Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.

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Still today I am receiving requests for reprints of the book, but unfortunately it is out of print. Therefore, since the book still seems to receive some attention, I p- posed to Springer Verlag to provide a free online edition. I am very happy that Springer agreed. Except for the correction of some typographical errors, the online edition is just a copy of the printed version, no updates have been made. In particular, Table 13.1 gives the status of TSPLIB at the time of publishing the book. For accessing TSPLIB the link http://www.iwr.uni-heidelberg.de/iwr/comopt/software/TSPLIB95/ should be used instead of following the procedure described in Chapter 13. Heidelberg, January 2001 Gerhard Reinelt Preface More than ?fteen years ago, I was faced with the following problem in an assignment for a class in computer science. A brewery had to deliver beer to ?ve stores, and the task was to write a computer program for determining the shortest route for the truck driver to visit all stores and return to the brewery. All my attemps to ?nd a reasonable algorithm failed, I could not help enumerating all possible routes and then select the best one.

This book describes the rapidly developing field of interior point methods (IPMs). An extensive analysis is given of path-following methods for linear programming, quadratic programming and convex programming. These methods, which form a subclass of interior point methods, follow the central path, which is an analytic curve defined by the problem. Relatively simple and elegant proofs for polynomiality are given. The theory is illustrated using several explicit examples. Moreover, an overview of other classes of IPMs is given. It is shown that all these methods rely on the same notion as the path-following methods: all these methods use the central path implicitly or explicitly as a reference path to go to the optimum.