Exponential lower bounds for finding Brouwer fixed points
Journal of Complexity
Nonlinear differential equations and dynamical systems
Nonlinear differential equations and dynamical systems
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
On optimality of Krylov's information when solving linear operator equations
Journal of Complexity
An efficient method for locating and computing periodic orbits of nonlinear mappings
Journal of Computational Physics
Approximating fixed points of weakly contracting mappings
Journal of Complexity
Optimal solution of nonlinear equations
Optimal solution of nonlinear equations
A two-dimensional bisection envelope algorithm for fixed points
Journal of Complexity
A recursive algorithm for the infinity-norm fixed point problem
Journal of Complexity
On algorithms for discrete and approximate brouwer fixed points
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Algorithm 848: A recursive fixed-point algorithm for the infinity-norm case
ACM Transactions on Mathematical Software (TOMS)
A note on two fixed point problems
Journal of Complexity
Matching algorithmic bounds for finding a Brouwer fixed point
Journal of the ACM (JACM)
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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.