A Two-Dimensional Bisection Envelope Algorithm for Fixed Points

Abstract

In this paper we present a new algorithm for the two-dimensional fixed point problem f(x)=x on the domain [0, 1] × [0, 1], where f is a Lipschitz continuous function with respect to the infinity norm, with constant 1. The computed approximation x satisfies ||f(x) − x||. [ e for a specified tolerance e < 0.5. The upper bound on the number of required function evaluations is given by 2Klog 2 (1/e)L+1. Similar bounds were derived for the case of the 2-norm by Z.

Computational complexity of fixed points

“…We summarize first some fixed point properties of the functions in F a,b . The proofs of these theorems are found in [40].…”

“…Several algorithms have been invented since then, including Vol. 9 (2009) Computational complexity of fixed points 253 restart methods [31], homotopy methods [17] (see also [2,18,54]), ellipsoid methods [49,8,24], and bisection-envelope methods [42,43,40,41,39]. The point of departure of our paper is the classical Banach's fixed point theorem.…”

“…The computed approximation satisfies a residual criterion given a specified tolerance. The BEDFix algorithm improves the BEFix algorithm presented in [40] by utilizing "deep cuts", that is, eliminating additional segments of the feasible domain which cannot contain a fixed point. The upper bound on the number of required function evaluations is the same for BEDFix and BEFix, but our experiments indicate that BEDFix significantly improves the average-case performance for all of the numerical tests that we carried out.…”

“…We also provide a d-dimensional Remark 5.5 that we believe will be crucial in the construction of optimal multivariate algorithms. [40] in which we developed a Bisection Envelope Fixed point algorithm (BEFix), and the paper [41] in which we developed a Bisection Envelope Deep-cut Fixed point algorithm (BEDFix) for computing fixed points of two-dimensional nonexpansive functions. Those algorithms exhibit the optimal worst case complexity of 2 log(1/ ) + 1, but the BEDFix algorithm is more efficient for all our numerical tests.…”

“…We stress that the above problem has infinite worst case complexity with respect to the absolute error criterion [40,39] even in the two-dimensional case. We formulate this in We believe that the above result also holds for the case of L = 1, which indeeed is the case for the second norm (see Section 5.3).…”