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A First Course in Optimization Theory
Book · June 1996
DOI: 10.1017/CBO9780511804526 · Source: RePEc
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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
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viii
Contents
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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
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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
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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
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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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Contents
4.5 A Proof of the First-Order Conditions
4.6 A Proof of the Second-Order Conditions
4.7 Exercises
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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
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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
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x
Contents
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
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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
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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
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Contents
9.3
xi
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
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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
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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
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Contents
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
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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
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Appendix B The Real Line
B.I Construction of the Real Line
B.2 Properties of the Real Line
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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
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Bibliography
Index
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publication