Explicit and Implicit Enforcing - Randomized Optimization

This tutorial is aiming at some randomized approaches for geometric optimization that have been developed over the last ten to fifteen years. The methods are simple, sometimes even the analysis is. The simplicity of the methods entails that they use very little of the underlying structure of the problems to tackle, and, as a consequence, they are applicable to a whole range of problems. Most prominently, they have led to new bounds for combinatorial solutions to linear programming. But with little adaption necessary – basically few problem specific primitive operations have to be supplied – the methods can be used for computing smallest enclosing balls of point sets, smallest volume enclosing ellipsoids, the distance between polytopes (given either as convex hulls or as intersection of halfspaces), and for several other problems. If the dimension of the input instance is constant, then several linear time methods are available, and some of them even behave ‘well’ when the dimension grows; at least, no better provable behavior is known. Many of the methods can be interpreted as variants of the simplex algorithm, with appropriately chosen randomized pivot rules. After the good news, the even better news. It seems that there is ample space for improvement. The fact that the methods are so general is nice, but we would be more than happy to use more of the structure, if we only knew how. The ultimate goal is to derive a polynomial combinatorial linear programming algorithm.