書籍介紹
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。
目次
Preface 1
Acknowledgment 2
Introduction 3
1 Linear Algebra (I) : Vocabulary 7
1.1 Vector space Rn and its properties 7
1.2 Subspaces of Rn 8
1.3 Linear independence 12
1.4 Basis and dimension 15
1.5 Inner product of Rn 18
1.6 Gram-Schmidt process 21
1.7 Exercises for Chapter 1 24
2 Linear Algebra (II) : Ranks 26
2.1 Review on matrices 26
2.2 Solving equations by Gaussian eliminations 28
2.3 Applications of Gaussian eliminations 31
2.4 Rank 33
2.5 Determinant 36
2.6 Inverse 40
2.7 Exercises for Chapter 2 42
3 Linear Algebra (III) : Definiteness 45
3.1 Some special matrices 45
3.2 Motivation : Complete the squares 48
3.3 Eigenvalues and eigenvectors 49
3.4 Diagonalization of symmetric matrices 52
3.5 Definiteness 56
3.6 Sylvester’s criterion 57
3.7 Connection with quadratic forms 61
3.8 Second derivative test and generalization 63
3.9 Proof of Sylvester’s criterion 66
3.10 Exercises for Chapter 3 69
4 Constrained Optimization (I) 72
4.1 Optimization : Equality constraints 72
4.2 Non-degenerate constraint qualifications (NDCQ) 77
4.3 Worked example : Equality constraints 80
4.4 Optimization : Inequality constraints 81
4.5 NDCQ for inequality constrained problems 87
4.6 Proof of complementary slackness 89
4.7 Proof of non-negativity of multipliers 91
4.8 Worked example : Inequality constraints 94
4.9 Worked examples : Linear programming on R2 96
4.10 Exercises for Chapter 4 102
5 Constrained Optimization (II) 104
5.1 Optimization : Mixed constraints 104
5.2 Worked examples : Mixed constraints 105
5.3 Optimization : Minimization problems 108
5.4 Worked example : Minimization problems 110
5.5 Optimization : Kuhn-Tucker’s formulation 112
5.6 Proof of Kuhn-Tucker’s FOC 114
5.7 NDCQ in Kuhn-Tucker’s formulation 115
5.8 Worked examples : Kuhn-Tucker’s formulation 117
5.9 Exercises for Chapter 5 121
6 Envelope Theorems 123
6.1 Motivation : Linear budget constraint problem 123
6.2 Envelope Theorem for equality constraints 124
6.3 Worked example : Envelope Theorem 128
6.4 Applications : Shadow prices 129
6.5 Envelope Theorem for unconstrained problems 131
6.6 Envelope Theorem for inequality constraints 133
6.7 Applications in Economics 134
6.8 Proofs of Envelope Theorems 136
6.9 Exercises for Chapter 6 139
7 Second Order Conditions 141
7.1 Motivation : Bordered Hessian matrices 141
7.2 Bordered Hessian matrices 144
7.3 Second order conditions : Statement 147
7.4 Worked examples : Second order conditions 154
7.5 Second derivative test for unconstrained problems 157
7.6 Sketch of the proofs of SOC 160
7.7 Exercises for Chapter 7 166
Appendix. Differential Calculus 169
A. Partial derivatives 169
B. Chain rule for multivariable functions 172
C. Elementary results of optimization in multivariables 177
Answers to Selected Exercises 183
Chapter 1 183
Chapter 2 184
Chapter 3 185
Chapter 4 187
Chapter 5 191
編／著／譯者簡介
Kwok-Wing Tsoi & Ya-Ju Tsai: Project Assistant Professors (Teaching) in the Department of Mathematics at National Taiwan University
（蔡國榮、蔡雅如：國立臺灣大學數學系助理教授）
分類
其他詳細資訊
- 適用對象：成人(學術性)
- 關鍵詞：calculus（微積分），constrained optimization（制限最佳化）
- 附件：無附件
- 頁/張/片數：204
授權資訊
- 著作財產權管理機關或擁有者：國立臺灣大學出版中心
- 取得授權資訊：聯絡處室：國立臺灣大學出版中心
姓名：王沿竣
電話：02-33663991~3 #17
地址：台北市羅斯福路四段1號