Min-max functions

Gunawardena, Jeremy

doi:10.1007/bf01440235

Cited by 78 publications

(47 citation statements)

References 19 publications

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“…There are natural generalizations of conditions (18) and (19) under which the cited results can still be proved. Namely, one can require these conditions to be valid for some iteration of the operator B.…”

Section: Theorem 11 ([23])mentioning

confidence: 94%

Untitled

Kolokoltsov¹

2001

Journal of Mathematical Sciences

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Consider the set A = R ∪ {+∞} with the binary operations ⊕ = max and = + and denote by A n the set of vectors v = (v 1 , ..., v n ) with entries in A. Let the generalised sum u ⊕ v of two vectors denote the vector with entries u j ⊕ v j , and the product a v of an element a ∈ A and a vector v ∈ A n denote the vector with the entries a vj. With these operations, the set A n provides the simplest example of an idempotent semimodule. The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation. Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces. We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics.

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Section: Theorem 11 ([23])mentioning

confidence: 94%

Untitled

Kolokoltsov¹

2001

Journal of Mathematical Sciences

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“…More generally, we shall say that a map from R d to R d is a homogeneous min-max self-map if its coordinates are of the form of Definition 4.1. This definition is inspired by the min-max functions considered by Gunawardena [Gun94] and Olsder [Ols91]. The terms of this form comprise the semidifferentials of the min-max functions considered there.…”

Section: A Policy Iteration Algorithm To Compute the Smallest Fixed Pmentioning

confidence: 99%

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

Adjé¹,

Gaubert²,

Goubault³

2014

Journal of Mathematical Analysis and Applications

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The problem of computing the smallest fixed point of an orderpreserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.

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“…One can also express different offsets on different arguments of max; for instance max(x + 5, y − 3) z can be written as max(x, y ) + 5 z ∧ y + 8 = y, where y is fresh. Furthermore, since max(e 1 Another less trivial equivalence of MAP is with the problem of deciding the existence of super fixpoints of min-max functions [8]. A min-max function is a function f : Z n → Z n whose coordinates are min-max expressions, i.e., terms in the grammar A more significant relationship is with the problem of computing earliest job start times for the systems of AND/OR precedence constraints of [9].…”

Section: Simple Equivalences With Mapmentioning

confidence: 99%