A FIRST COURSE IN OPTIMIZATION THEORY

Transcription

1 A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface Acknowledgements page xiii xvii 1 Mathematical Preliminaries Notation and Preliminary Definitions Integers, Rationals, Reals, M" Inner Product, Norm, Metric Sets and Sequences in W Sequences and Limits Subsequences and Limit Points Cauchy Sequences and Completeness Suprema, Infima, Maxima, Minima Monotone^equences in R The Lim Sup and Lim Inf Open Balls, Open Sets, Closed Sets Bounded Sets and Compact Sets Convex Combinations and Convex Sets Unions, Intersections, and Other Binary Operations Matrices Sum, Product, Transpose Some Important Classes of Matrices Rank of a Matrix The Determinant The Inverse Calculating the Determinant Functions Continuous Functions Differentiable and Continuously Differentiable Functions 43 vn

3 viii Contents ^ Partial Derivatives and Differentiability Directional Derivatives and Differentiability Higher Order Derivatives Quadratic Forms: Definite and Semidefinite Matrices Quadratic Forms and Definiteness Identifying Definiteness and Semidefiniteness Some Important Results Separation Theorems The Intermediate and Mean Value Theorems The Inverse and Implicit Function Theorems Exercises 66 2 Optimization in R" Optimization Problems in R" Optimization Problems in Parametric Form Optimization Problems: Some Examples Utility Maximization Expenditure Minimization Profit Maximization Cost Minimization Consumption-Leisure Choice Portfolio Choice Identifying Pareto Optima Optimal Provision of Public Goods Optimal Commodity Taxation Objectives of Optimization Theory A Roadmap Exercises 88 3 Existence of Solutions: The Weierstrass Theorem The Weierstrass Theorem The Weierstrass Theorem in Applications A Proof of the Weierstrass Theorem Exercises 97 4 Unconstrained Optima "Unconstrained" Optima First-Order Conditions Second-Order Conditions Using the First- and Second-Order Conditions 104

4 Contents 4.5 A Proof of the First-Order Conditions A Proof of the Second-Order Conditions Exercises Equality Constraints and the Theorem of Lagrange Constrained Optimization Problems Equality Constraints and the Theorem of Lagrange Statement of the Theorem The Constraint Qualification The Lagrangean Multipliers Second-Order Conditions Using the Theorem of Lagrange A "Cookbook" Procedure Why the Procedure Usually Works When It Could Fail A Numerical Example Two Examples from Economics An Illustration from Consumer Theory An Illustration from Producer Theory Remarks A Proof of the Theorem of Lagrange A Proof of the Second-Order Conditions Exercises Inequality Constraints and the Theorem of Kuhn and Tucker The Theorem of Kuhn and Tucker Statement of the Theorem The Constraint Qualification The Kuhn-Tucker Multipliers Using the Theorem of Kuhn and Tucker A "Cookbook" Procedure Why the Procedure Usually Works When It Could Fail A Numerical Example Illustrations from Economics An Illustration from Consumer/Theory An Illustration from Producer Theory The General Case: Mixed Constraints A Proof of the Theorem of Kuhn and Tucker Exercises 168 ix

5 x Contents 7x Convex Structures in Optimization Theory Convexity Defined Concave and Convex Functions Strictly Concave and Strictly Convex Functions Implications of Convexity Convexity and Continuity Convexity and Differentiability Convexity and the Properties of the Derivative Convexity and Optimization Some General Observations Convexity and Unconstrained Optimization Convexity and the Theorem of Kuhn and Tucker Using Convexity in Optimization A Proof of the First-Derivative Characterization of Convexity A Proof of the Second-Derivative Characterization of Convexity A Proof of the Theorem of Kuhn and Tucker under Convexity Exercises Quasi-Convexity and Optimization Quasi-Concave and Quasi-Convex Functions Quasi-Convexity as a Generalization of Convexity Implications of Quasi-Convexity Quasi-Convexity and Optimization Using Quasi-Convexity in Optimization Problems A Proof of the First-Derivative Characterization of Quasi-Convexity , A Proof of the Second-Derivative Characterization of Quasi-Convexity A Proof of the Theorem of Kuhn and Tucker under Quasi-Convexity Exercises Parametric Continuity: The Maximum Theorem Correspondences Upper-and Lower-Semicontinuous Correspondences Additional Definitions A Characterization of Semicontinuous Correspondences Semicontinuous Functions and Semicontinuous Correspondences Parametric Continuity: The Maximum Theorem The Maximum Theorem The Maximum Theorem under Convexity 237

6 Contents 9.3 An Application to Consumer Theory Continuity of the Budget Correspondence The Indirect Utility Function and Demand Correspondence An Application to Nash Equilibrium Normal-Form Games The Brouwer/Kakutani Fixed Point Theorem Existence of Nash Equilibrium Exercises Supermodularity and Parametric Monotonicity Lattices and Supermodularity Lattices Supermodularity and Increasing Differences Parametric Monotonicity An Application to Supermodular Games Supermodular Games The Tarski Fixed Point Theorem Existence of Nash Equilibrium A Proof of the Second-Derivative Characterization of Supermodularity Exercises Finite-Horizon Dynamic Programming Dynamic Programming Problems Finite-Horizon'Dynamic Programming Histories, Strategies, and the Value Function Markovian Strategies Existence of an Optimal Strategy An Example: The Consumption-Savings Problem Exercises Stationary Discounted Dynamic Programming Description of the Framework Histories, Strategies, and the Value Function The Bellman Equation A Technical Digression Complete Metric Spaces and Cauchy Sequences Contraction Mappings Uniform Convergence 289 xi

7 xii Contents Existence of an Optimal Strategy A Preliminary Result Stationary Strategies Existence of an Optimal Strategy An Example: The Optimal Growth Model The Model Existence of Optimal Strategies Characterization of Optimal Strategies Exercises 309 Appendix A Set Theory and Logic: An Introduction 315 A.I Sets, Unions, Intersections 315 A.2 Propositions: Contrapositives and Converses 316 A.3 Quantifiers and Negation 318 A.4 Necessary and Sufficient Conditions 320 Appendix B The Real Line 323 B.I Construction of the Real Line 323 B.2 Properties of the Real Line 326 Appendix C Structures on Vector Spaces 330 C.I Vector Spaces 330 C.2 Inner Product Spaces 332 C.3 Normed Spaces 333 C.4 /Metric Spaces 336 / C.4.1 Definitions 336 C.4.2 Sets and Sequences in Metric Spaces 337 C.4.3 Continuous Functions on Metric Spaces 339 C.4.4 Separable Metric Spaces 340 C.4.5 Subspaces 341 C.5 Topological Spaces 342 C.5.1 Definitions 342 C.5.2 Sets and Sequences in Topological Spaces 343 C.5.3 Continuous Functions on Topological Spaces 343 C.5.4 Bases ^ 343 C.6 Exercises 345 Bibliography 349 Index 351