Mathematical Preliminaries
1.1 Notation and Preliminary Definitions
1.1.1 Integers, Rationals, Reals, Rn
1.1.2 Inner Product, Norm, Metric
1.2 Sets and Sequences in Rn
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 Sequences 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
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
2 Optimization in R
2.1 Optimization Problems in Rn
2.2 Optimization Problems in Parametric Form
2.3 Optimization Problems: Some Examples
2.5 A Roadmap
2.6 Exercises
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
4 Unconstrained Optima
4.1 "Unconstrained" Optima
4.2 First-Order Conditions
4.3 Second-Order Conditions
4.4 Using the First- and Second-Ordei Conditions
4.5 A Proof of the First-Order Conditions
4.6 A Proof of the Second-Order Conditions
4.7 Exercises
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
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
7 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
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
9 Parametric Continuity: The Maximum Theorem
10 Supermodularity and Parametric Monotomicity
11 Finite-Horizon Dynamic Programming
12 Stationary Discounted Dynamic Programming
Appendix A Set Theory and Logic: An Introduction
Appendix B The Real Line
Bibliography
Index
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