## ＜電子ブック＞Stochastic Approximation Methods for Constrained and Unconstrained Systems

責任表示 by Harold J. Kushner, Dean S. Clark Kushner, Harold J Clark, Dean S SpringerLink (Online service) English (英語) Springer New York 1978- New York, NY, United States シリーズ Applied Mathematical Sciences ; 26 The book deals with a powerful and convenient approach to a great variety of types of problems of the recursive monte-carlo or stochastic approximation type. Such recu- sive algorithms occur frequentl...y in stochastic and adaptive control and optimization theory and in statistical esti- tion theory. Typically, a sequence {X } of estimates of a n parameter is obtained by means of some recursive statistical th st procedure. The n estimate is some function of the n_l estimate and of some new observational data, and the aim is to study the convergence, rate of convergence, and the pa- metric dependence and other qualitative properties of the - gorithms. In this sense, the theory is a statistical version of recursive numerical analysis. The approach taken involves the use of relatively simple compactness methods. Most standard results for Kiefer-Wolfowitz and Robbins-Monro like methods are extended considerably. Constrained and unconstrained problems are treated, as is the rate of convergence problem. While the basic method is rather simple, it can be elaborated to allow a broad and deep coverage of stochastic approximation like problems. The approach, relating algorithm behavior to qualitative properties of deterministic or stochastic differ ential equations, has advantages in algorithm conceptualiza tion and design. It is often possible to obtain an intuitive understanding of algorithm behavior or qualitative dependence upon parameters, etc., without getting involved in a great deal of deta~l.続きを見る I. Introduction1.1. General Remarks1.2. The Robbins-Monro Process1.3. A 'Continuous' Process Version of Section 21.4. Regulation of a Dynamical System; a simple example1.5. Function Minimization: The Kiefer-Wolfowitz Procedure1.6. Constrained Problems1.7. An Economics ExampleII. Convergence w.p.1 for Unconstrained Systems2.1. Preliminaries and Motivation2.2. The Robbins-Monro and Kiefer-Wolfowitz Algorithms: Conditions and Discussion2.3. Convergence Proofs for RM and KW-like Procedures2.4. A General Robbins-Monro Process: 'Exogenous Noise'2.5. A General RM Process; State Dependent Noise2.5.1. Extensions and Localizations of Theorem 2.5.22.6. Some Applications2.7. Mensov-Rademacher EstimatesIII. Weak Convergence of Probability MeasuresIV. Weak Convergence for Unconstrained Systems4.1. Conditions and General Discussion4.2. The Robbins-Monro and Kiefer-Wolfowitz Procedures4.3. A General Robbins-Monro Process: Exogenous Noise4.4. A General RM Process: State Dependent Noise4.5. The Identification Problem4.6. A Counter-Example to Tightness4.7. Boundedness of {Xn} and Tightness of {Xn(-)}V. Convergence w.p.1 For Constrained Systems5.1. A Penalty-Multiplier Algorithm for Equality Constraints5.2. A Lagrangian Method for Inequality Constraints5.3. A Projection Algorithm5.4. A Penalty-Multiplier Method for Inequality ConstraintsVI. Weak Convergence: Constrained Systems6.1. A Multiplier Type Algorithm for Equality Constraints6.2. The Lagrangian Method6.3. A Projection Algorithm6.4. A Penalty-Multiplier Algorithm for Inequality ConstraintsVII. Rates of Convergence7.1. The Problem Formulation7.2. Conditions and Discussions7.3. Rates of Convergence for Case 1, the KW Algorithm7.4. Discussion of Rates of Convergence for Two KW Algorithms.続きを見る http://hdl.handle.net/2324/1000935591 Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

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レコードID 3646320 QA1-939 510 Mathematics. Mathematics. Mathematics, general. ssj0001298586 9780387903415(print) 9781468493528 2020.06.27 2020.06.28