线性代数及其在规划中的应用 英文版 作者：郭树里，韩丽娜主编 出版时间：2017年版 内容简介 全书从结构上分为三个部分。靠前部分主要介绍群、环、矩阵的基本理论。靠前章着重介绍集合、部分序、函数、单射，双射、满射以及方程的解等概念以及一些基本结论；第二章是介绍群的理论，是全书比较难的章节，也是线性代数的中心问题之一。特别环同胚映射、模同胚映射是群同胚映射的特殊情形。第二章，第三章涉及的环、矩阵等都是线性代数的中心议题。第三章主要谈环——具有乘法运算的加法群，其加法、乘法运算满足分配律。理想是正则子群，环是群的同态等。第四章主要介绍了逆矩阵，转置矩阵，初等矩阵，系统方程以及行列式的秩等。重要的结论包括相似矩阵有相同的秩，相同的迹，以及相同的特征多项式。一个有限张成的向量空间自同态有明确定义的行列式,迹和特征多项式等。第二部分包括了向量分析、张量分析中基本概念、向量空间、线性变换、矩阵的秩以及张量代数。这部分内容展示给工程类、理科类学生矢量空间，张量空间新进展以及这些概念系统化的发展过程。这部分知识覆盖了向量空间、线性变换、矩阵的秩、张量代数，写作目的是展示工程实践中需要了解的主要概念、基本结论。第三部分主要介绍规划理论，特点如下。第十章介绍了线性规划中时变多项式算法的理论与方法，重要的是内点法。第十一章介绍了线性规划中一些广义必要条件，例如优问题中一阶、二阶必要条件以及不考虑导数条件下的0阶条件、非约束条件下下降法、收敛性分析、线性规划以及非线性规划中牛顿法。第十二章介绍了约束条件下线性规划必要条件的全局性理论，展示了0阶条件、一般非线性规划中的内点法、惩罚函数以及障碍函数法。本部分的一个重要特色是分别从原空间、对偶空间中展示全局或局部性结论。读者选择上，本书适用于具有数学、工程类或者理科专业高年级学生、研究生、教师、工程师。专业选择上，本书适应于系统分析、算子研究、数值化分析、管理科学以及其他应用学科。 目录 Part Ⅰ Chapter 1 Background and Fundamentals of Mathematics 1.1 Basic Concepts 1.2 Relations 1.3 Functions 1.4 The Integers 1.4.1 Long Division 1.4.2 Relatively Prime 1.4.3 Prime 1.4.4 The Unique Factorization Theorem Chapter 2 Groups 2.1 Groups 2.2 Subgroups 2.3 Normal Subgroups 2.4 Homomorphisms 2.5 Permutations 2.6 Product of Groups Chapter 3 Rings 3.1 Commutative Rings 3.2 Units 3.3 The Integers Mod N 3.4 Ideals and Quotient Rings 3.5 Homomorphism 3.6 Polynomial Rings 3.6.1 The Division Algorithm 3.6.2 Associate 3.7 Product of Rings 3.8 Characteristic 3.9 Boolean Rings Chapter 4 Matrices and Matrix Rings 4.1 Elementary Operations and Elementary Matrices 4.2 Systems of Equations 4.3 Determinants 4.4 Similarity Part Ⅱ Chapter 5 Vector Spaces 5.1 The Axioms for a Vector Space 5.2 Linear Independence,Dimension,and Basis 5.3 Intersection,Sum and Direct Sum of Subspaces 5.4 Factor Space 5.5 Inner Product Spaces 5.6 Orthonormal Bases and Orthogonal Complements 5.7 Reciprocal Basis and Change of Basis Chapter 6 Linear Transformations 6.1 Definition of Linear Transformation 6.2 Sums and Products of Liner Transformations 6.3 Special Types of Linear Transformations 6.4 The Adjoint of a Linear Transformation 6.5 Component Formulas Chapter 7 Determinants And Matrices 7.1 The Generalized Kronecker Deltas and the Summation Convention 7.2 Determinants 7.3 The Matrix of a Linear Transformation 7.4 Solution of Systems of Linear Equation 7.5 Special Matrices Chapter 8 Spectral Decompositions 8.1 Direct Sum of Endomorphisms 8.2 Eigenvectors and Eigenvalues 8.3 The Characteristic Polynomial 8.4 Spectral Decomposition for Hermitian Endomorphisms 8.5 Illustrative Examples 8.6 The Minimal Polynomial 8.7 Spectral Decomposition for Arbitrary Endomorphisms Chapter 9 Tensor Algebra 9.1 Linear Functions,the Dual Space 9.2 The Second Dual Space, Canonical Isomorphisms Part Ⅲ Chapter 10 Linear Programming 10.1 Basic Properties of Linear Programs 10.2 Many Computational Procedures to Simplex Method 10.3 Duality 10.3.1 Dual Linear Programs 10.3.2 The Duality Theorem 10.3.3 Relations to the Simplex Procedure 10.4 Interior-point Methods 10.4.1 Elements of Complexity Theory 10.4.2 The Analytic Center 10.4.3 The Central Path 10.4.4 Solution Strategies Chapter 11 Unconstrained Problems 11.1 Transportation and Network Flow Problems 11.1.1 The Transportation Problem 11.1.2 The Northwest Comer Rule 11.1.3 Basic Network Concepts 11.1.4 Maximal Flow 11.2 Basic Properties of Solutions and Algorithms 11.2.1 First-order Necessary Conditions 11.2.2 Second-order Conditions 11.2.3 Minimization and Maximization of Convex Functions 11.2.4 Zeroth-order Conditions 11.2.5 Global Convergence of Descent Algorithms 11.2.6 Speed of Convergence 11.3 Basic Descent Methods 11.3.1 Fibonacci and Golden Section Search 11.3.2 Closedness of Line Search Algorithms 11.3.3 Line Search 11.3.4 The Steepest Descent Method 11.3.5 Coordinate Descent Methods 11.4 Conjugate Direction Methods 11.4.1 Conjugate Directions 11.4.2 Descent Properties of the Conjugate Direction Method 11.4.3 The Conjugate Gradient Method 11.4.4 The C -G Method as an Optimal Process Chapter 12 Constrained Minimization 12.1 Quasi-Newton Methods 12.1.1 Modified Newton Method 12.1.2 Scaling 12.1.3 Memoryless Quasi-Newton Methods 12.2 Constrained Minimization Conditions 12.2.1 Constraints 12.2.2 Tangent Plane 12.2.3 First-order Necessary Conditions ( Equality Constraints) 12.2.4 Second-order Conditions 12.2.5 Eigenvalues in Tangent Subspace 12.2.6 Inequality Constraints 12.2.7 Zeroth-order Conditions and Lagrange Multipliers 12.3 Primal Methods 12.3.1 Feasible Direction Methods 12.3.2 Active Set Methods 12.3.3 The Gradient Projection Method 12.3.4 Convergence Rate of the Gradient Projection Method 12.3.5 The Reduced Gradient Method 12.4 Penalty and Barrier Methods 12.4.1 Penalty Methods 12.4.2 Barrier Methods 12.4.3 Properties of Penalty and Barrier Functions 12.5 Dual and Cutting Plane Methods 12.5. 1 Global Duality 12.5.2 Local Duality 12.5.3 Dual Canonical Convergence Rate 12.5.4 Separable Problems 12.5.5 Decomposition 12.5.6 The Dual Viewpoint 12.5.7 Cutting Plane Methods 12.5.8 Kelley' s Convex Cutting Plane Algorithm 12.5.9 Modifications 12.6 Primal-dual Methods 12.6.1 The Standard Problem 12.6.2 Strategies 12.6.3 A Simple Merit Function 12.6.4 Basic Primal-dual Methods 12.6.5 Modified Newton Methods 12.6.6 Descent Properties 12.6.7 Interior Point Methods Bibliography
