Calculus: Applications in Constrained Optimization

簡介

本書以「限制條件下的最佳化問題」為主題，旨在探討傳統微積分課程中的雙變量函數二階導數檢驗和等式約束下的Lagrange乘子法推廣。由於最佳化理論在許多學科中經常被應用（尤其是經濟學），因此在每一主題中都有向讀者實例示範相關定理的應用，期能幫助讀者在理解理論之餘，也能將理論靈活運用在實務面上。

Calculus: Applications in Constrained Optimization provides an accessible yet mathematically rigorous introduction to constrained optimization, designed for undergraduate students who have some experiences with multivariable calculus. Based on a successful course at National Taiwan University (NTU) ── Calculus 4: Applications in Economics and Management, this book connects foundational calculus with contemporary techniques of optimization used in economics, management, and data science.

Classical tools such as Lagrange multipliers and second derivative tests are extended into a general framework that covers both equality and inequality constraints. Readers will also learn to identify degenerate cases and apply second-order conditions in multivariable settings. Key concepts from linear algebra are introduced and integrated throughout.

Each chapter concludes with a carefully structured set of exercises:
● Type (A) questions test basic understanding;
● Type (B) questions reinforce key examples;
● Type (C) questions challenge students to synthesise ideas across topics.

Whether used as a course text or for self-study, this book provides a concise, structured, and student-friendly guide to the essential ideas and methods of constrained optimization.

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目次

Preface
Acknowledgment
Introduction
1　Linear Algebra (I): Vocabulary
　1.1　Vector space Rn and its properties
　1.2　Subspaces of Rn
　1.3　Linear independence
　1.4　Basis and dimension
　1.5　Inner product of Rn
　1.6　Gram-Schmidt process
　1.7　Exercises for Chapter 1
2　Linear Algebra (II): Ranks
　2.1　Review on matrices
　2.2　Solving equations by Gaussian eliminations
　2.3　Applications of Gaussian eliminations
　2.4　Rank
　2.5　Determinant
　2.6　Inverse
　2.7　Exercises for Chapter 2
3　Linear Algebra (III): Definiteness
　3.1　Some special matrices
　3.2　Motivation: Complete the squares
　3.3　Eigenvalues and eigenvectors
　3.4　Diagonalization of symmetric matrices
　3.5　Definiteness
　3.6　Sylvester's criterion
　3.7　Connection with quadratic forms
　3.8　Second derivative test and generalization
　3.9　Proof of Sylvester's criterion
　3.10　Exercises for Chapter 3
4　Constrained Optimization (I)
　4.1　Optimization: Equality constraints
　4.2　Non-degenerate constraint qualifications (NDCQ)
　4.3　Worked example: Equality constraints
　4.4　Optimization: Inequality constraints
　4.5　NDCQ for inequality constrained problems
　4.6　Proof of complementary slackness
　4.7　Proof of non-negativity of multipliers
　4.8　Worked example: Inequality constraints
　4.9　Worked examples: Linear programming on R2
　4.10　Exercises for Chapter 4
5　Constrained Optimization (II)
　5.1　Optimization: Mixed constraints
　5.2　Worked examples: Mixed constraints
　5.3　Optimization: Minimization problems
　5.4　Worked example: Minimization problems
　5.5　Optimization: Kuhn-Tucker's formulation
　5.6　Proof of Kuhn-Tucker's FOC
　5.7　NDCQ in Kuhn-Tucker's formulation
　5.8　Worked examples : Kuhn-Tucker's formulation
　5.9　Exercises for Chapter 5
6　Envelope Theorems
　6.1　Motivation: Linear budget constraint problem
　6.2　Envelope Theorem for equality constraints
　6.3　Worked example: Envelope Theorem
　6.4　Applications: Shadow prices
　6.5　Envelope Theorem for unconstrained problems
　6.6　Envelope Theorem for inequality constraints
　6.7　Applications in Economics
　6.8　Proofs of Envelope Theorems
　6.9　Exercises for Chapter 6
7　Second Order Conditions
　7.1　Motivation : Bordered Hessian matrices
　7.2　Bordered Hessian matrices
　7.3　Second order conditions: Statement
　7.4　Worked examples: Second order conditions
　7.5　Second derivative test for unconstrained problems
　7.6　Sketch of the proofs of SOC
　7.7　Exercises for Chapter 7
Appendix. Di!erential Calculus
　A. Partial derivatives
　B. Chain rule for multivariable functions
　C. Elementary results of optimization in multivariables
Answers to Selected Exercises
Bibliography
Index