Topical and sub-topical functions, downward sets and abstract convexity

Rubinov, A. M.; Singer, Ivan

doi:10.1080/02331930108844567

Cited by 76 publications

(85 citation statements)

References 4 publications

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“…It is geometrically obvious that we cannot obtain the convex function of Figure 1 as the sup of affine functions. (Linear functions, however, lead to an interesting theory if we consider max-plus concave functions instead of max-plus convex functions, see Rubinov and Singer [RS01], and downward sets instead of max-plus convex sets, see Martínez-Legaz, Rubinov, and Singer [MLRS02]. )…”

Section: Introductionmentioning

confidence: 99%

Max-plus convex sets and functions

Cohen¹,

Gaubert²,

Quadrat³

et al. 2005

Idempotent Mathematics and Mathematical Physics

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Abstract. We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if K is a conditionally complete idempotent semifield, with completionK, a convex function K n →K which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of K n , which extends earlier results of Zimmermann, Samborski, and Shpiz.

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Section: Introductionmentioning

confidence: 99%

Max-plus convex sets and functions

Cohen¹,

Gaubert²,

Quadrat³

et al. 2005

Idempotent Mathematics and Mathematical Physics

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show abstract

“…Of course, topical maps also include max-plus linear maps sending R n to R, which can be represented in a dual way. The following observation was made by Rubinov and Singer [23], and, independently by Gunawardena and Sparrow (personal communication).…”

Section: Property 1 (Semimodule Of Convex Functions)mentioning

confidence: 97%

Max-Plus Convex Geometry

Gaubert

Katz

2006

Relations and Kleene Algebra in Computer Science

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Abstract. Max-plus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including max-plus versions of the separation theorem, existence of linear and non-linear projectors, max-plus analogues of the Minkowski-Weyl theorem, and the characterization of the analogues of "simplicial" cones in terms of distributive lattices.

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“…It is known that a self-map Ψ of R d can be represented as the Shapley operator of some stochastic game -that does not satisfy necessarily assumption (A) -if and only if it preserves the standard partial order of R d and commutes with the addition of a constant [24]. Moreover, the transition probabilities can be even required to be degenerate (deterministic), see [41,21].…”

Section: Definitions and Fundamental Propertiesmentioning

confidence: 99%

Definable Zero-Sum Stochastic Games

2015

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Definable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games.We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally subanalytic Shapley operator.Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given.

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Topical and sub-topical functions, downward sets and abstract convexity

Cited by 76 publications

References 4 publications

Max-plus convex sets and functions

Max-plus convex sets and functions

Max-Plus Convex Geometry

Definable Zero-Sum Stochastic Games

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