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E-grāmata: Stochastic Multi-Stage Optimization: At the Crossroads between Discrete Time Stochastic Control and Stochastic Programming

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Stochastic Multi-Stage Optimization: At the Crossroads between Discrete Time Stochastic Control and Stochastic ProgrammingPalielināt
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The focus of the present volume is stochastic optimization of dynamical systems in discrete time where - by concentrating on the role of information regarding optimization problems - it discusses the related discretization issues. There is a growing need to tackle uncertainty in applications of optimization. For example the massive introduction of renewable energies in power systems challenges traditional ways to manage them. This book lays out basic and advanced tools to handle and numerically solve such problems and thereby is building a bridge between Stochastic Programming and Stochastic Control. It is intended for graduates readers and scholars in optimization or stochastic control, as well as engineers with a background in applied mathematics.

I Preliminaries.- 1.Issues and Problems in Decision Making under Uncertainty.- 2.Open-Loop Control: The Stochastic Gradient Method.- II Decision under Uncertainty and the Role of Information.- 3.Tools for Information Handling.- 4.Information and Stochastic Optimization Problems.- Optimality Conditions for SOC Problems.- III Discretization and Numerical Methods.- 6.Discretization Methodology for Problems with SIS.- 7.Numerical Algorithms.- IV Convergence Analysis.- 8.Convergence Issues in Stochastic Optimization.- V Advanced Topics.- 9.Multi-Agent Decision Problems.- Dual Effect for Multi-Agent Stochastic I-O Systems.- VI Appendices.- A. Basics in Analysis and Optimization.- B. Basics in Probability.- References.- Index.

Recenzijas

I consider the book as a guide to the different aspects of stochastic optimization and the most important motive for distinguishing it from other similar textbooks and monographs is a great emphasis put on the role of information. (Jerzy Ombach, zbMATH 1336.90066, 2016)

Part I Preliminaries
1 Issues and Problems in Decision Making Under Uncertainty
3(24)
1.1 Introduction
3(4)
1.1.1 Decision Making as Constrained Optimization Problems
3(1)
1.1.2 Facing Uncertainty
4(1)
1.1.3 The Role of Information in the Presence of Uncertainty
5(2)
1.2 Problem Formulations and Information Structures
7(5)
1.2.1 Stochastic Optimal Control (SOC)
7(3)
1.2.2 Stochastic Programming (SP)
10(2)
1.3 Examples
12(4)
1.3.1 A Basic Example in Static Information
12(1)
1.3.2 The Communication Channel
13(3)
1.3.3 Witsenhausen's Celebrated Counterexample
16(1)
1.4 Discretization Issues
16(9)
1.4.1 Problems with Static Information Structure (SIS)
17(1)
1.4.2 Working Out an Example
18(7)
1.5 Conclusion
25(2)
2 Open-Loop Control The Stochastic Gradient Method
27(38)
2.1 Introduction
27(1)
2.2 Open-Loop Optimization Problems
28(3)
2.2.1 Problem Statement
28(2)
2.2.2 Sample Approximation in Stochastic Optimization
30(1)
2.3 Stochastic Gradient Method Overview
31(8)
2.3.1 Stochastic Gradient Algorithm
31(3)
2.3.2 Connection with Stochastic Approximation
34(5)
2.4 Convergence Analysis
39(6)
2.4.1 Auxiliary Problem Principle
40(1)
2.4.2 Stochastic Auxiliary Problem Principle Algorithm
41(1)
2.4.3 Convergence Theorem
42(3)
2.4.4 Conclusions
45(1)
2.5 Efficiency and Averaging
45(5)
2.5.1 Stochastic Newton Algorithm
45(3)
2.5.2 Stochastic Gradient Algorithm with Averaging
48(1)
2.5.3 Sample Average Approximation
49(1)
2.6 Practical Considerations
50(6)
2.6.1 Stopping Criterion
51(1)
2.6.2 Tuning the Standard Algorithm
51(3)
2.6.3 Robustness of the Averaged Algorithm
54(2)
2.7 Conclusion
56(1)
2.8 Appendix
57(8)
2.8.1 Robbins-Siegmund Theorem
57(1)
2.8.2 A Technical Lemma
58(1)
2.8.3 Proof of Theorem 2.17
59(6)
Part II Decision Under Uncertainty and the Role of Information
3 Tools for Information Handling
65(1)
3 1 Introduction
65(30)
3.2 Basic Facts on Binary Relations and on Lattices
66(5)
3.2.1 Binary Relations
66(4)
3.2.2 Lattices
70(1)
3.3 Partitions and Fields Approach
71(9)
3.3.1 The Lattice of Partitions/Equivalence Relations
71(3)
3.3.2 The Lattice of π-Fields (Partition Fields)
74(4)
3.3.3 The Lattice of σ-Fields
78(2)
3.4 Mapping Measurability Approach
80(7)
3.4.1 Measurability of Mappings w.r.t. Partitions
80(1)
3.4.2 Measurability of Mappings w.r.t. π-Fields
81(5)
3.4.3 Measurability of Mappings w.r.t. σ-Fields
86(1)
3.5 Conditional Expectation and Optimization
87(6)
3.5.1 Conditional Expectation w.r.t. a Partition
87(3)
3.5.2 Interchanging Minimization and Conditional Expectation
90(2)
3.5.3 Conditional Expectation as an Optimal Value of a Minimization Problem
92(1)
3.6 Conclusion
93(2)
4 Information and Stochastic Optimization Problems
95(38)
4.1 Introduction
95(1)
4.2 The Witsenhausen Counterexample
96(5)
4.2.1 A Simple Linear Quadratic Control Problem
96(2)
4.2.2 Problem Transformation Exploiting Sequentiality
98(2)
4.2.3 The Dual Effect of the Initial Decision
100(1)
4.3 Other Information Patterns
101(3)
4.3.1 Full Noise Observation
101(1)
4.3.2 Classical Information Pattern
102(1)
4.3.3 Markovian Information Pattern
103(1)
4.3.4 Past Control Observation
103(1)
4.3.5 The Witsenhausen Counterexample
104(1)
4.4 State Model and Dynamic Programming (DP)
104(7)
4.4.1 State Model
105(1)
4.4.2 State Feedbacks, Decisions, State and Control Maps
106(2)
4.4.3 Criterion
108(1)
4.4.4 Stochastic Optimization Problem
109(1)
4.4.5 Stochastic Dynamic Programming
109(2)
4.5 Sequential Optimization Problems
111(21)
4.5.1 Sequential Optimal Stochastic Control Problem
112(3)
4.5.2 Optimal Stochastic Control Problem in Standard Form
115(4)
4.5.3 What Is a State?
119(1)
4.5.4 Dynamic Programming Equations
119(13)
4.6 Conclusion
132(1)
5 Optimality Conditions for Stochastic Optimal Control (SOC) Problems
133(22)
5.1 Introduction
133(1)
5.2 SOC Problems, Formulation and Assumptions
134(4)
5.2.1 Dynamics
135(1)
5.2.2 Cost Function
135(1)
5.2.3 Constraints
136(1)
5.2.4 The Stochastic Programming (SP) Version
137(1)
5.3 Optimality Conditions for the SP Formulation
138(3)
5.3.1 Projection on the Feasible Set
138(2)
5.3.2 Stationary Conditions
140(1)
5.4 Optimality Conditions for the SOC Formulation
141(5)
5.4.1 Computation of the Cost Gradient
141(3)
5.4.2 Optimality Conditions with Non-adapted Co-States
144(1)
5.4.3 Optimality Conditions with Adapted Co-States
145(1)
5.5 The Markovian Case
146(5)
5.5.1 Markovian Setting and Assumptions
146(1)
5.5.2 Optimality Conditions with Non-adapted Co-States
147(2)
5.5.3 Optimality Conditions with Adapted Co-States
149(1)
5.5.4 Optimality Conditions from a Functional Point of View
150(1)
5.6 Conclusions
151(4)
Part III Discretization and Numerical Methods
6 Discretization Methodology for Problems with Static Information Structure (SIS)
155(26)
6.1 Quantization
156(4)
6.1.1 Set-Theoretic Quantization
156(1)
6.1.2 Optimal Quantization in Normed Vector Spaces
157(3)
6.2 A Systematic Approach to Discretization
160(14)
6.2.1 The Problematics of Discretization
160(1)
6.2.2 The Approach Inspired by Pennanen's Work
161(7)
6.2.3 A Constructive Proposal
168(6)
6.3 A Handicap of the Scenario Tree Approach
174(5)
6.3.1 How to Sample Noises to Get Scenario Trees
174(1)
6.3.2 Variance Analysis
175(4)
6.4 Conclusion
179(2)
7 Numerical Algorithms
181(30)
7.1 Introduction
181(2)
7.2 A Simple Benchmark Problem
183(4)
7.2.1 Formulation
183(3)
7.2.2 Numerical and Functional Data
186(1)
7.3 Manipulating Functions with a Computer and Implementation in Dynamic Programming (DP)
187(5)
7.3.1 The DP Equation
187(1)
7.3.2 Discrete Representation of a Function
188(2)
7.3.3 The Discrete DP Equation
190(2)
7.3.4 Application to the Benchmark Problem
192(1)
7.4 Resolution by the Scenario Tree Technique
192(8)
7.4.1 General Considerations
193(1)
7.4.2 Formulation of the Problem over a Scenario Tree
194(2)
7.4.3 Optimality Conditions and Resolution
196(1)
7.4.4 About Feedback Synthesis
197(1)
7.4.5 Results Obtained for the Benchmark Problem
198(2)
7.5 The Particle Method
200(6)
7.5.1 Algorithm
201(3)
7.5.2 Results Obtained for the Benchmark Problem and Comments
204(2)
7.6 Conclusion
206(5)
Part IV Convergence Analysis
8 Convergence Issues in Stochastic Optimization
211(44)
8.1 Introduction
211(2)
8.2 Convergence Notions
213(6)
8.2.1 Epi-Convergence and Mosco Convergence
213(4)
8.2.2 Convergence of Subfields
217(2)
8.3 Operations on Integrands
219(8)
8.3.1 Multifunctions
219(1)
8.3.2 Integrands
220(2)
8.3.3 Upper Integral
222(1)
8.3.4 Conditional Expectation of a Normal Integrand
223(1)
8.3.5 Interchange of Minimization and Integration
224(3)
8.4 Application to Open-Loop Optimization Problems
227(1)
8.5 Application to Closed-Loop Optimization Problems
228(24)
8.5.1 Main Convergence Theorem
228(3)
8.5.2 Revisiting a Basic Example in Static Information
231(2)
8.5.3 Discussion About Related Works
233(2)
8.5.4 Revisiting the Example of Sect. 1.4.2
235(10)
8.5.5 Companion Propositions to Theorem 8.42
245(7)
8.6 Conclusion
252(3)
Part V Multi-Agent Systems
9 Multi-Agent Decision Problems
255(38)
9.1 Introduction
255(1)
9.2 Witsenhausen Intrinsic Model
256(6)
9.2.1 The Extensive Space of Decisions and States of Nature
256(3)
9.2.2 Information Fields and Policies
259(3)
9.3 Causality and Solvability for Stochastic Control Systems
262(3)
9.3.1 Solvability and Solvability/Measurability
262(2)
9.3.2 Causality
264(1)
9.3.3 Solvability, Causality and "State"
264(1)
9.4 Four Binary Relations Between Agents
265(9)
9.4.1 The Precedence Relation B
265(2)
9.4.2 The Subsystem Relation
267(3)
9.4.3 The Information-Memory Relation M
270(2)
9.4.4 The Decision-Memory Relation D
272(2)
9.5 A Typology of Stochastic Control Systems
274(12)
9.5.1 A Typology of Systems
274(3)
9.5.2 Examples of Systems with Two Agents
277(5)
9.5.3 Partially Nested and Sequential Systems
282(3)
9.5.4 Summary Table
285(1)
9.6 Policy Independence of Conditional Expectations and Dynamic Programming
286(6)
9.6.1 Policy Independence of Conditional Expectations
286(2)
9.6.2 Application to Decomposition by Dynamic Programming
288(4)
9.7 Conclusion
292(1)
10 Dual Effect for Multi-Agent Stochastic Input-Output Systems
293(16)
10.1 Introduction
293(1)
10.2 Multi-Agent Stochastic Input-Output Systems (MASIOS)
294(6)
10.2.1 Definition of Multi-Agent Stochastic Input-Output Systems
294(1)
10.2.2 Control Laws
295(2)
10.2.3 Precedence and Memory-Communication Relations
297(1)
10.2.4 A Typology of MASIOS
298(2)
10.3 No Open-Loop Dual Effect and No Dual Effect Control Laws
300(7)
10.3.1 No Open-Loop Dual Effect (NOLDE)
301(1)
10.3.2 No Dual Effect Control Laws
301(1)
10.3.3 Characterization of No Dual Effect Control Laws
302(5)
10.4 Conclusion
307(2)
Appendix A Basics in Analysis and Optimization 309(18)
Appendix B Basics in Probability 327(22)
References 349(8)
Index 357
Professor Pierre Carpentiers primary research areas are Decomposition and Coordination for the Optimization of Large-Scale Systems in the stochastic framework, with a special interest in numerical methods. He is currently working at the applied mathematics unit UMA, ENSTA ParisTech, France. Professor J. Ph. Chanceliers research contributions have been in the fields of Stochastic Optimization, Control and Computer Languages for Numerical Computations. He currently holds a position at the applied mathematics center research CERMICS, École des Ponts ParisTech, France. The main research contributions of Professor Guy Cohen have been in the theory of Decomposition and Coordination for the Optimization of Large-Scale Systems, in the development of a linear theory of a certain class of Discrete Event Systems based on the use of the so-called Max-Plus algebra, and more recently in numerical methods for Stochastic Optimal Control. He is currently a Researcher Emeritus. Professor Michel De Laras main theoretical research fields are control theory and stochastic control. With regard to applications, he specializes in developing mathematical methods for the sustainable management of natural resources, concentrating on renewable energy and biodiversity. He currently holds a position at the applied mathematics center research CERMICS, École des Ponts ParisTech, France.
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