Nonlinear Perron-Frobenius problem on an ordered Banach space

Ogiwara, Toshiko

doi:10.4099/math1924.21.43

Cited by 20 publications

(11 citation statements)

References 13 publications

(8 reference statements)

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“…Since this holds for all f ∈ int(C + (S)), this establishes the first inequality in (8). The proof of the second inequality in ( 8) is similar, based on [37,Lemma 3.1.7 (iii)]. This concludes the proof of Theorem 1.…”

Section: The Perron-frobenius Eigenvaluesupporting

confidence: 59%

“…A great deal is known about analogs of the Perron-Frobenius theorem for increasing positively one-homogeneous maps on finite dimensional vector spaces, see the recent book [30]. When the map is on an ordered Banach space (and one talks about a Krein-Rutman theorem rather than a Perron-Frobenius theorem, in view of the seminal work in [29]), we rely on Theorem 3.1.1, Proposition 3.1.5, and Lemma 3.1.7 of [37], as seen in the proof below (see also [36], [33]). These results in [37] are themselves stated in a much broader context than the special case of the Banach space C(S) and the order structure defined by the cone C + (S), with S a compact metric space, which suffices for our purposes.…”

Section: T (N) Is Compact (This Holds Under the Weaker Assumptions (mentioning

confidence: 99%

“…When the map is on an ordered Banach space (and one talks about a Krein-Rutman theorem rather than a Perron-Frobenius theorem, in view of the seminal work in [29]), we rely on Theorem 3.1.1, Proposition 3.1.5, and Lemma 3.1.7 of [37], as seen in the proof below (see also [36], [33]). These results in [37] are themselves stated in a much broader context than the special case of the Banach space C(S) and the order structure defined by the cone C + (S), with S a compact metric space, which suffices for our purposes. The recent papers [32] and [12] claim even stronger nonlinear Krein-Rutman theorems.…”

Section: T (N) Is Compact (This Holds Under the Weaker Assumptions (mentioning

confidence: 99%

See 2 more Smart Citations

A Variational Formula for Risk-Sensitive Reward

Anantharam¹,

Borkar²

2017

SIAM J. Control Optim.

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We derive a variational formula for the optimal growth rate of reward in the infinite horizon risk-sensitive control problem for discrete time Markov decision processes with compact metric state and action spaces, extending a formula of Donsker and Varadhan for the Perron-Frobenius eigenvalue of a positive operator. This leads to a concave maximization formulation of the problem of determining this optimal growth rate.

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Section: The Perron-frobenius Eigenvaluesupporting

confidence: 59%

Section: T (N) Is Compact (This Holds Under the Weaker Assumptions (mentioning

confidence: 99%

Section: T (N) Is Compact (This Holds Under the Weaker Assumptions (mentioning

confidence: 99%

See 1 more Smart Citation

A Variational Formula for Risk-Sensitive Reward

Anantharam¹,

Borkar²

2017

SIAM J. Control Optim.

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“…A nonlinear variant of the Krein-Rutman theorem [17] then asserts that under some technical hypotheses, a unique positive principal eigenvalue and a corresponding unique (up to a scalar multiple) positive eigenvector for T exist. Our interest is in the following nonlinear scenario arising in risk-sensitive control: Consider…”

Section: Discrete Time Problemsmentioning

confidence: 99%

`Controlled' versions of the Collatz-Wielandt and Donsker-Varadhan formulae

Arapostathis¹,

Borkar²

2019

Preprint

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“…H{v + v') < Hv + Hv'). Further, followingOgiwara (1995), pp. 47-49, we make use of the following nonstandard definition of an eigenvalue: A G R+ is called an eigenvalue of H if Hv = Xv for some v G 5J+, v ^0.Some of our results are based on the following assumption (which can be shown to be equivalent to saying that the operator H is compact).…”

mentioning

confidence: 99%

Decision Theory and Multi-Agent Planning

Riccia¹,

DuBois²,

Kruse³

et al. 2006

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Planning is an area that deals with sequential decision problems. Starting from an initial state, the decision maker is interested in finding a sequence of actions in order to achieve a set of predefined goals. Planning is a generic human activity. From an algorithmic point of view it goes back to the early 60's with the General Problem Solver (GPS), which was invented and propagated (unsuccessfully) by the famous Nobel Price winner Herbert Simon. GPS was the first automated planner published in the literature. Typically, this type of planner assumes a deterministic world that can be handled by unconditional, ever successful actions. Despite its limitations, it has had a strong impact on follow-up research in Artificial Intelligence. A further branch of research, strongly overlapping with the planning domain of AI was Dynamic Programming and Markov Decision Theory, developed by researchers affiliated to Operations Research. They consider multi-stage decision making under uncertainty with actions depending on the current state and time. It is of interest to note that Bayesian Belief Network and Influence Diagram Methods have their roots in Dynamic Programming. Section 2 "Prediction, Planning and Decision"Belief, preferences and utility are the cornerstones or prerequisites of prediction, planning and decision making. Therefore, after having read section 2 the reader will be ready for details about such activities. As human decision makers are not free of emotions, it is worthwhile to have a look at this aspect, too.In "Efficient computation of project characteristics in a series-parallel activity network with interval durations" Pawel Zielinski addresses an instance of the most basic of all scheduling problems, when a set of activities is supposed to be performed while respecting some precedence constraints. The corresponding predictive planning problem aims at determining conditions for a minimal time execution of the set of activities, detecting the critical ones which may delay the end of the project. The originality of the paper lies in the way uncertainty pervading the activity durations is modelled. Assuming ill-known durations is a realistic assumption for predictive high-level planning tasks. Here, this uncertainty is modelled by simple intervals. The surprising feature of this problem is that while the deterministic version is straightforward (and has been solved more than 40 years ago), the presence of interval-valued time estimates representing uncertainty makes it NP-Hard. As a consequence the author deals with the special (but very common in practice) case of series-parallel networks, where the problem remains of polynomial complexity. Relying on special data structures for tree processing, the author manages to cut down this complexity in a drastic way for the computation of latest starting times and floats of activities, thus providing a very efficient implementation of the scheduling algorithm. deal with "Graphical Models for Industrial Planning on Complex Domains". The authors focus on industrial ...

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Nonlinear Perron-Frobenius problem on an ordered Banach space

Cited by 20 publications

References 13 publications

A Variational Formula for Risk-Sensitive Reward

A Variational Formula for Risk-Sensitive Reward

`Controlled' versions of the Collatz-Wielandt and Donsker-Varadhan formulae

Decision Theory and Multi-Agent Planning

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