Spectral theorem for convex monotone homogeneous maps, and ergodic control

Abstract

Abstract. We consider convex maps f : R n → R n that are monotone (i.e., that preserve the product ordering of R n ), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f , when it is non-empty, is isomorphic to a convex inf-subsemilattice of R n , whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f . This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f . We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

A policy iteration algorithm for zero-sum stochastic games with mean payoff

“…The present algorithm extends the ones which have been developed by the authors and Gunawardena [7,4] in the case of deterministic games. Its proof exploits earlier results of Akian and the second author [1], on the structure of the fixed point set of a convex order-preserving additively homogeneous map. The existence of a polynomial time algorithm to compute χ(f ) is an open question [5], even in the deterministic case [2,11].…”

“…quelques définitions et résultats de [1]. Supposons qu'il existe au moins un vecteur harmonique, u. Nous définissons le sous-différentiel de g au point u, ∂g(u) : …”

“…We say that a vector u ∈ R n is harmonic (with respect to g) if g(u) = u, and that it is super-harmonic if g(u) ≤ u. Let us recall some definitions and results of [1]. Assume that there exists at least one harmonic vector, u.…”

Iteration of order preserving subhomogeneous maps on a cone

“…In this paper we investigate this problem for continuous maps f : K → K that are, in addition, order preserving and subhomogeneous. In particular, we prove in Theorem 2.1 that each bounded orbit of f converges to a periodic orbit and that the period of each periodic point of f is bounded by (1) β N = max…”

“…Order preserving subhomogeneous maps have been studied intensively in nonlinear Perron-Frobenius theory. They arise in various fields, such as optimal control and game theory [1,24,29], idempotent analysis [17,23], the analysis of monotone dynamical systems [15,16,18,19,32,33], and discrete event systems [4,12,13]. In this list we have quoted only a few recent works and we suggest the reader to consult [25,26] for further references.…”