工程系统优化问题的数学建模与算法详解

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"《应用优化：工程系统的形式化与算法》是一本专为解决工程问题而设计的教材，它强调将实际问题转化为精确的数学表达是数值问题解决的第一步。书中详细阐述了如何逐步构建问题模型，以便利用现有的软件工具求解。作者关注五种主要的优化问题类型：线性方程组、非线性方程组、无约束优化、等式约束优化和不等式约束优化，这些类别涵盖了工程实践中广泛遇到的优化场景。在每一类问题中，作者不仅提供了理论概念的介绍，还通过丰富的案例研究进行深入剖析。这些工程实例不仅展示了问题的形成过程，还启发了高效算法的设计，有时涉及问题的重构，使其更易于处理。书中的内容不仅包括详细的步骤和方法，还有大量的习题和练习，非常适合工程学或运筹学专业的学生学习优化理论和实践操作。书中注重实践导向，鼓励读者将理论知识应用于实际工程问题，帮助他们掌握优化技术的核心技巧，以提升工程决策的效率和准确性。无论是初次接触优化的学员，还是寻求提升现有技能的专业人士，都能从中受益匪浅。这是一本结合理论与实践，既适合教学也适于自我学习的优化问题处理指南。"

List of illustrations xv

11.1 Resolution of vector into two perpendicular vectors. 431

11.2 Three bus-power system state estimation problem. 439

11.3 Three-bus power system with spread out measurements. 440

12.1 Production from three machines. 449

12.2 Part of feasible set for least-cost production case study. 451

12.3 Average production cost versus production for a typical machine. 454

12.4 Production cost versus production for typical machine. 454

12.5 Three-bus power system with a bus having neither load nor generation. 458

12.6 Zero injection bus re-interpreted as an exact measurement. 458

13.1 Feasible descent directions for a function at a point. 465

13.2 No feasible descent directions for a function at a point. 466

13.3 Tangent plane to contour set of function. 467

13.4 Tangent plane to contour set of function. 469

13.5 Descent directions for a function at a point, none maintaining feasibility. 470

13.6 Feasible set strictly contained in tangent plane to contour set of function. 472

13.7 Contour sets for Lagrangian evaluated with Lagrange multipliers. 479

13.8 Contour sets for Lagrangian evaluated at dual variables. 479

13.9 Contour sets for non-convex objective. 480

13.10 Non-convex objective function. 493

13.11 Convex penalized objective function. 493

14.1 Tangent plane to a set. 532

14.2 Feasible point and tangent plane. 534

14.3 Feasible points and directions along the corresponding tangent planes. 535

14.4 Movement from an infeasible point. 536

14.5 Points that satisfy the ﬁrst-order necessary conditions. 540

14.6 State estimation problem with zero injection buses. 556

15.1 Feasible set for least-cost production case study. 562

15.2 A data communications network. 564

15.3 Origin destination pair joined by a path consisting of two links. 568

15.4 Data values with outlier and least-squares ﬁt. 574

15.5 Example patterns and hyperplane that separates them. 578

15.6 Schematic diagram of gates and latches joined by interconnect. 585

15.7 Dimensions of segment of interconnect. 585

15.8 Equivalent resistive capacitive circuit for interconnect. 589

15.9 Data communications network with three nodes and four directed links. 603

16.1 Feasible set and minimizer for example problem. 610

16.2 Hyperbolic approximation to the complementary slackness constraint. 612

16.3 Contour sets of function. 614

16.4 Contour sets of function on feasible set. 616

16.5 Changes in the working set. 622

16.6 Trajectory of iterates using active set algorithm. 625

16.7 Barrier objective for non-negativity constraint. 632

16.8 Effect on barrier function as barrier parameter becomes small. 633

16.9 Trajectory of minimizers of barrier problem for example. 637

16.10 A ﬁxed tolerance will prevent convergence to a minimizer. 644

16.11 Progress of primal dual interior point algorithm in primal coordinates. 654

Preface

There are many excellent books on optimization and it is important to justify the

need for yet another one. The motivation for this book stems from my observa-

tions of the orientation of typical optimization texts used in optimization courses

compared to the needs of students in our engineering program. Many optimiza-

tion books and courses concentrate on the design of algorithms, with less attention

to adapting a problem to make it amenable to solution by an existing algorithm.

That is, many optimization books are about how to design optimization algorithms,

about how to write optimization software, or about how to apply optimization soft-

ware to an existing problem formulation.

While this book is about the solution of simultaneous equations and optimization

problems, it is not primarily about how to design algorithms, write software, or

solve existing problems. Instead, it is about how to formulate new problems so that

they can be solved by existing software.

Given the fabulous panoply of well-written optimization software available to-

day, the skill of formulating a problem so that it is solvable with standard software

is the most widely applicable skill for most engineers in our graduate program.

Increasingly, the “scarce resource” is the ability to formulate problems, rather than

the hardware or optimization software itself. This book is primarily designed for

people who have a simultaneous equations problem or an optimization problem in

mind and who want to formulate it so that it is solvable with standard software.

Educators in various disciplines have recognized the need to focus on problem

formulation. Business schools have been the most agile in shifting the empha-

sis of optimization and operations research courses from algorithms towards ap-

plications. This book is an attempt to make a similar shift that concentrates on

engineering applications to illustrate the process of formulating engineering opti-

mization problems and then solving them with existing general-purpose software.

The book also discusses simultaneous equations problems, since many signal pro-

cessing and other engineering applications involve the solution of equations.

xvii

xviii Preface

This book began as the notes for a one semester elective course that I teach at

The University of Texas at Austin. The course is taken by many of our incoming

graduate students and is also accessible to advanced undergraduates. A typical

class has approximately thirty to ﬁfty students with a background in electrical or

other engineering.

The slides that I use in teaching this course are available for downloading from

www.cambridge.org/9780521855648. Over two hundred pages of worked solu-

tions to selected exercises are available to instructors. The book can also be used

as a reference for engineers who want to formulate an optimization problem and

apply optimization software to it.

The level of development may appear somewhat formal to a typical ﬁrst year

engineering graduate student and certainly requires a mathematical background.

Calculus and Analytic Geometry by Thomas and Finney [114], together with the

material in the downloadable Appendix A of mathematical preliminaries, provide

such background. Nevertheless, the deﬁnitions and proofs have been deliberately

spelled out to make them accessible and provide motivation, even at the expense of

stating the “obvious,” particularly in the early chapters. In the interest of clarity and

of elementary development, many of the theorems are not stated in their sharpest

form, with more general versions cited in the references.

Engineering students often have considerable past experience with tools such

as M

ATLAB [74], but little experience in using optimization packages. Use of

the M

ATLAB Optimization Toolbox [17] in this book builds on typical student

experience and avoids the considerable “start-up costs” of introducing students to

a completely new software package.

I have beneﬁted from extensive feedback from students at The University of

Texas at Austin who have taken this course. Students and graders who have been

particularly helpful with making suggestions, correcting mistakes, and helping

me to improve the presentation include: Seung Jun Baek, Seyeong Choi, Jerome

Froment-Curtil, Philippe Girolami, Sergey Gorinsky, Hyun-moo Kim, Aditya Lele,

Caleb Lo, and Lin Xu. Of course, remaining errors are my own responsibility. If

you come across any errors or have any comments about the material, please send

me email, baldick@mail.utexas.edu, and I will endeavor to improve the presen-

tation for the next edition. I would also like to hear about descriptions of novel

problems that might ﬁt into this framework.

In teaching this material, one of the most delightful comments I have received is

“I ﬁnally understand the need for the formal development.” If you are unconvinced

at the beginning, I hope that by the end of the book you are a convert to precise

mathematical descriptions.

Ross Baldick, Austin, Texas, November 2005.

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