Mathematical Programming at Oberwolfach (Mathematical programming study 14): H. Konig et al.

Description

From May 6 to 12, 1979 a conference entitled "Mathematische Optimierung" was held at the Mathematisches Forschungsinstitut Oberwolfach. This meeting, which was directed by the editors of this Study, was the first one at Oberwolfach which was exclusively devoted to the subject of mathematical programming. It was attended by leading scholars who came from 10 different countries. Since its start in the late forties, mathematical programming has experienced a tremendous growth over the last three decades. Beginning with military and economic applications, it is by now among the most important fields of applied mathematics with extensive use in engineering, natural sciences, economics, and biological sciences. During the developing period of mathematical programming two main streams of research emerged, namely a continuous approach (non- linear optimization) with relations to control theory, calculus of variations, and approximation theory, on the one hand, and combinatoriai or discrete optimization with relations to combinatorics, graph theory and modern discrete mathematics, on the other hand. It has caused some concern in the field that these main areas tend to diverge, using different mathematical tools and doing research in different areas of mathematics. Thus, one of the main aims of the conference was to bring scholars together from these different areas and to emphasize unifying aspects.

A great contribution to bridging these two main directions was the appearance of the ellipsoidal algorithm by L.G. Khachian. We can proudly state that this method was firstly brought to the attention of the scientific community outside of Russia at this meeting at Oberwolfach and its theoretical as well as practical consequences were already discussed there at informal meetings. Consequently we are very happy to have the famous paper of P. Gacs and L. Lovasz in this volume, which has made this polynomial algorithm for linear programming understandable and widely known in the western world. To the best of our knowledge there was no other phenomenon in mathematics which has caused so much attention in the general public and on frontpages of leading newspapers and magazines.

This overemphasis on publicity has certainly led to many wrong implications about the ellipsoid method and its practical use. Nevertheless, this approach, which is essentially a special rank-one-updating formula--well-known in continuous optimization for years- has opened up new insights about algorithms and complexity questions in combinatorial optimization. On the other hand it might be worthwhile to discover the combinatorial backbone of this method which at first glance is of purely geometrical nature. This then might even lead to a substantial practical application of it. But right now it is too speculative to make a final judgement of this work, which is still very much under development.

The 16 papers of this volume reflect the different areas of mathematical programming as well as the interaction of the two above-mentioned main streams. It contains papers about polyhedral theory, characterization of adjacency, facial structures, algorithms for matroid properties, questions concerning duality and complexity, higher order optimality conditions, variable metric methods as well as numerical aspects of optimization methods.

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