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Approximative methods for monotone systems of min-max-polynomial equations. (English) Zbl 1153.65337
Aceto, Luca (ed.) et al., Automata, languages and programming. 35th international colloquium, ICALP 2008, Reykjavik, Iceland, July 7–11, 2008. Proceedings, Part I. Berlin: Springer (ISBN 978-3-540-70574-1/pbk). Lecture Notes in Computer Science 5125, 698-710 (2008).
Summary: A monotone system of min-max-polynomial equations (min-max-MSPE) over the variables \(X _{1},\dots ,X _{n }\) has for every \(i\) exactly one equation of the form \(X _{i } = f _{i }(X _{1},\dots ,X _{n })\) where each \(f _{i }(X _{1},\dots ,X _{n })\) 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. Min-max-MSPEs generalize MSPEs for which convergence speed results of Newton’s method were established. 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 \(\epsilon \)-optimal positional strategies for the player who wants to maximize the outcome in a recursive stochastic game.
For the entire collection see [Zbl 1142.68001].

65H10 Numerical computation of solutions to systems of equations
91A15 Stochastic games, stochastic differential games
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