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