Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

Our ability to solve large, important combinatorial optimization problems has improved dramatically in the decade. The availability of reliable software, extremely fast and inexpensive hardware and high-level languages that make the modeling of complex problems much faster have led to a much greater demand for optimization tools. This paper highlights the major breakthroughs and then describes some very exciting future oppommities. Previously, large research projects required major data collection efforts, expensive mainframes and substantial analyst manpower. Now, we can solve much larger problems on personal computers, much of the necessary data is routinely collected and tools exist to speed up both the modeling and the post-optimality analysis. With the information-technology revolution taking place currently, we now have the oppommity to have our tools embedded into supply-chain systems that determine production and distribution schedules, process-design and location-allocation decisions. These tools can be used industry-wide with only minor modifications being done by each user.

For many combinatorial optimization problems investigations of associated polyhedra have led to enormous successes with respect to both theoretical insights into the structures of the problems as well as to their algorithmic solvability. Among these problems are quite prominent NP-hard ones, like, e.g., the traveling salesman problem, the stable set problem, or the maximum cut problem. In this chapter we overview the polyhedral work that has been done on the quadratic assignment problem (QAP). Our treatment includes a brief introduction to the methods of polyhedral combinatorics in general, descriptions of the most important polyhedral results that are known about the QAP, explanations of the techniques that are used to prove such results, and a discussion of the practical results obtained by cutting plane algorithms that exploit the polyhedral knowledge. We close by some remarks on the perspectives of this kind of approach to the QAP.

The first purpose of this paper is to tell the history of John von Neumann’s development of the minimax theorem for two-person zero-sum games from his first proof of the theorem in 1928 until 1944 when he gave a completely different proof in the first coherent book on game theory. I will argue that von Neumann’s conception of this theorem