Stochastic Approximation Methods for Constrained and Unconstrained Systems I. Introduction 1.1. General Remarks 1.2. The Robbins Monro Process 1.3. A “Continuous” Process Version of Section 2 1.4. Regulation of a Dynamical System; a simple example 1.5. Function Minimization: The Kiefer Wolfowitz Procedure 1.6. Constrained Problems 1.7. An Economics Example II. Convergence w.p.1 for Unconstrained Systems 2.1. Preliminaries and Motivation 2.2. The Robbins Monro and Kiefer Wolfowitz Algorithms: Conditions and Discussion 2.3. Convergence Proofs for RM and KW like Procedures 2.4. A General Robbins Monro Process: “Exogenous Noise” 2.5. A General RM Process; State Dependent Noise 2.5.1. Extensions and Localizations of Theorem 2.5.2 2.6. Some Applications 2.7. Mensov Rademacher Estimates III. Weak Convergence of Probability Measures IV. Weak Convergence for Unconstrained Systems 4.1. Conditions and General Discussion 4.2. The Robbins Monro and Kiefer Wolfowitz Procedures 4.3. A General Robbins Monro Process: Exogenous Noise 4.4. A General RM Process: State Dependent Noise 4.5. The Identification Problem 4.6. A Counter Example to Tightness 4.7. Boundedness of {Xn} and Tightness of {Xn(•)} V. Convergence w.p.1 For Constrained Systems 5.1. A Penalty Multiplier Algorithm for Equality Constraints 5.2. A Lagrangian Method for Inequality Constraints 5.3. A Projection Algorithm 5.4. A Penalty Multiplier Method for Inequality Constraints VI. Weak Convergence: Constrained Systems 6.1. A Multiplier Type Algorithm for Equality Constraints 6.2. The Lagrangian Method 6.3. A Projection Algorithm 6.4. A Penalty Multiplier Algorithm for Inequality Constraints VII. Rates of Convergence 7.1. The Problem Formulation 7.2. Conditions and Discussions 7.3. Rates of Convergence for Case 1 the KW Algorithm 7.4. Discussion of Rates of Convergence for Two KW Algorithms. 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 frequently 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.