Acta ApplicandaeMathematicae3 (1985)
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Book Review A. Bachem, M, Gr0tschel, and B. Korte (eds.): MathematicalProgramming- The State of the Art (Bonn 1982), Springer-Verlag, 1983, 655 pp. The eleventh Mathematical Programming Symposium, held in Bonn from 23 to 27 August 1982, provided an impressive demonstration of the status that mathematical programming has now acquired within applied mathematics. About 900 participants from all continents gathered to attend 500 invited and contributed lectures, to witness several award ceremonies, and to listen to 23 carefully selected tutorials. Twenty-one of those have been written up and form the core of this book, preceded by the text of the usual festive welcoming addresses by various officials and followed by the scientific program of the meeting. Although one wonders if these two latter items of information are worth preserving in this form, there is no doubt at all that the formula adopted by the editors/organizers for these proceedings is infinitely more attractive than the standard one in which a huge collection of poorly refereed contributions is assembled and the public is left with the unpleasant task of distinguishing between the good, the bad and the ugly. In the Bonn Proceedings, 21 experts have been found willing to discuss recent developments in their special areas of interest and to provide references for further study. They have generally done a very good job. In a sense, the title of the book is misleading. Not only are these not conference proceedings in the traditional sense, but also they do not really provide an overview of the state of the art as much as review of important recent results that presuppose a fair amount of advance knowledge on the part of the reader. Very few people will be able to read all the reviews with equal ease; some of them are written very concisely and speak to the initiated more than to the interested outsider. With a few exceptions, this book is not primarily meant to attract new people to the area. Indeed, 900 participants is enough for any conference. The diversity in the 21 tutorials is impressive. They range from the historical (reminiscences by the one truly grand old man of the area, G. B. Dantzig) via the traditional (penalty functions, trust region methods, variable metric methods and conjugate gradient methods for (non)linear programming by R. Fletcher, J. J. Mor6, M. J. D. Powell and J. Stoer) to the nondifferentiable (generalized gradient methods by N. Z. Shor), from the infinite dimensional (semi-inf'mite programming with - surprisingly - applications by S. A. Gustafson and K. O. Kortanek) to the combinatorial (min-max inequalities by A. Schrijver), from the abstract (generalized subgradients by R. T. Rockafellar, generalized equations by S. M. Robinson) to the applied (matroid theory by M. Iri, scheduling theory by E. L. Lawler, game theory by J. Rosenmtlller) and from
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the solidly deterministic (the FKG inequality by R. L. Graham, polyhedral counting theory by L. J. Billera) to the stochastic (the average speed of the simplex method by S. Smale, stochastic programming by R. J. B. Wets).Some contributions are particularly striking in that they show how mathematical programming continues to reach out to new branches of mathematics and vice-versa, such as those by E. L. Allgower and K. Georg on homotopy methods and L. Lovasz on the relation between submodular set functions, concavity and convexity. Some are striking in that they show how much room there still is for fresh, ingenious ideas in areas of traditional concern, such as those by W. R. Pulleyblank on algorithmic applications of polyhedral theory in combinatorial optimization and R. B. Schnabel on conic and tensor methods, two new local models for use in nonlinear programming. Is anything missing? To a large extent, that must be a matter of taste. But a contribution from the area of control theory and mathematical systems theory could have been reasonably expected, if only because these areas are so clearly bordering on the budding mathematical programming empire. And for an area that owes so much of its expansion to the rise of the computer, there is surprisingly small emphasis on what mathematical programming software has to offer and how it should be developed and evaluated. However, these are details. This is a book written by mathematical programmers or those ready to join the crowd; it is not so much a work of propaganda as a self-assured statement about the wonderful things that have happened and more of it to come. The more one delves into the paradigm of optimization, the more one realizes its fertility and depth, to the point that mathematical programming now contributes to many of its founding mathematical disciplines. Virtually every contribution in this book ends on a warning note of problems yet to be solved and obstacles yet to be overcome. There is no doubt that at the next Mathematical Programming Symposium (in Boston, 1985) that challenge will be seen to have generated an impressive response.
Erasmus University, Rotterdam
A . H . G . RINNOOY KAN
