Title Neural approximations for optimal control and decision / Riccardo Zoppoli, Marcello Sanguineti, Giorgio Gnecco, Thomas Parisini

Published Cham : Springer, 2020

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Description 1 online resource (532 pages) Series Communications and Control Engineering Communications and control engineering. Contents Intro -- Preface -- Contents -- 1 The Basic Infinite-Dimensional or Functional Optimization Problem -- 1.1 General Comments on Infinite-Dimensional or Functional Optimization -- 1.1.1 IDO and FDO Problems -- 1.1.2 From the Ritz Method to the Extended Ritz Method (ERIM) -- 1.1.3 Approximation of Functions -- 1.1.4 From Function Approximation to Approximate Infinite-Dimensional Optimization -- 1.1.5 Relationships with Parametrized Control Approaches -- 1.2 Contents and Structure of the Book -- 1.3 Infinite-Dimensional Optimization -- 1.3.1 Statement of the Problem 1.3.2 Finite-Dimensional Versus Infinite-Dimensional Optimization -- 1.4 General Conventions and Assumptions -- 1.4.1 Existence and Uniqueness of Minimizers -- 1.4.2 Other Definitions, Conventions, and Assumptions -- 1.5 Examples of Infinite-Dimensional Optimization Problems -- 1.5.1 A Deterministic Continuous-Time Optimal Control Problem -- 1.5.2 A Continuous-Time Network Flow Problem -- 1.5.3 A T-Stage Stochastic Optimal Control Problem -- 1.5.4 An Optimal Estimation Problem -- 1.5.5 A Static Team Optimal Control Problem -- References 2 From Functional Optimization to Nonlinear Programming by the Extended Ritz Method -- 2.1 Fixed-Structure Parametrized (FSP) Functions -- 2.2 The Sequence of Nonlinear Programming Problems Obtained by FSP Functions of Increasing Complexity -- 2.2.1 The Case of Problem P -- 2.2.2 The Case of Problem PM -- 2.3 Solution of the Nonlinear Programming Problem Pn -- 2.4 Optimizing FSP Functions -- 2.5 Polynomially Complex Optimizing FSP Functions -- 2.5.1 The Growth of the Dimension d -- 2.5.2 Polynomial and Exponential Growths of the Model Complexity n with the Dimension d -- 2.6 Approximating Sets 2.7 Polynomially Complex Approximating Sequences of Sets -- 2.7.1 The Worst-Case Error of Approximation of Functions -- 2.7.2 Polynomial and Exponential Growth of the Model Complexity n with the Dimension d -- 2.8 Connections Between Approximating Sequences and Optimizing Sequences -- 2.8.1 From mathcalSd-Approximating Sequences of Sets to Pd-Optimizing Sequences of FSP Functions -- 2.8.2 From Pd-Optimizing Sequences of FSP Functions to Polynomially Complex Pd-Optimizing Sequences of FSP Functions -- 2.8.3 Final Remarks -- 2.9 Notes on the Practical Application of the ERIM -- References 3 Some Families of FSP Functions and Their Properties -- 3.1 Linear Combinations of Fixed-Basis Functions -- 3.2 One-Hidden-Layer Networks -- 3.2.1 The Structure of OHL Networks -- 3.2.2 A More Abstract View -- 3.2.3 Tensor-Product, Ridge, and Radial Constructions -- 3.3 Multi-Hidden-Layer Networks -- 3.4 Terminology -- 3.5 Kernel Smoothing Models -- 3.6 Density Properties -- 3.6.1 mathscrC- and mathscrLp-Density Properties -- 3.6.2 The Case of Ridge OHL Networks -- 3.6.3 The Case of Radial OHL Networks -- 3.6.4 Multiple Hidden Layers and Multiple Outputs Summary Neural Approximations for Optimal Control and Decision provides a comprehensive methodology for the approximate solution of functional optimization problems using neural networks and other nonlinear approximators where the use of traditional optimal control tools is prohibited by complicating factors like non-Gaussian noise, strong nonlinearities, large dimension of state and control vectors, etc. Features of the text include: " a general functional optimization framework; " thorough illustration of recent theoretical insights into the approximate solutions of complex functional optimization problems; " comparison of classical and neural-network based methods of approximate solution; " bounds to the errors of approximate solutions; " solution algorithms for optimal control and decision in deterministic or stochastic environments with perfect or imperfect state measurements over a finite or infinite time horizon and with one decision maker or several; " applications of current interest: routing in communications networks, traffic control, water resource management, etc.; and " numerous, numerically detailed examples. The authors diverse backgrounds in systems and control theory, approximation theory, machine learning, and operations research lend the book a range of expertise and subject matter appealing to academics and graduate students in any of those disciplines together with computer science and other areas of engineering Notes 3.7 From Universality of Approximation to Moderate Model Complexity Bibliography Includes bibliographical references and index Notes Print version record Subject Control theory -- Mathematical models Mathematical optimization. Control theory -- Mathematical models Mathematical optimization Form Electronic book Author Sanguineti, Marcello Gnecco, Giorgio Parisini, Thomas ISBN 9783030296933 3030296938

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