A Condition Number for Polyhedral Conic Systems
The second part of this book is essentially a self-contained course on linear programming. Unlike the vast majority of expositions of this subject, our account is “condition-based.” It emphasizes the numerical aspects of linear programming and derives probabilistic (average and smoothed) analyses of the relevant algorithms by reducing the object of these analyses from the algorithm to the condition number of the underlying problem.
In this chapter we begin the development of our course. We do so based on a particular problem, the feasibility of polyhedral conic systems. Briefly stated, the feasibility problem we consider is whether a polyhedral cone given by homogeneous linear inequalities is nontrivial (i.e., has a point other than the coordinate origin). An idea pioneered by Renegar is, in these situations, to define conditioning in terms of distance to ill-posedness. The main character in this chapter, the condition number Open image in new window —here A is the matrix stipulating the linear inequalities—is defined in these terms. As the chapter evolves, we see that it can, in addition, be characterized in a number of different ways. The last section of the chapter shows that Open image in new window is a natural parameter in the analysis of some classical simple algorithms to find points in feasible cones. In subsequent chapters, it will feature in the analysis of more sophisticated algorithms. The characterizations we just mentioned will turn out to be helpful in these analyses.