Calculus：Applications in Constrained Optimization
暫譯: 微積分：受限最佳化的應用
Kwok-Wing Tsoi, Ya-Ju Tsai
- 出版商: 國立臺灣大學出版中心
- 出版日期: 2025-08-11
- 定價: $500
- 售價: 9.0 折 $450
- 語言: 英文
- 頁數: 204
- 裝訂: 平裝
- ISBN: 6267768112
- ISBN-13: 9786267768112
-
相關分類:
微積分 Calculus
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商品描述
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.
Preface
Constrained optimization is a critical and contemporary subject across many disciplines. For example, many principles in classical economical theory are developed based on optimization theory to allocate scarce resources optimally. More recently, theory in optimization has been developed rapidly to catch up with the recent trend of Artificial Intelligence and Machine Learning. The increasing demand for knowledge in optimization led the Department of Mathematics at National Taiwan University to offer an 32-hour elective course on constrained optimization (titled ‘Calculus 4 – Applications in Economics and Management’) in the spring of 2019. The course is intended for students who have just finished courses in classical multivariable (differential) calculus.
The aforementioned course aims to extend and refine the elementary forms of the second derivative test for functions of two variables and the method of Lagrange multipliers for problems with equality constraints, as introduced in students’ earlier courses. To give a little more detail, these ideas were further developed to handle optimization problems involving both equality and inequality constraints, to identify degenerate points arising from such constraints, to derive Envelope Theorems that capture the sensitivity of the optimal value with respect to variations in the constraints and to formulate second-order conditions for constrained optimization problems in spaces of arbitrary dimension. The course has been well-received by both students and faculty members.
To achieve this goal, we will require a substantial background from linear algebra which is the formal language and theory of vectors and matrices. These concepts will be introduced in the beginning. Indeed, optimization theory is one of the many instances in mathematics where (linear) algebra and calculus intersect and enrich each other. The tools of both disciplines work in harmony to deepen our understanding of constrained optimization problems.
Although much literature and textbooks already exist on the subject, most are either too advanced or, at the other extreme, lack sufficient mathematical rigor. This book aims to introduce the basics of optimization theory in an intuitive manner that is accessible to undergraduate students who have acquired a standard course in multivariable differential calculus while maintaining some level of mathematical rigor. The authors believe that understanding both applications and mathematical foundations of optimization theory is crucial for preparing students of this generation, correctly, for more advanced courses and future challenges.
商品描述(中文翻譯)
微積分：受限最佳化的應用提供了一個可接近但數學上嚴謹的受限最佳化入門，旨在為具有多變數微積分經驗的本科生設計。這本書基於國立台灣大學（NTU）成功的課程──微積分4：經濟學與管理的應用，將基礎微積分與當代經濟學、管理學和數據科學中使用的最佳化技術相連結。
經典工具如拉格朗日乘數法和二階導數測試被擴展到一個涵蓋等式和不等式約束的一般框架中。讀者還將學習識別退化情況並在多變數環境中應用二階條件。線性代數中的關鍵概念在整個過程中被引入並整合。
每一章結尾都有一組精心結構的練習題：
● 類型 (A) 問題測試基本理解；
● 類型 (B) 問題加強關鍵範例；
● 類型 (C) 問題挑戰學生在主題間綜合思考。
無論是作為課程教材還是自學，這本書都提供了一個簡明、結構化且對學生友好的指南，涵蓋受限最佳化的基本概念和方法。
前言
受限最佳化是許多學科中一個關鍵且當代的主題。例如，許多經典經濟理論中的原則是基於最佳化理論來最佳配置稀缺資源。最近，最佳化理論迅速發展，以跟上人工智慧和機器學習的最新趨勢。對最佳化知識的需求日益增加，促使國立台灣大學數學系在2019年春季開設了一門32小時的選修課程，名為「微積分4 – 經濟學與管理的應用」。該課程旨在為剛完成經典多變數（微分）微積分課程的學生設計。
上述課程旨在擴展和精煉針對兩變數函數的二階導數測試的基本形式，以及針對具有等式約束的問題的拉格朗日乘數法，這些內容在學生的早期課程中已經介紹過。更具體地說，這些概念進一步發展以處理涉及等式和不等式約束的最佳化問題，識別由此類約束產生的退化點，推導捕捉最佳值對約束變化敏感性的包絡定理，並在任意維度的空間中為受限最佳化問題制定二階條件。該課程受到學生和教職員的好評。
為了實現這一目標，我們需要有相當的線性代數背景，這是向量和矩陣的正式語言和理論。這些概念將在一開始介紹。事實上，最佳化理論是數學中許多線性代數和微積分交叉並相互豐富的例子之一。這兩個學科的工具協同工作，加深我們對受限最佳化問題的理解。
儘管已有許多文獻和教科書存在於這一主題上，但大多數要麼過於高深，要麼在另一極端上缺乏足夠的數學嚴謹性。本書旨在以直觀的方式介紹最佳化理論的基礎，使其對已修習標準多變數微分微積分課程的本科生可接近，同時保持一定程度的數學嚴謹性。作者相信，理解最佳化理論的應用和數學基礎對於正確地為這一代學生準備更高級的課程和未來挑戰至關重要。
作者簡介
Kwok-Wing Tsoi（蔡國榮）
Kwok-Wing Tsoi is a Project Assistant Professor (Teaching) in the Department of Mathematics at National Taiwan University, where he plays a pivotal role in the development and instruction of foundational mathematics courses. He received his Ph.D. in algebraic number theory from King’s College London in 2018. In recognition of his dedication to teaching, he has been honored with the King’s Education Award (2019) and the NTU Distinguished Teaching Award (2021).
Ya-Ju Tsai（蔡雅如）
Ya-Ju Tsai is a Project Assistant Professor (Teaching) in the Department of Mathematics at National Taiwan University. She earned her Ph.D. in harmonic analysis from the University of California, Los Angeles in 2005. She serves as the chief coordinator of the university-wide unified calculus courses and has over a decade of teaching experience. Her contributions to education have been recognized with the NTU Distinguished Teaching Award (2024).
作者簡介(中文翻譯)
蔡國榮（Kwok-Wing Tsoi）
蔡國榮是國立台灣大學數學系的專案助理教授（教學），在基礎數學課程的開發與教學中扮演著關鍵角色。他於2018年在倫敦國王學院獲得代數數論的博士學位。因其對教學的奉獻，他獲得了國王教育獎（2019年）和國立台灣大學傑出教學獎（2021年）。
蔡雅如（Ya-Ju Tsai）
蔡雅如是國立台灣大學數學系的專案助理教授（教學）。她於2005年在加州大學洛杉磯分校獲得調和分析的博士學位。她擔任全校統一微積分課程的首席協調員，並擁有超過十年的教學經驗。她在教育方面的貢獻獲得了國立台灣大學傑出教學獎（2024年）。
目錄大綱
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 multivariable
Answers to Selected Exercises
Bibliography
Index
目錄大綱(中文翻譯)
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 multivariable
Answers to Selected Exercises
Bibliography
Index
