Abstract
We consider the complexity of finding a fixed point whose existence is guaranteed by an order-theoretic fixed point theorem, e.g., Caristi's fixed point theorem or Brøndsted's fixed point theorem. The problem of finding a fixed point guaranteed by some fixed point theorem is often used to characterize a complexity class. The best well-known such a problem is the problem of finding a Brouwer's fixed point. In contrast to this problem, there has been relatively little computational complexity analysis of computing a Caristi's fixed point and computing a Brøndsted's fixed point. In this paper, we show that the problem of finding a Caristi's fixed point is PLS-complete. Furthermore, we prove that the problem of finding a Brøndsted's fixed point is CLS-hard and belongs to PPAD.
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