Nonlinear semigroups and infinite horizon
                    optimization

Yakovenko, S.Yu.; Kontorer, L.

doi:10.1090/advsov/013/10

Cited by 9 publications

(5 citation statements)

References 8 publications

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“…, n} to R, or equivalently an element of R n . We set , states that such a strategy is optimal both for the ergodic control problem and for all the finite horizon problems with final reward φ = u, which means, loosely speaking, that taking φ = u makes it possible for the player to behave (optimally) in the short term as he would in the long term (see [YK92] for more details on the economic interpretation).…”

Section: Stochastic Control Interpretationmentioning

confidence: 99%

Spectral theorem for convex monotone homogeneous maps, and ergodic control

Akian

Gaubert

2003

Nonlinear Analysis: Theory, Methods & Applications

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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.

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Section: Stochastic Control Interpretationmentioning

confidence: 99%

Spectral theorem for convex monotone homogeneous maps, and ergodic control

Akian

Gaubert

2003

Nonlinear Analysis: Theory, Methods & Applications

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“…Later, under upper (lower) semicontinuity, some extensions to general Banach state spaces were obtained by e.g. Kolokoltsov [50], Yakovenko and L.A. Kontorer [88], Zaslavski [90,92,93], Mamedov [59].…”

Section: The Turnpike Theorems In the Deterministic Settingmentioning

confidence: 99%

Turnpike Theorems for Markov Games

Kolokoltsov

Yang

2012

Dyn Games Appl

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This paper has a two-folded purpose. First, we attempt to outline the development of the turnpike theorems in the the last several decades. Second, we study turnpike theorems in finite-horizon twoperson zero-sum Markov games on a general Borel state space. Utilising the Bellman (or Shapley) operator defined for this game, we prove the stochastic versions of the early turnpike theorem on the set of optimal strategies and the middle turnpike theorem on the distribution of the state space.

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“…In particular, we sketch the theory of infinite extremals, which is due essentially to S. Yakovenko [57], [58].…”

Section: Infinite Extremals and Turnpikes In Dynamic Optimisation Andmentioning

confidence: 99%

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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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Nonlinear semigroups and infinite horizon optimization

Cited by 9 publications

References 8 publications

Spectral theorem for convex monotone homogeneous maps, and ergodic control

Spectral theorem for convex monotone homogeneous maps, and ergodic control

Turnpike Theorems for Markov Games

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