Stochastic Approximation Algorithms and Applications

Kushner, Harold J.; Yin, George

doi:10.1007/978-1-4899-2696-8

Cited by 1,263 publications

(1,599 citation statements)

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“…Using our iterative procedure (5)- (6) with a gradient obtained through either (33) or (34), if an appropriate step size sequence is selected, the continuous state ρ converges to a point that satisfies Theorem 5.1 and has the optimal allocation [5,5,5,5] 0 in its neighborhood. To illustrate this, using (33) and η n = 0.05 n+1 , the states evolve as follows: 6 Generalization to arbitrary feasible sets…”

Section: Recovery Of Optimal Discrete Statesmentioning

confidence: 99%

“…Convergence of the projected algorithm has also been considered in the literature (e.g., [32], [33]). In this paper we shall follow a very similar line of proof based on results from martingale convergence arguments, a method that seems to have originated with Gladyshev [28].…”

Section: Convergence Analysismentioning

confidence: 99%

“…We let k.k denote the standard Euclidean norm on R N . The assumptions that we will need for the convergence theorem are the standard ones found in the literature (e.g., [32], [33]) except for (H1) and they are as follows:…”

Section: Convergence Analysismentioning

confidence: 99%

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Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems

Gokbayrak

Cassandras

2001

Journal of Optimization Theory and Applications

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We consider stochastic discrete optimization problems where the decision variables are non-negative integers. We propose and analyze an on-line control scheme which transforms the problem into a "surrogate" continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches while simultaneously updating both actual and surrogate system states. It is shown that the solution of the original problem is recovered as an element of the discrete state neighborhood of the optimal surrogate state. For the special case of separable cost functions, we show that this methodology becomes particularly efficient. Finally, convergence of the proposed algorithm is established under standard technical conditions and numerical results are included in the paper to illustrate the fast convergence properties of this approach.

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Section: Recovery Of Optimal Discrete Statesmentioning

confidence: 99%

Section: Convergence Analysismentioning

confidence: 99%

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Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems

Gokbayrak

Cassandras

2001

Journal of Optimization Theory and Applications

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“…Lemma A.2 (Theorem 2.3. of Kushner and Yin (2003)). Let {θ(i) : i ≥ 0} in H = r j=1 [a j , b j ] be a multivariate stochastic process satisfying,…”

Section: A2 Proofsmentioning

confidence: 95%

Monopoly Pricing in the Presence of Social Learning

Crapis¹,

et al. 2017

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A monopolist offers a product to a market of consumers with heterogeneous quality preferences. Although initially uninformed about the product quality, they learn by observing past purchase decisions and reviews of other consumers. Our goal is to analyze the social learning mechanism and its effect on the seller's pricing decision. This analysis borrows from the literature on social learning and on pricing and revenue management.Consumers follow a naive decision rule and, under some conditions, eventually learn the product's quality. Using mean-field approximation, the dynamics of this learning process are characterized for markets with high demand intensity. The relationship between the price and the speed of learning depends on the heterogeneity of quality preferences. Two pricing strategies are studied: a static price and a single price change. Properties of the optimal prices are derived. Numerical experiments suggest that pricing strategies that account for social learning may increase revenues considerably relative to strategies that do not.

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“…A formal analysis of the properties of the algorithm might be done in this framework and is left for further work. Background material can be found in [9,31,25].…”

Section: Variablesmentioning

confidence: 99%

Optimal Data Gathering Paths and Energy Balance Mechanisms in Wireless Networks

Jarry

Leone

Nikoletseas

et al. 2010

Distributed Computing in Sensor Systems

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Abstract. This paper studies the data gathering problem in wireless networks, where data generated at the nodes has to be collected at a single sink. We investigate the relationship between routing optimality and fair resource management. In particular, we prove that for energy balanced data propagation, Pareto optimal routing and flow maximization are equivalent, and also prove that flow maximization is equivalent to maximizing the network lifetime. We algebraically characterize the network structures in which energy balanced data flows are maximal. Moreover, we algebraically characterize communication links which are not used by an optimal flow. This leads to the characterization of minimal network structures supporting the maximal flows. We note that energy balance, although implying global optimality, is a local property that can be computed efficiently and in a distributed manner. We suggest online distributed algorithms for energy balance in different optimal network structures and numerically show their stability in particular setting. We remark that although the results obtained in this paper have a direct consequence in energy saving for wireless networks they do not limit themselves to this type of networks neither to energy as a resource. As a matter of fact, the results are much more general and can be used for any type of network and different types of resources. 3

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Stochastic Approximation Algorithms and Applications

Abstract: Library of CongressCataloging-in-Publication Data Kushner, Harold J. (Harold Joseph), 1933-Stochastic approximation algorithms and applications/Harold J. Kushner, G. George Yin. p. cm. -(Applications of mathematics; 35) Includes bibliographical references and index.

Cited by 1,263 publications

References 0 publications

Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems

Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems

Monopoly Pricing in the Presence of Social Learning

Optimal Data Gathering Paths and Energy Balance Mechanisms in Wireless Networks

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