最优化是在20世纪得到快速速发展的一门学科。随着计算机技术的发展，它在经济计划、工程设计、生产管理、交通运输、国防等重要领域得到了日益广泛的应用，它已受到政府部门、科研机构和产业部门的高度重视。

本书出自纽约大学著名教授之手，被国外众多大学用作教材或主要参考书。如普林斯顿大学、圣路易斯华盛顿大学、宾夕法尼亚大学、马里兰大学等。本书出版以来。已经重印了10多次，深受广大读者欢迎。

最优化是在20世纪得到快速发展的一门学科。本书介绍了最优化理论及其在经济学和相关学科中的应用，全书共分三个部分。第一部分研究了Rn中最优化问题的解的存在性以及如何确定这些解，第二部分探讨了最优化问题的解如何随着基本参数的变化而变化，最后一部分描述了有限维和无限维的动态规划。另外，还给出基础知识准备一章和三个附录，使得本书自成体系。

本书适合于高等院校经济学、工商管理、保险学、精算学等专业高年级本科生和研究生参考。

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

  2.3.1 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

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

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

9.3 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

10 Supermodularity and Parametric Monotonicity

i0.1 Lattices and Supermodularity

    10.1.1 Lattices

    I0.1.2 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

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

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

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

Appendix A Set Theory and Logic: An Introduction

A.1 Sets, Unions, Intersections

A.2 Propositions: Contrapositives and Converses

A.3 Quantifiers and Negation

A.4 Necessary and Sufficient Conditions

Appendix B The Real Line

B. 1 Construction of the Real Line

B.2 Properties of the Real Line

Appendix C Structures on Vector Spaces

C.1 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

Bibliography

Index