Continuous trajectories for primal-dual potential-reduction methods

This article considers continuous trajectories of the vector fields induced by two different primal-dual potential-reduction algorithms for solving linear programming problems. For both algorithms, it is shown that the associated continuous trajectories include the central path and the duality gap converges to zero along all these trajectories. For the algorithm of Kojima, Mizuno, and Yoshise, … Read more

Polynomial interior point cutting plane methods

Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the … Read more

A Linear Programming Approach to Semidefinite Programming Problems

Until recently, the study of interior point methods has dominated algorithmic research in semidefinite programming (SDP). From a theoretical point of view, these interior point methods offer everything one can hope for; they apply to all SDP’s, exploit second order information and offer polynomial time complexity. Still for practical applications with many constraints $k$, the … Read more

An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs

We consider the interior-point approach to sensitivity analysis in linear programming (LP) developed by the authors. We investigate the quality of the interior-point bounds under degeneracy. In the case of a special degeneracy, we show that these bounds have the same nice relationship with the optimal partition bounds as in the nondegenerate case. We prove … Read more

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