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  • countable state space  (6)
  • duality  (5)
  • 1
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Generalized convexity ; duality ; root term ; optimal solution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We extend the duality theorems for a class of nondifferentiable problems with Mond-Weir type duals.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-5217
    Keywords: Key words: Markov decision chains ; countable state space ; sensitive optimality ; unbounded costs.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract. We extend a result by Cavazos-Cadena and Lasserre on the existence of strong 1-optimal stationary policies in Markov decision chains with countable state spaces, uniformly ergodic transition probabilities and bounded costs to a larger class of models with unbounded costs and the so-called V-uniform ergodic transition structure.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Quasi-inverse ; epi-inverse ; hypo-inverse ; nondecreasing functions ; quasiconvex functions ; convex functions ; duality ; rearrangement ; modulus of continuity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung In dieser Arbeit wird systematisch die Umkehrung monoton nichtfallender Funktionenf: ℝ → ℝ ∪ {−∞, +∞} studiert. Die Ergebnisse bilden die Grundlage für eine neue Dualitätstheorie quasikonvexer Probleme [6]. Da jedoch die Fragestellung bei einer ganzen Anzahl weiterer Situationen auftritt, verdient sie eine gesonderte Behandlung. Anwendungen in der Topologie, Wahrscheinlichkeitstheorie, monotonen Umordnungen und in der konvexen Analysis werden aufgezeigt und skizziert.
    Notes: Abstract This work is devoted to a systematic study of the inversion of nondecreasing one variable extended real-valued functions. Its results are preparatory for a new duality theory for quasiconvex problem [6]. However the question arises in a variety of situations and as such deserves a separate treatment. Applications to topology, probability theory, monotone rearrangements, convex analysis are either pointed out or sketched.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: vectorial control-approximation problem ; vectorial location problem ; duality ; efficiency
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung In der Arbeit werden für eine Klasse von vektoriellen Steuer-Approximationsproblemen in reellen reflexiven Banachräumen vektorielle Dualprobleme konstruiert und Dualitätseigenschaften hergeleitet. Als Spezialfall ergeben sich entsprechende Aussagen für vektorielle Standortprobleme.
    Notes: Abstract The author formulates vectorial dual problems for a certain class of vectorial control-approximation problems in real reflexive Banach spaces. A number of propositions concerning duality are derived. Corresponding propositions are mentioned for the special case of the vectorial location problems.
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  • 5
    ISSN: 1432-5217
    Keywords: Constrained Markov Decision Processes ; countable state space ; finite horizon ; infinite horizon ; finite approximations ; asymptotic properties ; dynamic scheduling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We present in this paper several asymptotic properties of constrained Markov Decision Processes (MDPs) with a countable state space. We treat both the discounted and the expected average cost, with unbounded cost. We are interested in (1) the convergence of finite horizon MDPs to the infinite horizon MDP, (2) convergence of MDPs with a truncated state space to the problem with infinite state space, (3) convergence of MDPs as the discount factor goes to a limit. In all these cases we establish the convergence of optimal values and policies. Moreover, based on the optimal policy for the limiting problem, we construct policies which are almost optimal for the other (approximating) problems. Based on the convergence of MDPs with a truncated state space to the problem with infinite state space, we show that an optimal stationary policy exists such that the number of randomisations it uses is less or equal to the number of constraints plus one. We finally apply the results to a dynamic scheduling problem.
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  • 6
    ISSN: 1432-5217
    Keywords: Markov decision processes ; countable state space ; Linear programming ; duality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We present an Linear Programming formulation of MDPs with countable state and action spaces and no unichain assumption. This is an extension of the Hordijk and Kallenberg (1979) formulation in finite state and action spaces. We provide sufficient conditions for both existence of optimal solutions to the primal LP program and absence of duality gap. Then, existence of a (possibly randomized) average optimal policy is also guaranteed. Existence of a stationary average optimal deterministic policy is also investigated.
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  • 7
    ISSN: 1432-5217
    Keywords: Key words: Markov decision process ; extra constraints ; countable state space ; total cost
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract. The aim of this paper is to investigate the Lagrangian approach and a related Linear Programming (LP) that appear in constrained Markov decision processes (CMDPs) with a countable state space and total expected cost criteria (of which the expected discounted cost is a special case). We consider transient MDPs and MDPs with uniform Lyapunov functions, and obtain for these an LP which is the dual of another one that has been shown to provide the optimal values and stationary policies [3, 4]. We show that there is no duality gap between these LPs under appropriate conditions. In obtaining the Linear Program for the general transient case, we establish, in particular, a calculation approach for the value function of the CMDP based on finite state approximation. Unlike previous approaches for state approximations for CMDPs (most of which were derived for the contracting framework), we do not need here any Slater type condition. We finally present another type of LP that allows the computation of optimal mixed stationary-deterministic policies.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Optimization theory ; mathematical programming ; duality ; half spaces ; minimum norm duality ; separating hyperplanes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This note shows that half spaces play a very special role in the development of duality. In addition to the minimum norm duality, the duality in linear programming, and Wolfe's and Johri's formulations in nonlinear programming can all be derived via half spaces by following an identical five step procedure.
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  • 9
    ISSN: 1432-5217
    Keywords: Constrained Markov Decision Processes ; countable state space ; infinite horizon ; unbounded cost ; total cost criterion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper is the third in a series on constrained Markov decision processes (CMDPs) with a countable state space and unbounded cost. In the previous papers we studied the expected average and the discounted cost. We analyze in this paper the total cost criterion. We study the properties of the set of occupation measures achieved by different classes of policies; we then focus on stationary policies and on mixed deterministic policies and present conditions under which optimal policies exist within these classes. We conclude by introducing an equivalent infinite Linear Program.
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  • 10
    ISSN: 1432-5217
    Keywords: Key words: Stochastic games ; countable state space ; sensitive optimality ; unbounded payoffs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract. We consider stochastic games with countable state spaces and unbounded immediate payoff functions. Our assumptions on the transition structure of the game are based on a recent work by Meyn and Tweedie [19] on computable bounds for geometric convergence rates of Markov chains. The main results in this paper concern the existence of sensitive optimal strategies in some classes of zero-sum stochastic games. By sensitive optimality we mean overtaking or 1-optimality. We also provide a new Nash equilibrium theorem for a class of ergodic nonzero-sum stochastic games with denumerable state spaces.
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