Approximative methods for monotone systems of min-max-polynomial equations

Abstract

A monotone system of min-max-polynomial equations (min-max-MSPE) over the variables X1,...,Xn has for every i exactly one equation of the form Xi = fi (X1,...,Xn) where each fi (X1,...,Xn ) is an expression built up from polynomials with non-negative coefficients, minimum- and maximum-operators. The question of computing least solutions of min-max-MSPEs arises naturally in the analysis of recursive stochastic games [5,6,14]. Min-max-MSPEs generalize MSPEs for which convergence speed results of Newton's method are established in [11,3]. We present the first methods for approximatively computing least solutions of min-max-MSPEs which converge at least linearly. Whereas the first one converges faster, a single step of the second method is cheaper. Furthermore, we compute ε-optimal positional strategies for the player who wants to maximize the outcome in a recursive stochastic game. © 2008 Springer-Verlag.