数值优化：工程与科学领域的实用方法

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"Numerical Optimization 2nd 是一本英文版的高清教材，专注于数值优化这一在工程、科学和商业领域广泛应用的重要主题。本书是数值优化领域的经典著作，旨在提供当前最有效的连续优化方法的全面概述，特别关注适用于实际问题的算法。" 在当今的数据驱动世界中，优化技术是许多关键应用的基础，包括机器学习、数据分析和决策制定。《数值优化》第二版深入探讨了这一主题，旨在帮助读者理解并掌握解决复杂优化问题的工具和技术。该书可能涵盖以下几个核心知识点： 1. **连续优化基础**：介绍优化问题的基本概念，如目标函数、约束条件和局部最优解，以及如何识别和处理全局最优解。 2. **优化算法**：详述多种优化算法，包括梯度下降法、牛顿法、拟牛顿法、共轭梯度法、线性规划和非线性规划等。这些算法在求解不同的优化问题时各有优势和适用场景。 3. **数值计算方法**：讲解数值稳定性和误差分析，以及如何在计算过程中处理不精确性和浮点运算的问题。 4. **迭代过程与收敛性**：讨论迭代方法的收敛性理论，包括线性收敛和超线性收敛的概念，以及如何判断算法是否能有效地接近最优解。 5. **随机优化和概率方法**：可能涉及蒙特卡洛模拟和随机搜索策略，这些方法在处理不确定性和大规模问题时尤其有用。 6. **应用案例**：通过实际案例分析，展示优化技术在工程设计、物流管理、金融工程等领域的应用，帮助读者将理论知识应用于实践。 7. **补充材料**：可能包含变分不等式、互补问题、设施布局、随机规划等领域的扩展内容，这些是优化问题的子领域，对于深入理解优化问题至关重要。 8. **算法实现和编程**：可能介绍如何在编程环境中实现这些优化算法，如MATLAB、Python或R，以供读者进行实际操作和实验。 9. **性能分析和调试**：讲解如何评估优化算法的性能，以及如何诊断和改进算法的效率。 10. **最新发展**：由于是第二版，书中可能会涵盖近年来数值优化领域的最新研究进展和技术趋势。通过学习《数值优化》第二版，读者能够系统地掌握数值优化的理论和实践，提升解决实际问题的能力，并为在相关领域进一步研究和创新奠定坚实基础。这本书对从事科研、工程和商业分析的专业人士来说是一份宝贵的资源。

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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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