Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms

The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions – computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses. 1. Asyptotic convex geometry and Linear Programming Linear Programming studies the problem of maximizing a linear functional subject to linear constraints. Given an objective vector z ∈ R and constraint vectors a1, . . . , an ∈ R, we consider the linear program (LP) maximize 〈z, x〉 subject to 〈ai, x〉 ≤ 1, i = 1, . . . , n. This linear program has d unknowns, represented by x, and n constraints. Every linear program can be reduced to this form by a simple interpolation argument [36]. The feasible set of the linear program is the polytope P := {x ∈ R : 〈ai, x〉 ≤ 1, i = 1, . . . , n}. The solution of (LP) is then a vertex of P . We can thus look at (LP) from a geometric viewpoint: for a polytope P in R given by n faces, and for a vector z, find the vertex that maximizes the linear functional 〈z, x〉. The oldest and still the most popular method to solve this problem is the simplex method. It starts at some vertex of P and generates a walk on the edges of P toward the solution vertex. At each step, a pivot rule determines a choice of the next vertex; so there are many variants of the simplex method with different pivot rules. (We are not concerned here with how to find the initial vertex, which is a nontrivial problem in itself). Date: February 1, 2008. Supported by the Alfred P. Sloan Foundation and by NSF DMS grant 0401032.