Linear Programs and Related Problems. by Evar D. Nering; Albert W. TuckerReview by: R. W. CottleSIAM Review, Vol. 36, No. 4 (Dec., 1994), pp. 666-668Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2132733 .

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666 BOOK REVIEWS

illustrate how to put theory to work in some im- portant cases. Most of the exercises at the end of each chapter do not require computer help.

Following are a few of my criticisms. 1. Some of the notations are not conven-

tional; to list a few:

x+ instead of x. p. 6

S, instead of S, p. 6

x = p. 23 (here x looks exactly like a scalar, not a vector).

2. The bibliography is too light. 3. The author recommended using GLIM

(a British statistical software) and made an effort to explain its notations (see ?3.4.2). However, GLIM is designed specifically for the general- ized linear model and is somewhat outdated. For problems in linear models, software such as SAS and MINITAB are more popular and easier to use.

In real-life data analysis computer software becomes indispensible. Almost all its manuals were written based on the algebraic formulation of its theory in linear models, not the geomet- ric formulation adopted in this book. Such an incompatibility may limit its use as a textbook.

T. C. CHANG University of Cincinniiati

Large Sample Methods in Statistics: an In- troduction with Applications. B)' Praniab K. Sen and Juilio M. Singer. Chapman & Hall, New York, 1993. 382 pp. $59.95, cloth. ISBN 0-412- 0422 1-5.

Large sample theory plays an important role in mathematical statistics. The large sample method is an extremely useful tool in applied statistics. Unfortunately most of the knowledge in this area is scattered all over in articles and textbooks. To seek the right information for a potential user of- ten becomes a difficult task. An effort is made in this book to provide a solid justification for the ba- sic large sample theory with an emphasis toward its applications to a variety of problems. Chap- ters 6 and 7 contain some specific applications of asymptotic method in categorical data mod- els and regression models. These two models are the most commonly used families of models

in applied statistics; therefore, this book should greatly benefit those who intend to make good use of large sample theory in their professional activities.

The authors hope that this book could be used by advanced students in biostatistics or applied statistics. In viewing the mathematical sophisti- cation it is doubtful that those students can handle such a book or even need such a course. Practi- tioners often worry more about the sample size and size of error than the proof of the limiting theories. These two problems were hardly even addressed in this book.

T. C. CHANG Uniiversity of CinzcinnaCti

Invertible Point Transformations and Non- linear Differential Equations. By Willi-Hanis Steeb. World Scientific, Singapore, 1993. 180 pp. $32.00, cloth. ISBN 981-1355-7.

The book is a survey of techniques for determin- ing when a nonlinear (ordinary or partial) differ- ential equation is equivalent to a linear system or, more generally, is integrable and/or can be solved explicitly.

The invertible point transformations of the title simply refer to variable changes (time included) with nonvanishing Jacobians. At first these are used to obtain explicit solutions, but are then re- lated to Lie symmetries, the Cartan equivalence method, and the Painlev6 test.

The text is well written, and fairly elementary from a mathematical standpoint. The concepts are clearly illustrated; there are numerous ex- amples of interest to applied mathematicans and physicists.

RICHARD CHURCHILL Hunter College

Linear Programs and Related Problems. By Evar D. Nering anid Albert W Tucker. Academic Press, Boston, MA, 1993. xiv + 584 pp. $59.95. ISBN 0-12-515440-2.

Although this volume is unmistakeably a text- book, its intended audience is never precisely identified. It appears to have been written for (up- per division) undergraduates-and possibly for graduate students-in the mathematical sciences.

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BOOK REVIEWS 667

The book comes complete with a diskette (mine was 5.25 inches) and a set of solutions to its ex- ercises.

The diskette is for IBMs and compatibles with DOS 2.0 or higher. Altogether the files on this diskette consume about 237K bytes. The doc- umentation, such as it is, resides in a README file which when printed in 10-point type fits on two sheets of letter-size paper. About the two ex- ecutable files, simplex.exe and asn.exe, the README file says "they differ mainly in the algorithms used to handle the specific problems for which they are designed. They both use ex- tensive prompts on the screen to suggest what you can do next. They do not support the use of a mouse." Moreover, it says "the problems that can be handled are limited in size to those that can be displayed on a computer screen."

Liniear Programs and Related Problemiis is di- vided into the two parts suggested by its title. The coverage of linear programs is quite elemen- tary. Leading off with the standard set of linear programming applications, the authors move on to present tableau algebra done in the distinctive style of A.W. Tucker. Unlike many textbooks on this subject, the dual problem (though not the duality theorem) is introduced at an early stage. Throughout the book, much is made of duality as a unifying concept. The informal introduc- tion to the simplex method offered in the opening chapter is later developed in a separate chapter. Duality aside, the simplex method is the heart of any such text. This treatment is made more in- teresting by the rather immediate consideration of the degeneracy problem (cycling) and Bland's anti-cycling rule. The important chapter on the simplex method winds up with the existence du- ality theorem. In a chapter labeled "general lin- ear programs" the authors cover such topics as general dual linear programs, the two-phase sim- plex method, systems of linear inequalities, and complementary slackness. The first part of the book closes with a chapter on numerical consid- erations. Packed into this forty-one page chapter are such topics as the exponential Klee-Minty example (not done in general, but for n = 4 and n = 5), the revised simplex method, Gaussian elimination, a discussion of numerical accuracy, the ellipsoid algorithm, and the Karmarkar algo- rithm. Interestingly, the chapter has no figure illustrating either of the latter two topics.

The part of the book on "related problems" consists of chapters on matrix games, assignment

and matching problems, the transportation prob- lem, network flow problems, the transshipment problem, and nonlinear programs. The treatment of the material under these headings is also rather elementary, albeit somewhat less so in the chapter on nonlinear programming. In this final chapter of the book, the authors consider a version of the Karush-Kuhn-Tucker Theorem and a form of the constraint qualification. After this, the problems under consideration are all linearly constrained; in fact, they are either quadratic programs or lin- ear complementarity problems. These are, after all, problems related to linear programs; more- over, they can be-and are-treated by pivoting algorithms.

The book contains an adequate number of ex- ercises. These are grouped at the end of each chapter. Some of them require problem formu- lation, but most simply require the solution of a small numerical example of a problem (by a spe- cific method). It should be added that the prob- lems are carefully selected and finely crafted to illustrate various points. Nine of the chapters end with one of the most interesting features of this book: a separate category of material called "Questions." These are stated as if they were con- jectures, and the reader is asked to supply a proof or a counterexample for each. As the authors point out, "they are not conjectures in the sense that mathematicians use the term since it it known whether they are true or false." To those who do not wish to expend the effort required for a proof or counterexample, the authors suggest treating these conjectures as true or false questions. An- swers for the exercises and the questions appear toward the end of the book.

The book closes with a rather good (author and subject) index, just before which there come four very revealing pages labeled "Selected Bib- liography." With refreshing candor, the authors say "some citations in this bibliography are of historical interest. Some are sources for topics that we include in this book or for treatments that we present. Some extend or complement what we cover here." Looking at the selections, those familiar with the literature of linear and nonlin- ear programming will first marvel at the omis- sions and then come to realize just how personal a statement is made in this volume. Between its covers, this textbook does not pretend to be some- thing it isn't. Like the people of Garrison Keil- lor's Lake Wobegone, it seems to declare Sum11us quod sumnus: we are what we are. Even the sym- bol used to signal the end of each formal proof

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668 BOOK REVIEWS

shows a touch of individuality, if not whimsy. It is a charming miniature tableau looking something like Et .

R. W. COTTLE Staniford Uniiversity

Numerical Methods for Stochastic Control Problems in Continuous Time. By Harold J. Kutshnier and Paul G. Duipuiis. Springer-Verlag, New York, 1992. 439 pp. $49.00, cloth. Ap- plications for Math. 24. ISBN 0-387-97834-8.

The area of continuous time stochastic control has basically developed in the last three decades. The initial continuous time stochastic control problem that was solved is a controlled diffusion described by a family of linear stochastic differential equa- tions with a quadratic cost functional. The solu- tion of this problem follows easily from its deter- ministic counterpart. Unfortunately, this occur- rence is atypical or an anomaly. To find explicit solutions to other stochastic control problems is a formidable task. Even questions of existence and uniqueness of optimal controls require a sizable amount of machinery from probability. Only a relatively few stochastic control problems have explicit solutions (e.g., [I]-[4]). Even when an optimal control is naturally conjectured the task of verifying the optimality of this control can re- quire a sizable effort.

For the control of a diffusion process satisfy- ing a stochastic differential equation on a fixed time interval and an integral type cost functional the solution can be described by the (smooth) so- lution of a Hamilton-Jacobi (dynamic program- ming) equation assuming some conditions are satisfied, e.g., admissibility of the control ob- tained by the solution. Since this Hamilton- Jacobi equation is a nonlinear second order par- tial differential equation, solutions are difficult to find or even to verify their existence. In fact it can often be shown that the desired smoothness of the solution is not available which motivates the study of viscosity solutions.

Since the theory typically leads the compu- tational work as it has in stochastic control, the numerical methods have a shorter history. This book is a compilation of a family of the nu- merical methods for continuous time stochastic control problems. Stochastic control problems

include more difficult problems or less conven- tional problems than the aforementioned con- trolled diffusion. For example, there are ran- domly stopped problems, problems with absorb- ing or reflecting boundaries, impulsive or singu- lar control, controlled jump processes. For such problems the equation that is necessary to solve is more complicated than the Hamilton-Jacobi equation and less conventional from a numerical viewpoint. The numerical approach in this book is to approximate the continuous time stochas- tic process by a (finite state) Markov chain. The authors give a construction of the approximat- ing Markov chains and show (not surprisingly) that there is some choice in the selection of the Markov chain. Then the authors describe var- ious computational methods for the controlled Markov chains. The convergence proofs are given in a separate chapter so that a person who is only interested in using numerical methods for stochastic control problems can easily avoid the theoretical questions. The theoretical methods are based on weak convergence of probability measures. Kushner has a long history of work in stochastic control using weak convergence.

The authors readily acknowledge that the field of numerical methods for continuous time stochastic control is very young. Many questions can easily come to mind about the performance of the various numerical methods. However, this book provides an excellent source for ideas and methods for anyone who is doing numerical work in stochastic control and this book can be recom- mended as a required reference for any such per- son. Hopefully this book will stimulate interest in this area to develop more refined and perhaps better methods for these numerical questions.

REFERENCES

[1] A. BENSOUSSAN AND J. H. VAN SCHUPPEN, Opti- mal control ofpartially observable stochastic systems with an exponential-of-integral per- formance index, SIAM J. Control Optim., 23 (1985), pp. 599-613.

[2] T. E. DUNCAN, Some solvable stochastic control problems in noncompact symmetric spaces of rank one, Stochastics and Stochastic Reports, 35 (1991), pp. 129-142.

[3] U. G. HAUSSMAN, Some examples of optimal stochastic controls or: the stochastic max- imuim principle at work, SIAM Rev., 23 (1981), pp. 292-307.

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