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.
We present the PFix algorithm for the fixed point problem f ðxÞ ¼ x on a nonempty domain ½a; b; where dX1; a; bAR d ; and f is a Lipschitz continuous function with respect to the infinity norm, with constant qp1: The computed approximationx satisfies the residual criterion jj f ðxÞ Àxjj N pe; where e40: In general, the algorithm requires no more than P d i¼1 s i function component evaluations, where s Jmaxð1; log 2 ðjjb À ajj N =eÞÞn þ 1: This upper bound has order OðJlog d 2 ð1=eÞnÞ as e-0: For the domain ½0; 1 d with eo0:5 we prove a stronger result, i.e., an upper bound on the number of function component evaluations is ð dþrÀ1 rÀ1 Þ þ 2ð dþr rþ1 Þ; where r Jlog 2 ð1=eÞn: This bound approaches Oðr d =d!Þ as r-N ðe-0Þ and Oðd rþ1 =ðr þ 1Þ!Þ as d-N: We show that when qo1 the algorithm can also compute an approximationx satisfying the absolute criterion jjx À x à jj N pe; where x à is the unique fixed point of f : The complexity in this case resembles the complexity of the residual criterion problem, but with tolerance eð1 À qÞ instead of e: We show that when q41 the absolute criterion problem has infinite worst-case complexity when information consists of function evaluations. Finally, we report several numerical tests in which the actual number of evaluations is usually much smaller than the upper complexity bound. r
We present the BEDFix (Bisection Envelope Deep-cut Fixed point) algorithm for the problem of approximating a fixed point of a function of two variables. The function must be Lipschitz continuous with constant 1 with respect to the infinity norm; such functions are commonly found in economics and game theory. The computed approximation satisfies a residual criterion given a specified error tolerance. The BEDFix algorithm improves the BEFix algorithm presented in Shellman and Sikorski [2002] 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 numerical tests indicate that BEDFix significantly improves the average-case performance. In addition, we show how BEDFix may be used to solve the absolute criterion fixed point problem with significantly better performance than the simple iteration method, when the Lipschitz constant is less than but close to 1. BEDFix is highly efficient when used to compute residual solutions for bivariate functions, having a bound on function evaluations that is twice the logarithm of the reciprocal of the tolerance. In the tests described in this article, the number of evaluations performed by the method averaged 31 percent of this worst-case bound. BEDFix works for nonsmooth continuous functions, unlike methods that require gradient information; also, it handles functions with minimum Lipschitz constants equal to 1, whereas the complexity of simple iteration approaches infinity as the minimum Lipschitz constant approaches 1. When BEDFix is used to compute absolute criterion solutions, the worst-case complexity depends on the logarithm of the reciprocal of 1-q, where q is the Lipschitz constant, as well as on the logarithm of the reciprocal of the tolerance.
We present the PFix algorithm for approximating a fixed point of a function f that has arbitrary dimensionality, is defined on a rectangular domain, and is Lipschitz continuous with respect to the infinity norm with constant 1. PFix has applications in economics, game theory, and the solution of partial differential equations. PFix computes an approximation that satisfies the residual error criterion, and can also compute an approximation satisfying the absolute error criterion when the Lipschitz constant is less than 1. For functions defined on all rectangular domains, the worst-case complexity of PFix has order equal to the logarithm of the reciprocal of the tolerance, raised to the power of the dimension. Dividing this order expression by the factorial of the dimension yields the order of the worst-case bound for the case of the unit hypercube. PFix is a recursive algorithm, in that it uses solutions to a d -dimensional problem to compute a solution to a ( d + 1)-dimensional problem. A full analysis of PFix may be found in Shellman and Sikorski [2003b], and a C implementation is available through ACM ToMS.
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