f:M-M on it. As a generalization of the Banach fixed-point theorem, the Caristi-Kirk fixed-point theorem gives weaker conditions on (M,d) and f guaranteeing the existence of a fixed point of f. We show that every randomized algorithm that finds a fixed point must make the expected @W(|M|) oracle queries to (M,d) and f for some finite metric space (M,d) and some function f:M-M satisfying the conditions of the Caristi-Kirk fixed-point theorem.