Dynamic Programming

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ISBN: 9780486428093 | 0486428095
Cover: Paperback
Copyright: 3/4/2003

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An introduction to the mathematical theory of multistage decision processes, this text takes a "functional equation" approach to the discovery of optimum policies. Written by a leading developer of such policies, it presents a series of methods, uniqueness and existence theorems, and examples for solving the relevant equations. The text examines existence and uniqueness theorems, the optimal inventory equation, bottleneck problems in multistage production processes, a new formalism in the calculus of variation, strategies behind multistage games, and Markovian decision processes. Each chapter concludes with a problem set that Eric V. Denardo of Yale University, in his informative new introduction, calls "a rich lode of applications and research topics." 1957 edition. 37 figures.

Introduction to the Dover Edition

Preface

vii

A Multi-Stage Allocation Process

Introduction

(1)

A multi-stage allocation process

(1)

Discussion

(2)

Functional equation approach

(2)

Discussion

(1)

A multi-dimensional maximization problem

(1)

A ``smoothing'' problem

(1)

Infinite stage approximation

(1)

Existence and uniqueness theorems

(4)

Successive approximations

(1)

Approximation in policy space

(3)

Properties of the solution---I: Convexity

(1)

Properties of the solution---II: Concavity

(2)

Properties of the solution---III: Concavity

(3)

An ``ornery'' example

(1)

A particular example---I

(2)

A particular example---II

(1)

Approximation and stability

(1)

Time-dependent processes

(1)

Multi-activity processes

(2)

Multi-dimensional structure theorems

(1)

Locating the unique maximum of a concave function

(3)

Continuity and memory

(1)

Stochastic allocation processes

(1)

Functional equations

(1)

Stieltjes integrals

(21)

Exercises and research problems

(19)

Bibliography and comments

(2)

A Stochastic Multi-Stage Decision Process

Introduction

(1)

Stochastic gold-mining

(1)

Enumerative treatment

(1)

Functional equation approach

(1)

Infinite stage approximation

(1)

Existence and uniqueness

(1)

Approximation in policy space and monotone convergence

(1)

The solution

(3)

Discussion

(1)

Some generalizations

(2)

The form of f (x, y)

(1)

The problem for a finite number of stages

(2)

A three-choice problem

(2)

A stability theorem

(5)

Exercises and research problems

(2)

Bibliography and comments

(2)

The Structure of Dynamic Programming Processes

Introduction

(1)

Discussion of the two preceding processes

(2)

The principle of optimality

(1)

Mathematical formulation---I. A discrete deterministic process

(2)

Mathematical formulation---II. A discrete stochastic process

(1)

Mathematical formulation---III. A continuous deterministic process

(1)

Continuous stochastic processes

(1)

Generalizations

(1)

Causality and optimality

(1)

Approximation in policy space

(28)

Exercises and research problems

(25)

Bibliography and comments

115

(1)

Existence and Uniqueness Theorems

Introduction

116

(1)

A fundamental inequality

117

(2)

Equations of type one

119

(2)

Equations of type two

121

(1)

Monotone convergence

122

(1)

Stability theorems

123

(1)

Some directions of generalization

124

(1)

An equation of the third type

125

(4)

An ``optimal inventory'' equation

129

(23)

Exercises and research problems

132

(19)

Bibliography and comments

151

(1)

The Optimal Inventory Equation

Introduction

152

(1)

Formulation of the general problem

153

(4)

Finite total time period

154

(2)

Unbounded time period---discounted cost

156

(1)

Unbounded time period---partially expendable items

156

(1)

Unbounded time period---one period lag in supply

156

(1)

Unbounded time period---two period lag

157

(1)

A simple observation

157

(1)

Constant stock level---preliminary discussion

158

(1)

Proportional cost---one-dimensional case

159

(5)

Proportional cost---multi-dimensional case

164

(2)

Finite time period

166

(3)

Finite time---multi-dimensional case

169

(1)

Non-proportional penalty cost---red tape

169

(2)

Particular cases

171

(1)

The form of the general solution

171

(1)

Fixed costs

172

(1)

Preliminaries to a discussion of more complicated policies

173

(1)

Unbounded process---one period time lag

173

(3)

Convex cost function---unbounded process

176

(7)

Exercises and research problems

178

(4)

Bibliography and comments

182

(1)

Bottleneck Problems in Multi-Stage Production Processes

Introduction

183

(1)

A general class of multi-stage production problems

184

(3)

Discussion of the preceding model

187

(1)

Functional equations

188

(1)

A continuous version

189

(2)

Notation

191

(1)

Dynamic programming formulation

192

(1)

The basic functional equation

192

(1)

The resultant nonlinear partial differential equation

193

(1)

Application of the partial differential equation

193

(1)

A particular example

194

(3)

A dual problem

197

(3)

Verification of the solution given in § 10

200

(2)

Computational solution

202

(1)

Nonlinear problems

203

(4)

Exercises and research problems

204

(1)

Bibliography and comments

205

(2)

Bottleneck Problems: Examples

Introduction

207

(2)

Preliminaries

209

(2)

Delta-functions

211

(1)

The solution

211

(3)

The modified w solution

214

(1)

The equilibrium solution

215

(2)

A short-time w solution

217

(1)

Description of solution and proof

218

(4)

Bibliography and comments

221

(1)

A Continuous Stochastic Decision Process

Introduction

222

(1)

Continuous versions---I: A differential approach

223

(1)

Continuous versions---II: An integral approach

224

(1)

Preliminary discussion

224

(2)

Mixing at a point

226

(1)

Reformulation of the gold-mining process

227

(1)

Derivation of the differential equations

227

(1)

The variational procedure

228

(1)

The behavior of Ki

229

(1)

The solution for T = ∞

230

(1)

Solution for finite total time

231

(2)

The three-choice problem

233

(1)

Some lemmas and preliminary results

234

(1)

Mixed policies

235

(1)

The solution for infinite time, D > 0

236

(4)

D ≤ 0

240

(1)

The case r3 = r4

241

(1)

Nonlinear utility---two-choice problem

242

(3)

Bibliography and comments

244

(1)

A New Formalism in the Calculus of Variations

Introduction

245

(1)

A new approach

246

(2)

Max y ∫∞ o F (x, y) dt

248

(3)

Discussion

251

(1)

The two-dimensional case

251

(1)

Max y ∫T o F (x, y) dt

252

(1)

Max y ∫T o F (x, y) dt under the constraint 0 ≤ y ≤ x

253

(1)

Computational solution

254

(1)

Discussion

255

(1)

An example

256

(2)

A discrete version

258

(2)

A convergence proof

260

(3)

Max y ∫T o F (x, y, t) dt

263

(1)

Generalization and discussion

264

(1)

Integral constraints

265

(1)

Further remarks concerning numerical solution

266

(1)

Eigenvalue problems

267

(1)

The first formulation

268

(1)

An approximate solution

269

(1)

Second formulation

270

(1)

Discrete approximations

270

(1)

Successive approximation

271

(1)

Monotone approximation

272

(1)

Uniqueness of solution

273

(1)

Minimum maximum deviation

273

(10)

Exercises and research problems

274

(8)

Bibliography and comments

282

(1)

Multi-Stage Games

Introduction

283

(1)

A single-stage discrete game

284

(1)

The min-max theorem

285

(1)

Continuous games

286

(1)

Finite resources

287

(1)

Games of survival

287

(1)

Pursuit games

288

(1)

General formulation

289

(2)

The principle of optimality and functional equations

291

(1)

More general process

292

(2)

A basic lemma

294

(1)

Existence and uniqueness

295

(2)

Proof of results

297

(2)

Alternate proof of existence

299

(1)

Successive approximations in general

300

(1)

Effectiveness of solution

300

(2)

Further results

302

(1)

One-sided min-max

302

(1)

Existence and uniqueness for games of survival

303

(3)

An approximation

306

(1)

Non-zero-sum games --- games of survival

307

(1)

An approximate solution

308

(1)

Proof of the extended min-max theorem

308

(1)

A rationale for non-zero sum games

309

(8)

Exercises and research problems

310

(5)

Bibliography and comments

315

(2)

Markovian Decision Processes

Introduction

317

(1)

Markovian decision processes

317

(2)

Notation

319

(1)

A lemma

320

(1)

Existence and uniqueness---I

321

(3)

Existence and uniqueness---II

324

(1)

Existence and uniqueness---III

325

(1)

The Riccati equation

325

(1)

Approximation in policy space

326

(2)

Discrete versions

328

(2)

The recurrence relation

330

(2)

Min-max

332

(1)

Generalization of a Von Neumann result

332

(6)

Exercises and research problems

334

(2)

Bibliography and comments

336

(2)

Index of applications

338

(1)

Name and subject index

339

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Dynamic Programming

Summary

Author Biography

Table of Contents

Supplemental Materials