优化算法入门：科学与工程的实用指南

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"Numerical Optimization_2nd E" 是一本关于数值优化的书籍，适用于对解决优化问题感兴趣的读者，包括学生和从业者。优化在科学、工程、经济和工业领域的广泛应用使得理解优化算法变得至关重要。该书属于Springer Series in Operations Research and Financial Engineering系列，由多位知名编辑策划。该系列涵盖了一系列与运筹学和金融工程相关的著作，如性能分析、随机规划、变分不等式和互补问题等主题。书中的部分内容提到了其他相关书籍，例如Altiok的"Performance Analysis of Manufacturing Systems"关注制造业系统的性能分析，Bonnans和Shapiro的"Perturbation Analysis of Optimization Problems"探讨优化问题的扰动分析，以及Dantzig和Thapa的"Linear Programming"两卷本，深入介绍了线性规划的基础理论和扩展应用。此外，还提及了Facchinei和Pang的两卷本"Finite-Dimensional Variational Inequalities and Complementarity Problems"，专注于有限维变分不等式和互补问题，Fishman的两本书则分别关注离散事件模拟和蒙特卡洛方法，为模型构建、编程和分析提供了理论和算法。数值优化是利用数学和计算方法寻找函数的局部或全局最优解的过程。这个领域涉及到多种算法，如梯度下降法、牛顿法、拟牛顿法、遗传算法、粒子群优化等。这些算法在解决实际问题时，如工程设计、资源分配、投资组合优化、机器学习等，都有广泛的应用。线性规划是优化的一个基础分支，它处理的是目标函数和约束条件都是线性的问题。Dantzig和Thapa的著作为读者提供了线性规划的入门知识和高级理论。而变分不等式和互补问题则是非线性优化的重要组成部分，它们在处理多变量、非凸优化问题时起着关键作用。离散事件模拟(Discrete-Event Simulation)是研究系统动态行为的一种工具，特别是在物流和供应链管理中，能够模拟和预测不同决策对系统性能的影响。而蒙特卡洛方法(Monte Carlo)是一种基于随机抽样或统计试验的计算方法，广泛应用于各种概率问题和复杂系统分析。 "Numerical Optimization_2nd E"这本书及其所属系列，为读者提供了一个全面了解和掌握优化算法及其应用的平台，覆盖了从基本概念到高级技术的多个层面，对于想要在科学、工程或商业领域应用优化技术的人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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