Improved linear programming bounds for antipodal spherical codes

Let $S\subset[-1,1)$. A finite set $C=\{x_i\}_{i=1}^M\subset\Re^n$ is called a {\em spherical S-code} if $||x_i||=1$ for each $i$, and $x_i^T x_j\in S$, $i\ne j$. For $S=[-1,.5]$ maximizing $M=|C|$ is commonly referred to as the {\em kissing number} problem. A well-known technique based on harmonic analysis and linear programming can be used to bound $M$. We consider … Read more

Hyper-sparsity in the revised simplex method and how to exploit it

The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems … Read more

Notes on the Dual Simplex Method

0. The standard dual simplex method. 1. A more general and practical dual simplex method. 2. Phase I for the dual simplex method. 3. Degeneracy in the dual simplex method. 4. A generalized ratio test for the dual simplex method. Citation Draft, Department of Industrial Engineering andManagement Sciences, Northwestern University, 1994. Article Download View Notes … Read more

Warm start strategies in interior-point methods for linear programming

We study the situation in which, having solved a linear program with an interior-point method, we are presented with a new problem instance whose data is slightly perturbed from the original. We describe strategies for recovering a “warm-start” point for the perturbed problem instance from the iterates of the original problem instance. We obtain worst-case … Read more