数值最优化算法详解：前3章

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"Numerical Optimization 前3章" 是一本关于数值最优化算法的书籍，由Jorge Nocedal和Stephen J. Wright合作撰写，是第二版。这本书的前几章涵盖了数值优化的基础概念和核心算法，适用于工程、计算机科学、经济等领域。作者来自美国西北大学的电气与计算机工程系和威斯康星大学麦迪逊分校的计算机科学系。该书还附带部分习题解答，方便读者加深理解和应用。数值优化是寻找函数在特定条件下的极值（最大值或最小值）问题，是数学、工程和数据分析中的关键工具。前3章通常会包含以下内容： 1. **优化问题的定义**：介绍最优化问题的基本形式，包括连续和离散问题，无约束和约束优化，以及目标函数和决策变量的概念。 2. **一维搜索方法**：讲解线性搜索，如黄金分割法和二分法，这些方法用于在单变量函数上找到局部极值。 3. **梯度和方向导数**：介绍多元函数的梯度，它是函数在某一点的局部变化率最大方向，是优化中不可或缺的工具。 4. **最速下降法**：这是一种简单的优化算法，沿着梯度的负方向进行迭代，以减少目标函数的值。 5. **牛顿法和拟牛顿法**：牛顿法利用函数的二次泰勒展开式来更新迭代点，拟牛顿法则通过近似Hessian矩阵（二阶导数矩阵）来实现，这些方法通常比最速下降法收敛更快。 6. **线性规划基础**：简述线性规划问题，包括标准形式、图解法和单纯形法，它是实际应用中最常见的优化问题类型。 7. **共轭梯度法和高斯-塞德尔迭代**：讨论这些迭代方法，用于解决对称正定线性系统的求解，它们在大型稀疏系统中尤为有用。 8. **约束处理**：介绍如何处理约束条件，如拉格朗日乘子法和惩罚函数法，以及它们在处理等式和不等式约束时的作用。 9. **全局优化简介**：讨论如何寻找函数的全局最优解，因为局部最优解并不总是全局最优。 10. **数值稳定性与收敛性分析**：探讨算法的数值稳定性问题，以及如何评估算法的收敛性。以上是数值优化领域的一些基本知识点，这些内容为后续章节的深入学习和实际应用打下坚实基础。书中的习题解答则有助于读者巩固理论知识并提升解决实际问题的能力。

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Preface

This is a book for people interested in solving optimization problems. Because of the wide

(and growing) use of optimization in science, engineering, economics, and industry, it is

essential for students and practitioners alike to develop an understanding of optimization

algorithms. Knowledge of the capabilities and limitations of these algorithms leads to a better

understanding of their impact on various applications, and points the way to future research

on improving and extending optimization algorithms and software. Our goal in this book

is to give a comprehensive description of the most powerful, state-of-the-art, techniques

for solving continuous optimization problems. By presenting the motivating ideas for each

algorithm, we try to stimulate the reader’s intuition and make the technical details easier to

follow. Formal mathematical requirements are kept to a minimum.

Because of our focus on continuous problems, we have omitted discussion of impor-

tant optimization topics such as discrete and stochastic optimization. However, there are a

great many applications that can be formulated as continuous optimization problems; for

instance,

ﬁnding the optimal trajectory for an aircraft or a robot arm;

identifying the seismic properties of a piece of the earth’s crust by ﬁtting a model of

the region under study to a set of readings from a network of recording stations;

xviii P REFACE

designing a portfolio of investments to maximize expected return while maintaining

an acceptable level of risk;

controlling a chemical process or a mechanical device to optimize performance or

meet standards of robustness;

computing the optimal shape of an automobile or aircraft component.

Every year optimization algorithms are being called on to handle problems that

are much larger and complex than in the past. Accordingly, the book emphasizes large-

scale optimization techniques, such as interior-point methods, inexact Newton methods,

limited-memory methods, and the role of partially separable functions and automatic

differentiation. It treats important topics such as trust-region methods and sequential

quadratic programming more thoroughly than existing texts, and includes comprehensive

discussion of such “core curriculum” topics as constrained optimization theory, Newton

and quasi-Newton methods, nonlinear least squares and nonlinear equations, the simplex

method, and penalty and barrier methods for nonlinear programming.

The Audience

We intend that this book will be used in graduate-level courses in optimization, as of-

fered in engineering, operations research, computer science, and mathematics departments.

There is enough material here for a two-semester (or three-quarter) sequence of courses.

We hope, too, that this book will be used by practitioners in engineering, basic science, and

industry, and our presentation style is intended to facilitate self-study. Since the book treats

a number of new algorithms and ideas that have not been described in earlier textbooks, we

hope that this book will also be a useful reference for optimization researchers.

Prerequisites for this book include some knowledge of linear algebra (including nu-

merical linear algebra) and the standard sequence of calculus courses. To make the book as

self-contained as possible, we have summarized much of the relevant material from these ar-

eas in the Appendix. Our experience in teaching engineering students has shown us that the

material is best assimilated when combined with computer programming projects in which

the student gains a good feeling for the algorithms—their complexity, memory demands,

and elegance—and for the applications. In most chapters we provide simple computer

exercises that require only minimal programming proﬁciency.

Emphasis and Writing Style

We have used a conversational style to motivate the ideas and present the numerical

algorithms. Rather than being as concise as possible, our aim is to make the discussion ﬂow

in a natural way. As a result, the book is comparatively long, but we believe that it can be

read relatively rapidly. The instructor can assign substantial reading assignments from the

text and focus in class only on the main ideas.

A typical chapter begins with a nonrigorous discussion of the topic at hand, including

ﬁgures and diagrams and excluding technical details as far as possible. In subsequent sections,

P REFACE xix

the algorithms are motivated and discussed, and then stated explicitly. The major theoretical

results are stated, and in many cases proved, in a rigorous fashion. These proofs can be

skipped by readers who wish to avoid technical details.

The practice of optimization depends not only on efﬁcient and robust algorithms,

but also on good modeling techniques, careful interpretation of results, and user-friendly

software. In this book we discuss the various aspects of the optimization process—modeling,

optimality conditions, algorithms, implementation, and interpretation of results—but not

with equal weight. Examples throughout the book show how practical problems are formu-

lated as optimization problems, but our treatment of modeling is light and serves mainly

to set the stage for algorithmic developments. We refer the reader to Dantzig [86] and

Fourer, Gay, and Kernighan [112] for more comprehensive discussion of this issue. Our

treatment of optimality conditions is thorough but not exhaustive; some concepts are dis-

cussed more extensively in Mangasarian [198] and Clarke [62]. As mentioned above, we are

quite comprehensive in discussing optimization algorithms.

Topics Not Covered

We omit some important topics, such as network optimization, integer programming,

stochastic programming, nonsmooth optimization, and global optimization. Network and

integer optimization aredescribed in some excellent texts: for instance, Ahuja, Magnanti, and

Orlin [1] in the case of network optimization and Nemhauser and Wolsey [224], Papadim-

itriou and Steiglitz [235], and Wolsey [312] in the case of integer programming. Books on

stochastic optimization are only now appearing; we mention those of Kall and Wallace [174],

Birge and Louveaux [22]. Nonsmooth optimization comes in many ﬂavors. The relatively

simple structures that arise in robust data ﬁtting (which is sometimes based on the 

norm)

are treated by Osborne [232] and Fletcher [101]. The latter book also discusses algorithms

for nonsmooth penalty functions that arise in constrained optimization; we discuss these

brieﬂy, too, in Chapter 18. A more analytical treatment of nonsmooth optimization is given

by Hiriart-Urruty and Lemar

echal [170]. We omit detailed treatment of some important

topics that are the focus of intense current research, including interior-point methods for

nonlinear programming and algorithms for complementarity problems.

Additional Resource

The material in the book is complemented by an online resource called the NEOS

Guide, which can be found on the World-Wide Web at

http://www.mcs.anl.gov/otc/Guide/

The Guide contains information about most areas of optimization, and presents a number

of case studies that describe applications of various optimization algorithms to real-world

problems such as portfolio optimization and optimal dieting. Some of this material is

interactive in nature and has been used extensively for class exercises.

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