a first course in optimization theory - Semantic Scholar

A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University

CAMBRIDGE UNIVERSITY PRESS

Contents

Preface Acknowledgements

page xiii xvii

1 Mathematical Preliminaries 1.1 Notation and Preliminary Definitions 1.1.1 Integers, Rationals, Reals, M" 1.1.2 Inner Product, Norm, Metric 1.2 Sets and Sequences in W 1.2.1 Sequences and Limits 1.2.2 Subsequences and Limit Points 1.2.3 Cauchy Sequences and Completeness 1.2.4 Suprema, Infima, Maxima, Minima 1.2.5 Monotone^equences in R 1.2.6 The Lim Sup and Lim Inf 1.2.7 Open Balls, Open Sets, Closed Sets 1.2.8 Bounded Sets and Compact Sets 1.2.9 Convex Combinations and Convex Sets 1.2.10 Unions, Intersections, and Other Binary Operations 1.3 Matrices 1.3.1 Sum, Product, Transpose 1.3.2 Some Important Classes of Matrices 1.3.3 Rank of a Matrix 1.3.4 The Determinant 1.3.5 The Inverse 1.3.6 Calculating the Determinant 1.4 Functions 1.4.1 Continuous Functions 1.4.2 Differentiable and Continuously Differentiable Functions vn

1 2 2 4 7 7 10 11 14 17 18 22 23 23 24 30 30 32 33 35 38 39 41 41 43

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1.4.3 Partial Derivatives and Differentiability 1.4.4 Directional Derivatives and Differentiability 1.4.5 Higher Order Derivatives 1.5 Quadratic Forms: Definite and Semidefinite Matrices 1.5.1 Quadratic Forms and Definiteness 1.5.2 Identifying Definiteness and Semidefiniteness 1.6 Some Important Results 1.6.1 Separation Theorems 1.6.2 The Intermediate and Mean Value Theorems 1.6.3 The Inverse and Implicit Function Theorems 1.7 Exercises

46 48 49 50 50 53 55 56 60 65 66

2 Optimization in R" 2.1 Optimization Problems in R" 2.2 Optimization Problems in Parametric Form 2.3 Optimization Problems: Some Examples 2.3.1 Utility Maximization 2.3.2 Expenditure Minimization 2.3.3 Profit Maximization 2.3.4 Cost Minimization 2.3.5 Consumption-Leisure Choice 2.3.6 Portfolio Choice 2.3.7 Identifying Pareto Optima 2.3.8 Optimal Provision of Public Goods 2.3.9 Optimal Commodity Taxation 2.4 Objectives of Optimization Theory 2.5 A Roadmap 2.6 Exercises

74 74 77 78 78 79 80 80 81 81 82 83 84 85 86 88

3 Existence of Solutions: The Weierstrass Theorem 3.1 The Weierstrass Theorem 3.2 The Weierstrass Theorem in Applications 3.3 A Proof of the Weierstrass Theorem 3.4 Exercises

90 90 92 96 97

4 Unconstrained Optima 4.1 "Unconstrained" Optima 4.2 First-Order Conditions 4.3 Second-Order Conditions 4.4 Using the First- and Second-Order Conditions

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100 100 101 103 104

Contents 4.5 A Proof of the First-Order Conditions 4.6 A Proof of the Second-Order Conditions 4.7 Exercises

ix 106 108 110

5 Equality Constraints and the Theorem of Lagrange 5.1 Constrained Optimization Problems 5.2 Equality Constraints and the Theorem of Lagrange 5.2.1 Statement of the Theorem 5.2.2 The Constraint Qualification 5.2.3 The Lagrangean Multipliers 5.3 Second-Order Conditions 5.4 Using the Theorem of Lagrange 5.4.1 A "Cookbook" Procedure 5.4.2 Why the Procedure Usually Works 5.4.3 When It Could Fail 5.4.4 A Numerical Example 5.5 Two Examples from Economics 5.5.1 An Illustration from Consumer Theory 5.5.2 An Illustration from Producer Theory 5.5.3 Remarks 5.6 A Proof of the Theorem of Lagrange 5.7 A Proof of the Second-Order Conditions 5.8 Exercises

112 112 113 114 115 116 117 121 121 122 123 127 128 128 130 132 135 137 142

6 Inequality Constraints and the Theorem of Kuhn and Tucker 6.1 The Theorem of Kuhn and Tucker 6.1.1 Statement of the Theorem 6.1.2 The Constraint Qualification 6.1.3 The Kuhn-Tucker Multipliers 6.2 Using the Theorem of Kuhn and Tucker 6.2.1 A "Cookbook" Procedure 6.2.2 Why the Procedure Usually Works 6.2.3 When It Could Fail 6.2.4 A Numerical Example 6.3 Illustrations from Economics 6.3.1 An Illustration from Consumer/Theory 6.3.2 An Illustration from Producer Theory 6.4 The General Case: Mixed Constraints 6.5 A Proof of the Theorem of Kuhn and Tucker 6.6 Exercises

145 145 145 147 148 150 150 151 152 155 157 158 161 164 165 168

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7x Convex Structures in Optimization Theory 7.1 Convexity Defined 7.1.1 Concave and Convex Functions 7.1.2 Strictly Concave and Strictly Convex Functions 7.2 Implications of Convexity 7.2.1 Convexity and Continuity 7.2.2 Convexity and Differentiability 7.2.3 Convexity and the Properties of the Derivative 7.3 Convexity and Optimization 7.3.1 Some General Observations 7.3.2 Convexity and Unconstrained Optimization 7.3.3 Convexity and the Theorem of Kuhn and Tucker 7.4 Using Convexity in Optimization 7.5 A Proof of the First-Derivative Characterization of Convexity 7.6 A Proof of the Second-Derivative Characterization of Convexity 7.7 A Proof of the Theorem of Kuhn and Tucker under Convexity 7.8 Exercises

172 173 174 176 177 177 179 183 185 185 187 187 189 190 191 194 198

8 Quasi-Convexity and Optimization 8.1 Quasi-Concave and Quasi-Convex Functions 8.2 Quasi-Convexity as a Generalization of Convexity 8.3 Implications of Quasi-Convexity 8.4 Quasi-Convexity and Optimization 8.5 Using Quasi-Convexity in Optimization Problems 8.6 A Proof of the First-Derivative Characterization of Quasi-Convexity 8.7, A Proof of the Second-Derivative Characterization of Quasi-Convexity 8.8 A Proof of the Theorem of Kuhn and Tucker under Quasi-Convexity 8.9 Exercises

203 204 205 209 213 215 216

9 Parametric Continuity: The Maximum Theorem 9.1 Correspondences 9.1.1 Upper-and Lower-Semicontinuous Correspondences 9.1.2 Additional Definitions 9.1.3 A Characterization of Semicontinuous Correspondences 9.1.4 Semicontinuous Functions and Semicontinuous Correspondences 9.2 Parametric Continuity: The Maximum Theorem 9.2.1 The Maximum Theorem 9.2.2 The Maximum Theorem under Convexity

224 225 225 228 229

217 220 221

233 235 235 237

Contents 9.3

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An Application to Consumer Theory 9.3.1 Continuity of the Budget Correspondence 9.3.2 The Indirect Utility Function and Demand Correspondence 9.4 An Application to Nash Equilibrium 9.4.1 Normal-Form Games 9.4.2 The Brouwer/Kakutani Fixed Point Theorem 9.4.3 Existence of Nash Equilibrium 9.5 Exercises

240 240

10 Supermodularity and Parametric Monotonicity 10.1 Lattices and Supermodularity 10.1.1 Lattices 10.1.2 Supermodularity and Increasing Differences 10.2 Parametric Monotonicity 10.3 An Application to Supermodular Games 10.3.1 Supermodular Games 10.3.2 The Tarski Fixed Point Theorem 10.3.3 Existence of Nash Equilibrium 10.4 A Proof of the Second-Derivative Characterization of Supermodularity 10.5 Exercises

253 254 254 255 258 262 262 263 263 264 266

11 Finite-Horizon Dynamic Programming 11.1 Dynamic Programming Problems 11.2 Finite-Horizon'Dynamic Programming 11.3 Histories, Strategies, and the Value Function 11.4 Markovian Strategies 11.5 Existence of an Optimal Strategy 11.6 An Example: The Consumption-Savings Problem 11.7 Exercises

268 268 268 269 271 272 276 278

12 Stationary Discounted Dynamic Programming 12.1 Description of the Framework 12.2 Histories, Strategies, and the Value Function 12.3 The Bellman Equation 12.4 A Technical Digression 12.4.1 Complete Metric Spaces and Cauchy Sequences 12.4.2 Contraction Mappings 12.4.3 Uniform Convergence

281 281 282 283 286 286 287 289

242 243 243 244 246 247

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12.5 Existence of an Optimal Strategy 12.5.1 A Preliminary Result 12.5.2 Stationary Strategies 12.5.3 Existence of an Optimal Strategy 12.6 An Example: The Optimal Growth Model 12.6.1 The Model 12.6.2 Existence of Optimal Strategies 12.6.3 Characterization of Optimal Strategies 12.7 Exercises

291 292 294 295 298 299 300 301 309

Appendix A Set Theory and Logic: An Introduction A.I Sets, Unions, Intersections A.2 Propositions: Contrapositives and Converses A.3 Quantifiers and Negation A.4 Necessary and Sufficient Conditions

315 315 316 318 320

Appendix B The Real Line B.I Construction of the Real Line B.2 Properties of the Real Line

323 323 326

Appendix C Structures on Vector Spaces C.I Vector Spaces C.2 Inner Product Spaces C.3 Normed Spaces C.4 /Metric Spaces / C.4.1 Definitions C.4.2 Sets and Sequences in Metric Spaces C.4.3 Continuous Functions on Metric Spaces C.4.4 Separable Metric Spaces C.4.5 Subspaces C.5 Topological Spaces C.5.1 Definitions C.5.2 Sets and Sequences in Topological Spaces C.5.3 Continuous Functions on Topological Spaces C.5.4 Bases ^ C.6 Exercises

330 330 332 333 336 336 337 339 340 341 342 342 343 343 343 345

Bibliography Index

349 351