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Dive into the research topics where Harold J. Kushner is active.

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Featured researches published by Harold J. Kushner.


Siam Journal on Control and Optimization | 1990

Numerical methods for stochastic control problems in continuous time

Harold J. Kushner

A powerful and usable class of methods for numerically approximating the solutions to optimal stochastic control problems for diffusion, reflected diffusion, or jump-diffusion models is discussed. The basic idea involves uconsistent approximation of the model by a Markov chain, and then solving an appropriate optimization problem for the Murkoy chain model. A general method for obtaining a useful approximation is given. All the standard classes of cost functions can be handled here, for illustrative purposes, discounted and average cost per unit time problems with both reflecting and nonreflecting diffusions are concentrated on. Both the drift and the variance can be controlled. Owing to its increasing importance and to lack of material on numerical methods, an application to the control of queueing and production systems in heavy traffic is developed in detail. The methods of proof of convergence are relatively simple, using only some basic ideas in the theory of weak convergence of a sequence of probabi...


Journal of the American Statistical Association | 1997

Stochastic approximation algorithms and applications

Harold J. Kushner; G. Yin

Applications and issues application to learning, state dependent noise and queueing applications to signal processing and adaptive control mathematical background convergence with probability one - Martingale difference noise convergence with probability one - correlated noise weak convergence - introduction weak convergence methods for general algorithms applications - proofs of convergence rate of convergence averaging of the iterates distributed/decentralized and asynchronous algorithms.


Journal of the American Statistical Association | 1985

Approximation and weak convergence methods for random processes, with applications to stochastic systems theory

Harold J. Kushner

Control and communications engineers, physicists, and probability theorists, among others, will find this book unique. It contains a detailed development of approximation and limit theorems and methods for random processes and applies them to numerous problems of practical importance. In particular, it develops usable and broad conditions and techniques for showing that a sequence of processes converges to a Markov diffusion or jump process. This is useful when the natural physical model is quite complex, in which case a simpler approximation la diffusion process, for example) is usually made.The book simplifies and extends some important older methods and develops some powerful new ones applicable to a wide variety of limit and approximation problems. The theory of weak convergence of probability measures is introduced along with general and usable methods (for example, perturbed test function, martingale, and direct averaging) for proving tightness and weak convergence.Kushners study begins with a systematic development of the method. It then treats dynamical system models that have state-dependent noise or nonsmooth dynamics. Perturbed Liapunov function methods are developed for stability studies of nonMarkovian problems and for the study of asymptotic distributions of non-Markovian systems. Three chapters are devoted to applications in control and communication theory (for example, phase-locked loops and adoptive filters). Smallnoise problems and an introduction to the theory of large deviations and applications conclude the book.Harold J. Kushner is Professor of Applied Mathematics and Engineering at Brown University and is one of the leading researchers in the area of stochastic processes concerned with analysis and synthesis in control and communications theory. This book is the sixth in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky.


IEEE Transactions on Wireless Communications | 2004

Convergence of proportional-fair sharing algorithms under general conditions

Harold J. Kushner; Philip A. Whiting

We are concerned with the allocation of the base station transmitter time in time-varying mobile communications with many users who are transmitting data. Time is divided into small scheduling intervals, and the channel rates for the various users are available at the start of the intervals. Since the rates vary randomly, in selecting the current user there is a conflict between full use (by selecting the user with the highest current rate) and fairness (which entails consideration for users with poor throughput to date). The proportional fair scheduler of the Qualcomm High Data Rate system and related algorithms are designed to deal with such conflicts. The aim here is to put such algorithms on a sure mathematical footing and analyze their behavior. The available analysis, while obtaining interesting information, does not address the actual convergence for arbitrarily many users under general conditions. Such algorithms are of the stochastic approximation type and results of stochastic approximation are used to analyze the long-term properties. It is shown that the limiting behavior of the sample paths of the throughputs converges to the solution of an intuitively reasonable ordinary differential equation, which is akin to a mean flow. We show that the ordinary differential equation (ODE) has a unique equilibrium and that it is characterized as optimizing a concave utility function, which shows that PFS is not ad-hoc, but actually corresponds to a reasonable maximization problem. These results may be used to analyze the performance of PFS. The results depend on the fact that the mean ODE has a special form that arises in problems with certain types of competitive behavior. There is a large set of such algorithms, each one corresponding to a concave utility function. This set allows a choice of tradeoffs between the current rate and throughout. Extensions to multiple antenna and frequency systems are given. Finally, the infinite backlog assumption is dropped and the data is allowed to arrive at random. This complicates the analysis, but the same results hold.


Archive | 1990

Weak convergence methods and singularly perturbed stochastic control and filtering problems

Harold J. Kushner

1 Weak Convergence.- 0. Outline of the Chapter.- 1. Basic Properties and Definitions.- 2. Examples.- 3. The Skorohod Representation.- 4. The Function Space Ck [0, T].- 5. The Function Space Dk [0, T].- 6. Measure Valued Random Variables and Processes.- 2 Stochastic Processes: Background.- 0. Outline of the Chapter.- 1. Martingales.- 2. Stochastic Integrals and Itos Lemma.- 3. Stochastic Differential Equations: Bounds.- 4. Controlled Stochastic Differential Equations: Existence of Solutions.- 5. Representing a Martingale as a Stochastic Integral.- 6. The Martingale Problem.- 7. Jump-Diffusion Processes.- 8. Jump-Diffusion Processes: The Martingale Problem Formulation.- 3 Controlled Stochastic Differential Equations.- 0. Outline of the Chapter.- 1. Controlled S.D.E.s: Introduction.- 2. Relaxed Controls: Deterministic Case.- 3. Stochastic Relaxed Controls.- 4. The Martingale Problem Revisited.- 5. Approximations, Weak Convergence and Optimality.- 4 Controlled Singularly Perturbed Systems.- 0. Outline of the Chapter.- 1. Problem Formulation: Finite Time Interval.- 2. Approximation of the Optimal Controls and Value Functions.- 3. Discounted Cost and Optimal Stopping Problems.- 4. Average Cost Per Unit Time.- 5. Jump-Diffusion Processes.- 6. Other Approaches.- 5 Functional Occupation Measures and Average Cost Per Unit Time Problems.- 0. Outline of the Chapter.- 1. Measure Valued Random Variables.- 2. Limits of Functional Occupation Measures for Diffusions.- 3. The Control Problem.- 4. Singularly Perturbed Control Problems.- 5. Control of the Fast System.- 6. Reflected Diffusions.- 7. Discounted Cost Problem.- 6 The Nonlinear Filtering Problem.- 0. Outline of the Chapter.- 1. A Representation of the Nonlinear Filter.- 2. The Filtering Problem for the Singularly Perturbed System.- 3. The Almost Optimality of the Averaged Filter.- 4. A Counterexample to the Averaged Filter.- 5. The Near Optimality of the Averaged Filter.- 6. A Repair and Maintainance Example.- 7. Robustness of the Averaged Filters.- 8. A Robust Computational Approximation to the Averaged Filter.- 9. The Averaged Filter on the Infinite Time Interval.- 7 Weak Convergence: The Perturbed Test Function Method.- 0. Outline of the Chapter.- 1. An Example.- 2. The Perturbed Test Function Method: Introduction.- 3. The Perturbed Test Function Method: Tightness and Weak Convergence.- 4. Characterization of the Limits.- 8 Singularly Perturbed Wide-Band Noise Driven Systems.- 0. Outline of the Chapter.- 1. The System and Noise Model.- 2. Weak Convergence of the Fast System.- 3. Convergence to the Averaged System.- 4. The Optimality Theorem.- 5. The Average Cost Per Unit Time Problem.- 9 Stability Theory.- 0. Outline of the Chapter.- 1. Stability Theory for Jump-Diffusion Processes of Ito Type.- 2. Singularly Perturbed Deterministic Systems: Bounds on Paths.- 3. Singularly Perturbed Ito Processes: Tightness.- 4. The Linear Case.- 5. Wide Bandwidth Noise.- 6. Singularly Perturbed Wide Bandwidth Noise Driven Systems.- 10 Parametric Singularities.- 0. Outline of the Chapter.- 1. Singularly Perturbed Ito Processes: Weak Convergence.- 2. Stability.- References.- List of Symbols.


Siam Journal on Applied Mathematics | 1987

Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: Global minimization via Monte Carlo

Harold J. Kushner

The asymptotic behavior of the systems


Archive | 1978

Weak Convergence of Probability Measures

Harold J. Kushner; Dean S. Clark

X_{n + 1} = X_n + a_n b( {X_n ,\xi _n } ) + a_n \sigma ( X_n )\psi_n


Siam Journal on Control and Optimization | 1981

Asymptotic Properties of Stochastic Approximations with Constant Coefficients.

Harold J. Kushner; Hai Huang

and


Siam Journal on Control and Optimization | 1987

Asymptotic properties of distributed and communication stochastic approximation algorithms

Harold J. Kushner; G. Yin

dy = \bar b( y )dt + \sqrt {a( t )} \sigma ( y )dw


Siam Journal on Control and Optimization | 1993

Stochastic approximation with averaging of the iterates: optimal asymptotic rate of convergence for general processes

Harold J. Kushner; Jichuan Yang

is studied, where

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G. Yin

Wayne State University

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Robert T. Buche

North Carolina State University

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Amarjit Budhiraja

University of North Carolina at Chapel Hill

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Luiz Felipe Martins

Worcester Polytechnic Institute

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