要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。
科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。
这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
Preface
1 Introduction
Mathematical Formulation
Example： A Transportation Problem
Continuous versus Discrete Optimization
Constrained and Unconstrained Optimization
Global and Local Optimization
Stochastic and Deterministic Optimization
Optimization Algorithms
Convexity
Notes and References
2 Fundamentals of Unconstrained Optimization
2．1 What Is a Solution？
Recognizing a Local Minimum
Nonsmooth Problems
2．2 Overview of Algorithms
Two Strategies： Line Search and Trust Region
Search Directions for Line Search Methods
Models for Trust—Region Methods
Scaling
Rates of Convergence
R—Rates of Convergence
Notes and References
Exercises
3 Line Search Methods
3．1 Step Length
The Wolfe Conditions
The Goldstein Conditions
Sufficient Decrease and Backtracking
3．2 Convergence of Line Search Methods
3．3 Rate of Convergence
Convergence Rate of Steepest Descent
Quasi—Newton Methods
Newton's Method
Coordinate Descent Methods
3．4 Step—Length Selection Algorithms
Interpolation
The Initial Step Length
A Line Search Algorithm for the Wolfe Conditions
Notes and References
Exerases
4 Trust—Region Methods
Outline of the Algorithm
4．1 The Cauchy Point and Related Algorithms
The Cauchy Point
Improving on the Cauchy Point
The DoglegMethod
Two—Dimensional Subspace Minimization
Steihaug's Approach
4．2 Using Nearly Exact Solutions to the Subproblem
Characterizing Exact Solutions
Calculating Nearly Exact Solutions
The Hard Case
Proof of Theorem 4．3
4．3 Global Convergence
Reduction Obtained by the Cauchy Point
Convergence to Stationary Points
Convergence of Algorithms Based on Nearly Exact Solutions
4．4 Other Enhancements
Scaling
Non—Euclidean Trust Regions
Notes and References
Exercises
5 Conjugate Gradient Methods
5．1 The Linear Conjugate Gradient Method
Conjugate Direction Methods
Basic Properties of the Conjugate Gradient Method
A Practical Form of the Conjugate Gradient Method
Rate of Convergence
Preconditioning
Practical Preconditioners
5．2 Nonlinear Conjugate Gradient Methods
The Fletcher—Reeves Method
The Polak—Ribiere Method
Quadratic Termination and Restarts
Numerical Performance
Behavior of the Fletcher—Reeves Method
Global Convergence
Notes and References
Exerases
6 Practical Newton Methods
6．1 Inexact Newton Steps
6．2 Line Search Newton Methods
Line Search Newton—CG Method
Modified Newton's Method
6．3 Hessian Modifications
Eigenvalue Modification
Adding a Multiple of the Identity
Modified Cholesky Factorization
Gershgorin Modification
Modified Symmetric Indefinite Factorization
6．4 Trust—Region Newton Methods
Newton—Dogleg and Subspace—Minimization Methods
Accurate Solution of the Trust—Region Problem
Trust—Region Newton—CG Method
Preconditioning the Newton—CG Method
Local Convergence of Trust—Region Newton Methods
Notes and References
Exerases
7 Calculating Derivatives
7．1 Finite—Difference Derivative Approximations
Approximating the Gradient
Approximating a Sparse Jacobian
Approximatingthe Hessian
Approximating a Sparse Hessian
7．2 Automatic Differentiation
An Example
The Forward Mode
The Reverse Mode
Vector Functions and Partial Separability
Calculating Jacobians of Vector Functions
Calculating Hessians： Forward Mode
Calculating Hessians： Reverse Mode
Current Limitations
Notes and References
Exercises
8 Quasi—Newton Methods
8．1 The BFGS Method
Properties ofthe BFGS Method
Implementation
8．2 The SR1 Method
Properties of SRl Updating
8．3 The Broyden Class
Properties ofthe Broyden Class
8．4 Convergence Analysis
Global Convergence ofthe BFGS Method
Superlinear Convergence of BFGS
Convergence Analysis of the SR1 Method
Notes and References
Exercises
9 Large—Scale Quasi—Newton and Partially Separable Optimization
9．1 Limited—Memory BFGS
Relationship with Conjugate Gradient Methods
9，2 General Limited—Memory Updating
Compact Representation of BFGS Updating
SR1 Matrices
Unrolling the Update
9．3 Sparse Quasi—Newton Updates
9．4 Partially Separable Functions
A Simple Example
Internal Variables
9．5 Invariant Subspaces and Partial Separability
Sparsity vs．Partial Separability
Group Partial Separability
9．6 Algorithms for Partially Separable Functions
Exploiting Partial Separabilityin Newton's Method
Quasi—Newton Methods for Partially Separable Functions
Notes and References
Exercises
……
10 Nonlinear Least—Squares Problems
11 Nonlinear Equations
12 Theory of Constrained Optimization
13 Linear Programming： The Simplex Method
14 Linear Programming：Interior—Point Methods
15 Fundamentals of Algorithms for Nonlinear Constrained Optimization
16 Quadratic Programnung
17 Penalty， Barrier， and Augmented Lagrangian Methods
18 Sequential Quadratic Programming
A Background Material
References
Index
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